id sid tid token lemma pos american_scientific_journal-4481 1 1 51 51 NUM american_scientific_journal-4481 1 2 american american ADJ american_scientific_journal-4481 1 3 scientific scientific ADJ american_scientific_journal-4481 1 4 research research NOUN american_scientific_journal-4481 1 5 journal journal NOUN american_scientific_journal-4481 1 6 for for ADP american_scientific_journal-4481 1 7 engineering engineering NOUN american_scientific_journal-4481 1 8 , , PUNCT american_scientific_journal-4481 1 9 technology technology NOUN american_scientific_journal-4481 1 10 , , PUNCT american_scientific_journal-4481 1 11 and and CCONJ american_scientific_journal-4481 1 12 sciences science NOUN american_scientific_journal-4481 1 13 ( ( PUNCT american_scientific_journal-4481 1 14 asrjets asrjet NOUN american_scientific_journal-4481 1 15 ) ) PUNCT american_scientific_journal-4481 1 16 issn issn PROPN american_scientific_journal-4481 1 17 ( ( PUNCT american_scientific_journal-4481 1 18 print print NOUN american_scientific_journal-4481 1 19 ) ) PUNCT american_scientific_journal-4481 1 20 2313 2313 NUM american_scientific_journal-4481 1 21 - - SYM american_scientific_journal-4481 1 22 4410 4410 NUM american_scientific_journal-4481 1 23 , , PUNCT american_scientific_journal-4481 1 24 issn issn PROPN american_scientific_journal-4481 1 25 ( ( PUNCT american_scientific_journal-4481 1 26 online online ADJ american_scientific_journal-4481 1 27 ) ) PUNCT american_scientific_journal-4481 1 28 2313 2313 NUM american_scientific_journal-4481 1 29 - - SYM american_scientific_journal-4481 1 30 4402 4402 NUM american_scientific_journal-4481 1 31 Β© Β© PROPN american_scientific_journal-4481 1 32 global global ADJ american_scientific_journal-4481 1 33 society society NOUN american_scientific_journal-4481 1 34 of of ADP american_scientific_journal-4481 1 35 scientific scientific ADJ american_scientific_journal-4481 1 36 research research NOUN american_scientific_journal-4481 1 37 and and CCONJ american_scientific_journal-4481 1 38 researchers researcher NOUN american_scientific_journal-4481 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ VERB american_scientific_journal-4481 1 40 a a DET american_scientific_journal-4481 1 41 method method NOUN american_scientific_journal-4481 1 42 of of ADP american_scientific_journal-4481 1 43 finding find VERB american_scientific_journal-4481 1 44 the the DET american_scientific_journal-4481 1 45 distance distance NOUN american_scientific_journal-4481 1 46 between between ADP american_scientific_journal-4481 1 47 two two NUM american_scientific_journal-4481 1 48 places place NOUN american_scientific_journal-4481 1 49 on on ADP american_scientific_journal-4481 1 50 the the DET american_scientific_journal-4481 1 51 surface surface NOUN american_scientific_journal-4481 1 52 of of ADP american_scientific_journal-4481 1 53 the the DET american_scientific_journal-4481 1 54 earth earth NOUN american_scientific_journal-4481 1 55 f.t f.t PROPN american_scientific_journal-4481 1 56 . . PUNCT american_scientific_journal-4481 2 1 zohoraa zohoraa PROPN american_scientific_journal-4481 2 2 * * PROPN american_scientific_journal-4481 2 3 , , PUNCT american_scientific_journal-4481 2 4 m. m. NOUN american_scientific_journal-4481 2 5 sharmin sharmin PROPN american_scientific_journal-4481 2 6 akterb akterb VERB american_scientific_journal-4481 2 7 a a DET american_scientific_journal-4481 2 8 , , PUNCT american_scientific_journal-4481 2 9 bdepartment bdepartment NOUN american_scientific_journal-4481 2 10 of of ADP american_scientific_journal-4481 2 11 general general ADJ american_scientific_journal-4481 2 12 educational educational ADJ american_scientific_journal-4481 2 13 development development NOUN american_scientific_journal-4481 2 14 , , PUNCT american_scientific_journal-4481 2 15 daffodil daffodil ADJ american_scientific_journal-4481 2 16 international international ADJ american_scientific_journal-4481 2 17 university university NOUN american_scientific_journal-4481 2 18 , , PUNCT american_scientific_journal-4481 2 19 dhaka dhaka PROPN american_scientific_journal-4481 2 20 1207 1207 NUM american_scientific_journal-4481 2 21 , , PUNCT american_scientific_journal-4481 2 22 bangladesh bangladesh NOUN american_scientific_journal-4481 2 23 aemail aemail NOUN american_scientific_journal-4481 2 24 : : PUNCT american_scientific_journal-4481 2 25 fatema.ged@diu.edu.bd fatema.ged@diu.edu.bd PROPN american_scientific_journal-4481 2 26 bemail bemail NOUN american_scientific_journal-4481 2 27 : : PUNCT american_scientific_journal-4481 2 28 sharmin.ged@diu.edu.bd sharmin.ged@diu.edu.bd PROPN american_scientific_journal-4481 2 29 abstract abstract ADJ american_scientific_journal-4481 2 30 differential differential NOUN american_scientific_journal-4481 2 31 geometry geometry NOUN american_scientific_journal-4481 2 32 is be AUX american_scientific_journal-4481 2 33 the the DET american_scientific_journal-4481 2 34 language language NOUN american_scientific_journal-4481 2 35 of of ADP american_scientific_journal-4481 2 36 modern modern ADJ american_scientific_journal-4481 2 37 physics physics NOUN american_scientific_journal-4481 2 38 and and CCONJ american_scientific_journal-4481 2 39 provides provide VERB american_scientific_journal-4481 2 40 us we PRON american_scientific_journal-4481 2 41 with with ADP american_scientific_journal-4481 2 42 a a DET american_scientific_journal-4481 2 43 new new ADJ american_scientific_journal-4481 2 44 technique technique NOUN american_scientific_journal-4481 2 45 of of ADP american_scientific_journal-4481 2 46 solving solve VERB american_scientific_journal-4481 2 47 the the DET american_scientific_journal-4481 2 48 problems problem NOUN american_scientific_journal-4481 2 49 related relate VERB american_scientific_journal-4481 2 50 to to ADP american_scientific_journal-4481 2 51 the the DET american_scientific_journal-4481 2 52 calculation calculation NOUN american_scientific_journal-4481 2 53 of of ADP american_scientific_journal-4481 2 54 the the DET american_scientific_journal-4481 2 55 plane plane NOUN american_scientific_journal-4481 2 56 and and CCONJ american_scientific_journal-4481 2 57 surface surface NOUN american_scientific_journal-4481 2 58 with with ADP american_scientific_journal-4481 2 59 parameters parameter NOUN american_scientific_journal-4481 2 60 . . PUNCT american_scientific_journal-4481 3 1 there there PRON american_scientific_journal-4481 3 2 is be VERB american_scientific_journal-4481 3 3 a a DET american_scientific_journal-4481 3 4 lot lot NOUN american_scientific_journal-4481 3 5 of of ADP american_scientific_journal-4481 3 6 application application NOUN american_scientific_journal-4481 3 7 of of ADP american_scientific_journal-4481 3 8 differential differential ADJ american_scientific_journal-4481 3 9 geometry geometry NOUN american_scientific_journal-4481 3 10 in in ADP american_scientific_journal-4481 3 11 calculating calculate VERB american_scientific_journal-4481 3 12 some some PRON american_scientific_journal-4481 3 13 of of ADP american_scientific_journal-4481 3 14 the the DET american_scientific_journal-4481 3 15 geographical geographical ADJ american_scientific_journal-4481 3 16 problems problem NOUN american_scientific_journal-4481 3 17 . . PUNCT american_scientific_journal-4481 4 1 finding find VERB american_scientific_journal-4481 4 2 the the DET american_scientific_journal-4481 4 3 distance distance NOUN american_scientific_journal-4481 4 4 between between ADP american_scientific_journal-4481 4 5 my my PRON american_scientific_journal-4481 4 6 house house NOUN american_scientific_journal-4481 4 7 to to ADP american_scientific_journal-4481 4 8 my my PRON american_scientific_journal-4481 4 9 friend friend NOUN american_scientific_journal-4481 4 10 ’s ’s PART american_scientific_journal-4481 4 11 house house NOUN american_scientific_journal-4481 4 12 in in ADP american_scientific_journal-4481 4 13 the the DET american_scientific_journal-4481 4 14 next next ADJ american_scientific_journal-4481 4 15 block block NOUN american_scientific_journal-4481 4 16 is be AUX american_scientific_journal-4481 4 17 very very ADV american_scientific_journal-4481 4 18 easy easy ADJ american_scientific_journal-4481 4 19 but but CCONJ american_scientific_journal-4481 4 20 the the DET american_scientific_journal-4481 4 21 approach approach NOUN american_scientific_journal-4481 4 22 to to PART american_scientific_journal-4481 4 23 calculate calculate VERB american_scientific_journal-4481 4 24 the the DET american_scientific_journal-4481 4 25 distance distance NOUN american_scientific_journal-4481 4 26 between between ADP american_scientific_journal-4481 4 27 two two NUM american_scientific_journal-4481 4 28 places place NOUN american_scientific_journal-4481 4 29 holds hold VERB american_scientific_journal-4481 4 30 true true ADJ american_scientific_journal-4481 4 31 only only ADV american_scientific_journal-4481 4 32 to to ADP american_scientific_journal-4481 4 33 a a DET american_scientific_journal-4481 4 34 certain certain ADJ american_scientific_journal-4481 4 35 extent extent NOUN american_scientific_journal-4481 4 36 . . PUNCT american_scientific_journal-4481 5 1 in in ADP american_scientific_journal-4481 5 2 this this DET american_scientific_journal-4481 5 3 paper paper NOUN american_scientific_journal-4481 5 4 we we PRON american_scientific_journal-4481 5 5 have have AUX american_scientific_journal-4481 5 6 tried try VERB american_scientific_journal-4481 5 7 to to PART american_scientific_journal-4481 5 8 build build VERB american_scientific_journal-4481 5 9 a a DET american_scientific_journal-4481 5 10 method method NOUN american_scientific_journal-4481 5 11 for for ADP american_scientific_journal-4481 5 12 finding find VERB american_scientific_journal-4481 5 13 the the DET american_scientific_journal-4481 5 14 distance distance NOUN american_scientific_journal-4481 5 15 between between ADP american_scientific_journal-4481 5 16 two two NUM american_scientific_journal-4481 5 17 places place NOUN american_scientific_journal-4481 5 18 on on ADP american_scientific_journal-4481 5 19 the the DET american_scientific_journal-4481 5 20 surface surface NOUN american_scientific_journal-4481 5 21 of of ADP american_scientific_journal-4481 5 22 the the DET american_scientific_journal-4481 5 23 earth earth NOUN american_scientific_journal-4481 5 24 using use VERB american_scientific_journal-4481 5 25 differential differential ADJ american_scientific_journal-4481 5 26 geometry geometry NOUN american_scientific_journal-4481 5 27 . . PUNCT american_scientific_journal-4481 6 1 keywords keyword NOUN american_scientific_journal-4481 6 2 : : PUNCT american_scientific_journal-4481 6 3 great great ADJ american_scientific_journal-4481 6 4 circle circle NOUN american_scientific_journal-4481 6 5 ; ; PUNCT american_scientific_journal-4481 6 6 sphere sphere NOUN american_scientific_journal-4481 6 7 ; ; PUNCT american_scientific_journal-4481 6 8 latitude latitude NOUN american_scientific_journal-4481 6 9 and and CCONJ american_scientific_journal-4481 6 10 longitude longitude NOUN american_scientific_journal-4481 6 11 . . PUNCT american_scientific_journal-4481 7 1 1 1 X american_scientific_journal-4481 7 2 . . X american_scientific_journal-4481 7 3 introduction introduction NOUN american_scientific_journal-4481 7 4 it it PRON american_scientific_journal-4481 7 5 is be AUX american_scientific_journal-4481 7 6 imagined imagine VERB american_scientific_journal-4481 7 7 that that SCONJ american_scientific_journal-4481 7 8 the the DET american_scientific_journal-4481 7 9 earth earth NOUN american_scientific_journal-4481 7 10 is be AUX american_scientific_journal-4481 7 11 a a DET american_scientific_journal-4481 7 12 perfect perfect ADJ american_scientific_journal-4481 7 13 sphere sphere NOUN american_scientific_journal-4481 7 14 with with ADP american_scientific_journal-4481 7 15 an an DET american_scientific_journal-4481 7 16 axis axis NOUN american_scientific_journal-4481 7 17 around around ADP american_scientific_journal-4481 7 18 which which PRON american_scientific_journal-4481 7 19 it it PRON american_scientific_journal-4481 7 20 spins spin VERB american_scientific_journal-4481 7 21 . . PUNCT american_scientific_journal-4481 8 1 the the DET american_scientific_journal-4481 8 2 end end NOUN american_scientific_journal-4481 8 3 of of ADP american_scientific_journal-4481 8 4 the the DET american_scientific_journal-4481 8 5 axis axis NOUN american_scientific_journal-4481 8 6 are be AUX american_scientific_journal-4481 8 7 the the DET american_scientific_journal-4481 8 8 north north NOUN american_scientific_journal-4481 8 9 and and CCONJ american_scientific_journal-4481 8 10 south south ADJ american_scientific_journal-4481 8 11 pole pole NOUN american_scientific_journal-4481 8 12 . . PUNCT american_scientific_journal-4481 9 1 every every DET american_scientific_journal-4481 9 2 places place NOUN american_scientific_journal-4481 9 3 of of ADP american_scientific_journal-4481 9 4 the the DET american_scientific_journal-4481 9 5 earth earth NOUN american_scientific_journal-4481 9 6 has have VERB american_scientific_journal-4481 9 7 its its PRON american_scientific_journal-4481 9 8 latitude latitude NOUN american_scientific_journal-4481 9 9 and and CCONJ american_scientific_journal-4481 9 10 longitude longitude NOUN american_scientific_journal-4481 9 11 which which PRON american_scientific_journal-4481 9 12 are be AUX american_scientific_journal-4481 9 13 considered consider VERB american_scientific_journal-4481 9 14 as as ADP american_scientific_journal-4481 9 15 the the DET american_scientific_journal-4481 9 16 arcs arcs NOUN american_scientific_journal-4481 9 17 of of ADP american_scientific_journal-4481 9 18 the the DET american_scientific_journal-4481 9 19 circles circle NOUN american_scientific_journal-4481 9 20 . . PUNCT american_scientific_journal-4481 10 1 the the DET american_scientific_journal-4481 10 2 shortest short ADJ american_scientific_journal-4481 10 3 distance distance NOUN american_scientific_journal-4481 10 4 between between ADP american_scientific_journal-4481 10 5 two two NUM american_scientific_journal-4481 10 6 places place NOUN american_scientific_journal-4481 10 7 that that PRON american_scientific_journal-4481 10 8 covered cover VERB american_scientific_journal-4481 10 9 by by ADP american_scientific_journal-4481 10 10 great great ADJ american_scientific_journal-4481 10 11 circle circle NOUN american_scientific_journal-4481 10 12 as as ADP american_scientific_journal-4481 10 13 geodesics geodesic NOUN american_scientific_journal-4481 10 14 on on ADP american_scientific_journal-4481 10 15 a a DET american_scientific_journal-4481 10 16 sphere sphere NOUN american_scientific_journal-4481 10 17 are be AUX american_scientific_journal-4481 10 18 the the DET american_scientific_journal-4481 10 19 arcs arcs NOUN american_scientific_journal-4481 10 20 of of ADP american_scientific_journal-4481 10 21 great great ADJ american_scientific_journal-4481 10 22 circle circle NOUN american_scientific_journal-4481 11 1 [ [ X american_scientific_journal-4481 11 2 5 5 NUM american_scientific_journal-4481 11 3 ] ] PUNCT american_scientific_journal-4481 11 4 . . PUNCT american_scientific_journal-4481 12 1 with with ADP american_scientific_journal-4481 12 2 the the DET american_scientific_journal-4481 12 3 help help NOUN american_scientific_journal-4481 12 4 of of ADP american_scientific_journal-4481 12 5 spherical spherical ADJ american_scientific_journal-4481 12 6 trigonometry trigonometry NOUN american_scientific_journal-4481 12 7 and and CCONJ american_scientific_journal-4481 12 8 by by ADP american_scientific_journal-4481 12 9 using use VERB american_scientific_journal-4481 12 10 latitude latitude NOUN american_scientific_journal-4481 12 11 and and CCONJ american_scientific_journal-4481 12 12 longitude longitude NOUN american_scientific_journal-4481 12 13 we we PRON american_scientific_journal-4481 12 14 can can AUX american_scientific_journal-4481 12 15 calculate calculate VERB american_scientific_journal-4481 12 16 the the DET american_scientific_journal-4481 12 17 distance distance NOUN american_scientific_journal-4481 12 18 and and CCONJ american_scientific_journal-4481 12 19 direction direction NOUN american_scientific_journal-4481 12 20 from from ADP american_scientific_journal-4481 12 21 one one NUM american_scientific_journal-4481 12 22 place place NOUN american_scientific_journal-4481 12 23 to to ADP american_scientific_journal-4481 12 24 another another PRON american_scientific_journal-4481 12 25 on on ADP american_scientific_journal-4481 12 26 the the DET american_scientific_journal-4481 12 27 surface surface NOUN american_scientific_journal-4481 12 28 of of ADP american_scientific_journal-4481 12 29 the the DET american_scientific_journal-4481 12 30 earth earth NOUN american_scientific_journal-4481 12 31 with with ADP american_scientific_journal-4481 12 32 respect respect NOUN american_scientific_journal-4481 12 33 to to ADP american_scientific_journal-4481 12 34 the the DET american_scientific_journal-4481 12 35 geographical geographical ADJ american_scientific_journal-4481 12 36 north north NOUN american_scientific_journal-4481 12 37 pole pole NOUN american_scientific_journal-4481 12 38 or or CCONJ american_scientific_journal-4481 12 39 south south ADJ american_scientific_journal-4481 12 40 pole pole NOUN american_scientific_journal-4481 12 41 . . PUNCT american_scientific_journal-4481 13 1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-4481 14 1 * * PUNCT american_scientific_journal-4481 14 2 corresponding correspond VERB american_scientific_journal-4481 14 3 author author NOUN american_scientific_journal-4481 14 4 . . PUNCT american_scientific_journal-4481 15 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-4481 15 2 american american PROPN american_scientific_journal-4481 15 3 scientific scientific PROPN american_scientific_journal-4481 15 4 research research NOUN american_scientific_journal-4481 15 5 journal journal NOUN american_scientific_journal-4481 15 6 for for ADP american_scientific_journal-4481 15 7 engineering engineering NOUN american_scientific_journal-4481 15 8 , , PUNCT american_scientific_journal-4481 15 9 technology technology NOUN american_scientific_journal-4481 15 10 , , PUNCT american_scientific_journal-4481 15 11 and and CCONJ american_scientific_journal-4481 15 12 sciences science NOUN american_scientific_journal-4481 15 13 ( ( PUNCT american_scientific_journal-4481 15 14 asrjets asrjet NOUN american_scientific_journal-4481 15 15 ) ) PUNCT american_scientific_journal-4481 15 16 ( ( PUNCT american_scientific_journal-4481 15 17 2018 2018 NUM american_scientific_journal-4481 15 18 ) ) PUNCT american_scientific_journal-4481 15 19 volume volume NOUN american_scientific_journal-4481 15 20 49 49 NUM american_scientific_journal-4481 15 21 , , PUNCT american_scientific_journal-4481 15 22 no no DET american_scientific_journal-4481 15 23 1 1 NUM american_scientific_journal-4481 15 24 , , PUNCT american_scientific_journal-4481 15 25 pp pp ADV american_scientific_journal-4481 15 26 51 51 NUM american_scientific_journal-4481 15 27 - - SYM american_scientific_journal-4481 15 28 60 60 NUM american_scientific_journal-4481 15 29 52 52 NUM american_scientific_journal-4481 15 30 1.1 1.1 NUM american_scientific_journal-4481 15 31 . . PUNCT american_scientific_journal-4481 16 1 sphere sphere ADV american_scientific_journal-4481 16 2 a a DET american_scientific_journal-4481 16 3 sphere sphere NOUN american_scientific_journal-4481 16 4 is be AUX american_scientific_journal-4481 16 5 a a DET american_scientific_journal-4481 16 6 solid solid ADJ american_scientific_journal-4481 16 7 figure figure NOUN american_scientific_journal-4481 16 8 such such ADJ american_scientific_journal-4481 16 9 that that SCONJ american_scientific_journal-4481 16 10 every every DET american_scientific_journal-4481 16 11 point point NOUN american_scientific_journal-4481 16 12 of of ADP american_scientific_journal-4481 16 13 its its PRON american_scientific_journal-4481 16 14 surface surface NOUN american_scientific_journal-4481 16 15 is be AUX american_scientific_journal-4481 16 16 equally equally ADV american_scientific_journal-4481 16 17 distant distant ADJ american_scientific_journal-4481 16 18 from from ADP american_scientific_journal-4481 16 19 a a DET american_scientific_journal-4481 16 20 fixed fix VERB american_scientific_journal-4481 16 21 point point NOUN american_scientific_journal-4481 16 22 with with ADP american_scientific_journal-4481 16 23 in in ADP american_scientific_journal-4481 16 24 it it PRON american_scientific_journal-4481 16 25 , , PUNCT american_scientific_journal-4481 16 26 which which PRON american_scientific_journal-4481 16 27 is be AUX american_scientific_journal-4481 16 28 called call VERB american_scientific_journal-4481 16 29 center center NOUN american_scientific_journal-4481 16 30 of of ADP american_scientific_journal-4481 16 31 the the DET american_scientific_journal-4481 16 32 sphere sphere NOUN american_scientific_journal-4481 16 33 . . PUNCT american_scientific_journal-4481 17 1 the the DET american_scientific_journal-4481 17 2 straight straight ADJ american_scientific_journal-4481 17 3 line line NOUN american_scientific_journal-4481 17 4 drawn draw VERB american_scientific_journal-4481 17 5 through through ADP american_scientific_journal-4481 17 6 the the DET american_scientific_journal-4481 17 7 center center NOUN american_scientific_journal-4481 17 8 and and CCONJ american_scientific_journal-4481 17 9 terminated terminate VERB american_scientific_journal-4481 17 10 both both DET american_scientific_journal-4481 17 11 ways way NOUN american_scientific_journal-4481 17 12 by by ADP american_scientific_journal-4481 17 13 the the DET american_scientific_journal-4481 17 14 sphere sphere NOUN american_scientific_journal-4481 17 15 is be AUX american_scientific_journal-4481 17 16 called call VERB american_scientific_journal-4481 17 17 a a DET american_scientific_journal-4481 17 18 diameter diameter NOUN american_scientific_journal-4481 17 19 and and CCONJ american_scientific_journal-4481 17 20 any any DET american_scientific_journal-4481 17 21 straight straight ADJ american_scientific_journal-4481 17 22 line line NOUN american_scientific_journal-4481 17 23 joining join VERB american_scientific_journal-4481 17 24 the the DET american_scientific_journal-4481 17 25 center center NOUN american_scientific_journal-4481 17 26 of of ADP american_scientific_journal-4481 17 27 sphere sphere NOUN american_scientific_journal-4481 17 28 to to ADP american_scientific_journal-4481 17 29 any any DET american_scientific_journal-4481 17 30 point point NOUN american_scientific_journal-4481 17 31 on on ADP american_scientific_journal-4481 17 32 the the DET american_scientific_journal-4481 17 33 surface surface NOUN american_scientific_journal-4481 17 34 is be AUX american_scientific_journal-4481 17 35 called call VERB american_scientific_journal-4481 17 36 a a DET american_scientific_journal-4481 17 37 radius radius NOUN american_scientific_journal-4481 17 38 of of ADP american_scientific_journal-4481 17 39 the the DET american_scientific_journal-4481 17 40 sphere sphere NOUN american_scientific_journal-4481 17 41 . . PUNCT american_scientific_journal-4481 18 1 1.2 1.2 NUM american_scientific_journal-4481 18 2 . . PUNCT american_scientific_journal-4481 18 3 theorem theorem VERB american_scientific_journal-4481 18 4 1.3 1.3 NUM american_scientific_journal-4481 18 5 . . PUNCT american_scientific_journal-4481 19 1 if if SCONJ american_scientific_journal-4481 19 2 the the DET american_scientific_journal-4481 19 3 arc arc NOUN american_scientific_journal-4481 19 4 s s X american_scientific_journal-4481 19 5 of of ADP american_scientific_journal-4481 19 6 a a DET american_scientific_journal-4481 19 7 circle circle NOUN american_scientific_journal-4481 19 8 with with ADP american_scientific_journal-4481 19 9 radius radius NOUN american_scientific_journal-4481 19 10 r r NOUN american_scientific_journal-4481 19 11 subtends subtend VERB american_scientific_journal-4481 19 12 an an DET american_scientific_journal-4481 19 13 angles angles NOUN american_scientific_journal-4481 19 14 𝜽𝜽 𝜽𝜽 NOUN american_scientific_journal-4481 19 15 at at ADP american_scientific_journal-4481 19 16 the the DET american_scientific_journal-4481 19 17 center center NOUN american_scientific_journal-4481 19 18 then then ADV american_scientific_journal-4481 19 19 𝑺𝑺 𝑺𝑺 PROPN american_scientific_journal-4481 19 20 = = PUNCT american_scientific_journal-4481 19 21 π’“π’“πœ½πœ½ π’“π’“πœ½πœ½ NOUN american_scientific_journal-4481 19 22 where where SCONJ american_scientific_journal-4481 19 23 𝜽𝜽 𝜽𝜽 PROPN american_scientific_journal-4481 19 24 is be AUX american_scientific_journal-4481 19 25 measured measure VERB american_scientific_journal-4481 19 26 in in ADP american_scientific_journal-4481 19 27 radians radian NOUN american_scientific_journal-4481 19 28 figure figure VERB american_scientific_journal-4481 19 29 1 1 NUM american_scientific_journal-4481 19 30 2 2 NUM american_scientific_journal-4481 19 31 . . PUNCT american_scientific_journal-4481 19 32 spherical spherical ADJ american_scientific_journal-4481 19 33 triangle triangle NOUN american_scientific_journal-4481 19 34 a a DET american_scientific_journal-4481 19 35 spherical spherical ADJ american_scientific_journal-4481 19 36 triangle triangle NOUN american_scientific_journal-4481 20 1 [ [ X american_scientific_journal-4481 20 2 3 3 X american_scientific_journal-4481 20 3 ] ] PUNCT american_scientific_journal-4481 20 4 is be AUX american_scientific_journal-4481 20 5 the the DET american_scientific_journal-4481 20 6 proportion proportion NOUN american_scientific_journal-4481 20 7 of of ADP american_scientific_journal-4481 20 8 a a DET american_scientific_journal-4481 20 9 sphere sphere NOUN american_scientific_journal-4481 20 10 bounded bound VERB american_scientific_journal-4481 20 11 by by ADP american_scientific_journal-4481 20 12 three three NUM american_scientific_journal-4481 20 13 arcs arc NOUN american_scientific_journal-4481 20 14 of of ADP american_scientific_journal-4481 20 15 great great ADJ american_scientific_journal-4481 20 16 circles circle NOUN american_scientific_journal-4481 20 17 . . PUNCT american_scientific_journal-4481 21 1 the the DET american_scientific_journal-4481 21 2 arcs arc NOUN american_scientific_journal-4481 21 3 are be AUX american_scientific_journal-4481 21 4 its its PRON american_scientific_journal-4481 21 5 sides side NOUN american_scientific_journal-4481 21 6 and and CCONJ american_scientific_journal-4481 21 7 spherical spherical ADJ american_scientific_journal-4481 21 8 angles angle NOUN american_scientific_journal-4481 21 9 between between ADP american_scientific_journal-4481 21 10 the the DET american_scientific_journal-4481 21 11 arcs arc NOUN american_scientific_journal-4481 21 12 are be AUX american_scientific_journal-4481 21 13 its its PRON american_scientific_journal-4481 21 14 three three NUM american_scientific_journal-4481 21 15 angles angle NOUN american_scientific_journal-4481 21 16 . . PUNCT american_scientific_journal-4481 22 1 2.1 2.1 NUM american_scientific_journal-4481 22 2 . . PUNCT american_scientific_journal-4481 22 3 cosine cosine NOUN american_scientific_journal-4481 22 4 formula formula NOUN american_scientific_journal-4481 22 5 figure figure NOUN american_scientific_journal-4481 22 6 2 2 NUM american_scientific_journal-4481 22 7 to to PART american_scientific_journal-4481 22 8 find find VERB american_scientific_journal-4481 22 9 the the DET american_scientific_journal-4481 22 10 value value NOUN american_scientific_journal-4481 22 11 of of ADP american_scientific_journal-4481 22 12 cosine cosine NOUN american_scientific_journal-4481 22 13 of of ADP american_scientific_journal-4481 22 14 an an DET american_scientific_journal-4481 22 15 angle angle NOUN american_scientific_journal-4481 22 16 of of ADP american_scientific_journal-4481 22 17 a a DET american_scientific_journal-4481 22 18 spherical spherical ADJ american_scientific_journal-4481 22 19 triangle triangle NOUN american_scientific_journal-4481 22 20 in in ADP american_scientific_journal-4481 22 21 terms term NOUN american_scientific_journal-4481 22 22 of of ADP american_scientific_journal-4481 22 23 cosines cosine NOUN american_scientific_journal-4481 22 24 and and CCONJ american_scientific_journal-4481 22 25 sines sine NOUN american_scientific_journal-4481 22 26 of of ADP american_scientific_journal-4481 22 27 the the DET american_scientific_journal-4481 22 28 sides side NOUN american_scientific_journal-4481 22 29 [ [ X american_scientific_journal-4481 22 30 3 3 NUM american_scientific_journal-4481 22 31 ] ] PUNCT american_scientific_journal-4481 22 32 . . PUNCT american_scientific_journal-4481 23 1 abc abc PROPN american_scientific_journal-4481 23 2 is be AUX american_scientific_journal-4481 23 3 spherical spherical ADJ american_scientific_journal-4481 23 4 triangle triangle NOUN american_scientific_journal-4481 23 5 , , PUNCT american_scientific_journal-4481 23 6 o o X american_scientific_journal-4481 23 7 the the DET american_scientific_journal-4481 23 8 center center NOUN american_scientific_journal-4481 23 9 of of ADP american_scientific_journal-4481 23 10 sphere sphere NOUN american_scientific_journal-4481 23 11 . . PUNCT american_scientific_journal-4481 24 1 draw draw VERB american_scientific_journal-4481 24 2 two two NUM american_scientific_journal-4481 24 3 tangents tangent NOUN american_scientific_journal-4481 24 4 at at ADP american_scientific_journal-4481 24 5 a a PRON american_scientific_journal-4481 24 6 to to ADP american_scientific_journal-4481 24 7 the the DET american_scientific_journal-4481 24 8 arc arc NOUN american_scientific_journal-4481 24 9 ab ab PROPN american_scientific_journal-4481 24 10 and and CCONJ american_scientific_journal-4481 24 11 ac ac PROPN american_scientific_journal-4481 24 12 which which PRON american_scientific_journal-4481 24 13 intersect intersect VERB american_scientific_journal-4481 24 14 the the DET american_scientific_journal-4481 24 15 lines line NOUN american_scientific_journal-4481 24 16 ob ob NOUN american_scientific_journal-4481 24 17 and and CCONJ american_scientific_journal-4481 24 18 oc oc AUX american_scientific_journal-4481 24 19 extended extend VERB american_scientific_journal-4481 24 20 to to ADP american_scientific_journal-4481 24 21 d d PROPN american_scientific_journal-4481 24 22 and and CCONJ american_scientific_journal-4481 24 23 e e NOUN american_scientific_journal-4481 24 24 respectively respectively ADV american_scientific_journal-4481 24 25 . . PUNCT american_scientific_journal-4481 25 1 american american ADJ american_scientific_journal-4481 25 2 scientific scientific ADJ american_scientific_journal-4481 25 3 research research PROPN american_scientific_journal-4481 25 4 journal journal NOUN american_scientific_journal-4481 25 5 for for ADP american_scientific_journal-4481 25 6 engineering engineering NOUN american_scientific_journal-4481 25 7 , , PUNCT american_scientific_journal-4481 25 8 technology technology NOUN american_scientific_journal-4481 25 9 , , PUNCT american_scientific_journal-4481 25 10 and and CCONJ american_scientific_journal-4481 25 11 sciences science NOUN american_scientific_journal-4481 25 12 ( ( PUNCT american_scientific_journal-4481 25 13 asrjets asrjet NOUN american_scientific_journal-4481 25 14 ) ) PUNCT american_scientific_journal-4481 25 15 ( ( PUNCT american_scientific_journal-4481 25 16 2018 2018 NUM american_scientific_journal-4481 25 17 ) ) PUNCT american_scientific_journal-4481 25 18 volume volume NOUN american_scientific_journal-4481 25 19 49 49 NUM american_scientific_journal-4481 25 20 , , PUNCT american_scientific_journal-4481 25 21 no no DET american_scientific_journal-4481 25 22 1 1 NUM american_scientific_journal-4481 25 23 , , PUNCT american_scientific_journal-4481 25 24 pp pp ADV american_scientific_journal-4481 25 25 51 51 NUM american_scientific_journal-4481 25 26 - - SYM american_scientific_journal-4481 25 27 60 60 NUM american_scientific_journal-4481 25 28 53 53 NUM american_scientific_journal-4481 25 29 join join NOUN american_scientific_journal-4481 25 30 de de PROPN american_scientific_journal-4481 25 31 . . PROPN american_scientific_journal-4481 25 32 then then ADV american_scientific_journal-4481 25 33 ∠a ∠a VERB american_scientific_journal-4481 25 34 = = PROPN american_scientific_journal-4481 25 35 ∠dae ∠dae PROPN american_scientific_journal-4481 25 36 . . PUNCT american_scientific_journal-4481 26 1 let let VERB american_scientific_journal-4481 26 2 ∠𝐷𝐷𝐷𝐷𝐷𝐷 ∠𝐷𝐷𝐷𝐷𝐷𝐷 NUM american_scientific_journal-4481 26 3 = = SYM american_scientific_journal-4481 26 4 π‘Žπ‘Ž. π‘Žπ‘Ž. PROPN american_scientific_journal-4481 26 5 now now ADV american_scientific_journal-4481 26 6 , , PUNCT american_scientific_journal-4481 26 7 in in ADP american_scientific_journal-4481 26 8 the the DET american_scientific_journal-4481 26 9 triangle triangle NOUN american_scientific_journal-4481 26 10 doe doe NOUN american_scientific_journal-4481 26 11 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 PROPN american_scientific_journal-4481 26 12 = = SYM american_scientific_journal-4481 27 1 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 PROPN american_scientific_journal-4481 27 2 + + CCONJ american_scientific_journal-4481 27 3 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 PROPN american_scientific_journal-4481 27 4 βˆ’ βˆ’ NUM american_scientific_journal-4481 27 5 2𝐷𝐷𝐷𝐷.𝐷𝐷𝐷𝐷. 2𝐷𝐷𝐷𝐷.𝐷𝐷𝐷𝐷. NUM american_scientific_journal-4481 27 6 cos𝐷𝐷𝐷𝐷𝐷𝐷 cos𝐷𝐷𝐷𝐷𝐷𝐷 NOUN american_scientific_journal-4481 27 7 ( ( PUNCT american_scientific_journal-4481 27 8 1 1 NUM american_scientific_journal-4481 27 9 ) ) PUNCT american_scientific_journal-4481 27 10 again again ADV american_scientific_journal-4481 27 11 in in ADP american_scientific_journal-4481 27 12 the the DET american_scientific_journal-4481 27 13 triangle triangle NOUN american_scientific_journal-4481 27 14 dae dae NOUN american_scientific_journal-4481 27 15 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 NOUN american_scientific_journal-4481 27 16 = = SYM american_scientific_journal-4481 27 17 𝐴𝐴𝐷𝐷2 𝐴𝐴𝐷𝐷2 PROPN american_scientific_journal-4481 27 18 + + NUM american_scientific_journal-4481 27 19 𝐴𝐴𝐷𝐷2 𝐴𝐴𝐷𝐷2 PROPN american_scientific_journal-4481 27 20 βˆ’ βˆ’ PROPN american_scientific_journal-4481 28 1 2𝐴𝐴𝐷𝐷.𝐴𝐴𝐷𝐷. 2𝐴𝐴𝐷𝐷.𝐴𝐴𝐷𝐷. NUM american_scientific_journal-4481 28 2 cos𝐷𝐷𝐴𝐴𝐷𝐷 cos𝐷𝐷𝐴𝐴𝐷𝐷 VERB american_scientific_journal-4481 28 3 ( ( PUNCT american_scientific_journal-4481 28 4 2 2 NUM american_scientific_journal-4481 28 5 ) ) PUNCT american_scientific_journal-4481 28 6 subtracting subtract VERB american_scientific_journal-4481 28 7 ( ( PUNCT american_scientific_journal-4481 28 8 2 2 NUM american_scientific_journal-4481 28 9 ) ) PUNCT american_scientific_journal-4481 28 10 from from ADP american_scientific_journal-4481 28 11 ( ( PUNCT american_scientific_journal-4481 28 12 1 1 NUM american_scientific_journal-4481 28 13 ) ) PUNCT american_scientific_journal-4481 28 14 0 0 NUM american_scientific_journal-4481 29 1 = = SYM american_scientific_journal-4481 29 2 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 PROPN american_scientific_journal-4481 29 3 βˆ’ βˆ’ PROPN american_scientific_journal-4481 29 4 𝐴𝐴𝐷𝐷2 𝐴𝐴𝐷𝐷2 PROPN american_scientific_journal-4481 29 5 + + PROPN american_scientific_journal-4481 29 6 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 PROPN american_scientific_journal-4481 30 1 βˆ’ βˆ’ PROPN american_scientific_journal-4481 30 2 𝐴𝐴𝐷𝐷2 𝐴𝐴𝐷𝐷2 PROPN american_scientific_journal-4481 30 3 + + CCONJ american_scientific_journal-4481 30 4 2𝐴𝐴𝐷𝐷.𝐴𝐴𝐷𝐷. 2𝐴𝐴𝐷𝐷.𝐴𝐴𝐷𝐷. NUM american_scientific_journal-4481 30 5 cos𝐴𝐴 cos𝐴𝐴 NOUN american_scientific_journal-4481 30 6 βˆ’ βˆ’ PROPN american_scientific_journal-4481 30 7 2𝐷𝐷𝐷𝐷.𝐷𝐷𝐷𝐷. 2𝐷𝐷𝐷𝐷.𝐷𝐷𝐷𝐷. NUM american_scientific_journal-4481 30 8 cos cos NOUN american_scientific_journal-4481 30 9 π‘Žπ‘Ž π‘Žπ‘Ž VERB american_scientific_journal-4481 30 10 also also ADV american_scientific_journal-4481 30 11 the the DET american_scientific_journal-4481 30 12 angles angle NOUN american_scientific_journal-4481 30 13 oad oad NOUN american_scientific_journal-4481 30 14 and and CCONJ american_scientific_journal-4481 30 15 oae oae NOUN american_scientific_journal-4481 30 16 are be AUX american_scientific_journal-4481 30 17 right right ADJ american_scientific_journal-4481 30 18 angles angle NOUN american_scientific_journal-4481 30 19 , , PUNCT american_scientific_journal-4481 30 20 so so SCONJ american_scientific_journal-4481 30 21 that that SCONJ american_scientific_journal-4481 30 22 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 PROPN american_scientific_journal-4481 30 23 = = SYM american_scientific_journal-4481 30 24 𝐷𝐷𝐴𝐴2 𝐷𝐷𝐴𝐴2 PROPN american_scientific_journal-4481 30 25 + + CCONJ american_scientific_journal-4481 30 26 𝐴𝐴𝐷𝐷2 𝐴𝐴𝐷𝐷2 PROPN american_scientific_journal-4481 30 27 and and CCONJ american_scientific_journal-4481 30 28 𝐷𝐷𝐷𝐷2 𝐷𝐷𝐷𝐷2 PROPN american_scientific_journal-4481 30 29 = = SYM american_scientific_journal-4481 30 30 𝐷𝐷𝐴𝐴2 𝐷𝐷𝐴𝐴2 PROPN american_scientific_journal-4481 30 31 + + NUM american_scientific_journal-4481 30 32 𝐴𝐴𝐷𝐷2 𝐴𝐴𝐷𝐷2 PROPN american_scientific_journal-4481 30 33 . . PUNCT american_scientific_journal-4481 31 1 hence hence ADV american_scientific_journal-4481 31 2 we we PRON american_scientific_journal-4481 31 3 have have VERB american_scientific_journal-4481 31 4 0 0 NUM american_scientific_journal-4481 31 5 = = PUNCT american_scientific_journal-4481 31 6 𝐷𝐷𝐴𝐴2 𝐷𝐷𝐴𝐴2 PROPN american_scientific_journal-4481 31 7 + + X american_scientific_journal-4481 31 8 𝐷𝐷𝐴𝐴2 𝐷𝐷𝐴𝐴2 PROPN american_scientific_journal-4481 31 9 + + CCONJ american_scientific_journal-4481 31 10 2𝐴𝐴𝐷𝐷.𝐴𝐴𝐷𝐷. 2𝐴𝐴𝐷𝐷.𝐴𝐴𝐷𝐷. NUM american_scientific_journal-4481 31 11 cos𝐴𝐴 cos𝐴𝐴 NOUN american_scientific_journal-4481 31 12 βˆ’ βˆ’ PROPN american_scientific_journal-4481 31 13 2𝐷𝐷𝐷𝐷.𝐷𝐷𝐷𝐷. 2𝐷𝐷𝐷𝐷.𝐷𝐷𝐷𝐷. NUM american_scientific_journal-4481 31 14 cosπ‘Žπ‘Ž cosπ‘Žπ‘Ž NOUN american_scientific_journal-4481 31 15 changing change VERB american_scientific_journal-4481 31 16 sides side NOUN american_scientific_journal-4481 31 17 cos cos ADP american_scientific_journal-4481 31 18 π‘Žπ‘Ž π‘Žπ‘Ž X american_scientific_journal-4481 31 19 = = SYM american_scientific_journal-4481 31 20 𝐷𝐷𝐴𝐴 𝐷𝐷𝐴𝐴 PROPN american_scientific_journal-4481 31 21 𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷 VERB american_scientific_journal-4481 31 22 β‹… β‹… PROPN american_scientific_journal-4481 31 23 𝐷𝐷𝐴𝐴 𝐷𝐷𝐴𝐴 PROPN american_scientific_journal-4481 31 24 𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷 VERB american_scientific_journal-4481 31 25 + + CCONJ american_scientific_journal-4481 31 26 𝐴𝐴𝐷𝐷 𝐴𝐴𝐷𝐷 PROPN american_scientific_journal-4481 31 27 𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷 VERB american_scientific_journal-4481 31 28 β‹… β‹… PROPN american_scientific_journal-4481 31 29 𝐴𝐴𝐷𝐷 𝐴𝐴𝐷𝐷 PROPN american_scientific_journal-4481 31 30 𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷 NOUN american_scientific_journal-4481 31 31 cos cos NOUN american_scientific_journal-4481 31 32 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 31 33 therefore therefore ADV american_scientific_journal-4481 31 34 cos cos PROPN american_scientific_journal-4481 31 35 π‘Žπ‘Ž π‘Žπ‘Ž X american_scientific_journal-4481 31 36 = = SYM american_scientific_journal-4481 31 37 cos𝑏𝑏 cos𝑏𝑏 NOUN american_scientific_journal-4481 31 38 cos cos PROPN american_scientific_journal-4481 31 39 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 31 40 + + CCONJ american_scientific_journal-4481 31 41 sin sin NOUN american_scientific_journal-4481 31 42 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 31 43 sin sin NOUN american_scientific_journal-4481 31 44 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 31 45 cos𝐴𝐴 cos𝐴𝐴 NOUN american_scientific_journal-4481 31 46 similarly similarly ADV american_scientific_journal-4481 31 47 we we PRON american_scientific_journal-4481 31 48 can can AUX american_scientific_journal-4481 31 49 prove prove VERB american_scientific_journal-4481 31 50 that that SCONJ american_scientific_journal-4481 31 51 cos cos ADP american_scientific_journal-4481 31 52 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 31 53 = = PUNCT american_scientific_journal-4481 31 54 cos cos PROPN american_scientific_journal-4481 31 55 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 31 56 cos cos PROPN american_scientific_journal-4481 31 57 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 31 58 + + CCONJ american_scientific_journal-4481 31 59 sin sin NOUN american_scientific_journal-4481 31 60 𝑐𝑐 𝑐𝑐 NOUN american_scientific_journal-4481 31 61 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 31 62 cos𝐡𝐡 cos𝐡𝐡 PROPN american_scientific_journal-4481 31 63 and and CCONJ american_scientific_journal-4481 31 64 cos cos ADP american_scientific_journal-4481 31 65 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 31 66 = = SYM american_scientific_journal-4481 31 67 cosπ‘Žπ‘Ž cosπ‘Žπ‘Ž NOUN american_scientific_journal-4481 31 68 cos𝑏𝑏 cos𝑏𝑏 VERB american_scientific_journal-4481 31 69 + + CCONJ american_scientific_journal-4481 31 70 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 31 71 sin sin NOUN american_scientific_journal-4481 31 72 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 31 73 cos𝐢𝐢 cos𝐢𝐢 NOUN american_scientific_journal-4481 31 74 . . PUNCT american_scientific_journal-4481 32 1 2.2 2.2 NUM american_scientific_journal-4481 32 2 . . PUNCT american_scientific_journal-4481 33 1 sine sine ADJ american_scientific_journal-4481 33 2 formula formula NOUN american_scientific_journal-4481 33 3 to to PART american_scientific_journal-4481 33 4 prove prove VERB american_scientific_journal-4481 33 5 that that SCONJ american_scientific_journal-4481 33 6 the the DET american_scientific_journal-4481 33 7 angles angle NOUN american_scientific_journal-4481 33 8 of of ADP american_scientific_journal-4481 33 9 a a DET american_scientific_journal-4481 33 10 spherical spherical ADJ american_scientific_journal-4481 33 11 triangle triangle NOUN american_scientific_journal-4481 33 12 are be AUX american_scientific_journal-4481 33 13 proportional proportional ADJ american_scientific_journal-4481 33 14 to to ADP american_scientific_journal-4481 33 15 the the DET american_scientific_journal-4481 33 16 sines sine NOUN american_scientific_journal-4481 33 17 of of ADP american_scientific_journal-4481 33 18 the the DET american_scientific_journal-4481 33 19 opposite opposite ADJ american_scientific_journal-4481 33 20 sides side NOUN american_scientific_journal-4481 33 21 [ [ X american_scientific_journal-4481 33 22 3 3 NUM american_scientific_journal-4481 33 23 ] ] PUNCT american_scientific_journal-4481 33 24 . . PUNCT american_scientific_journal-4481 34 1 that that PRON american_scientific_journal-4481 34 2 is be AUX american_scientific_journal-4481 34 3 , , PUNCT american_scientific_journal-4481 34 4 american american ADJ american_scientific_journal-4481 34 5 scientific scientific ADJ american_scientific_journal-4481 34 6 research research NOUN american_scientific_journal-4481 34 7 journal journal NOUN american_scientific_journal-4481 34 8 for for ADP american_scientific_journal-4481 34 9 engineering engineering NOUN american_scientific_journal-4481 34 10 , , PUNCT american_scientific_journal-4481 34 11 technology technology NOUN american_scientific_journal-4481 34 12 , , PUNCT american_scientific_journal-4481 34 13 and and CCONJ american_scientific_journal-4481 34 14 sciences science NOUN american_scientific_journal-4481 34 15 ( ( PUNCT american_scientific_journal-4481 34 16 asrjets asrjet NOUN american_scientific_journal-4481 34 17 ) ) PUNCT american_scientific_journal-4481 34 18 ( ( PUNCT american_scientific_journal-4481 34 19 2018 2018 NUM american_scientific_journal-4481 34 20 ) ) PUNCT american_scientific_journal-4481 34 21 volume volume NOUN american_scientific_journal-4481 34 22 49 49 NUM american_scientific_journal-4481 34 23 , , PUNCT american_scientific_journal-4481 34 24 no no DET american_scientific_journal-4481 34 25 1 1 NUM american_scientific_journal-4481 34 26 , , PUNCT american_scientific_journal-4481 34 27 pp pp ADV american_scientific_journal-4481 34 28 51 51 NUM american_scientific_journal-4481 34 29 - - SYM american_scientific_journal-4481 34 30 60 60 NUM american_scientific_journal-4481 34 31 54 54 NUM american_scientific_journal-4481 34 32 figure figure NOUN american_scientific_journal-4481 34 33 3 3 NUM american_scientific_journal-4481 34 34 sin𝐴𝐴 sin𝐴𝐴 ADJ american_scientific_journal-4481 34 35 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 34 36 = = SYM american_scientific_journal-4481 34 37 sin𝐡𝐡 sin𝐡𝐡 PROPN american_scientific_journal-4481 34 38 sin sin NOUN american_scientific_journal-4481 34 39 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 34 40 = = PUNCT american_scientific_journal-4481 35 1 sin𝐢𝐢 sin𝐢𝐢 AUX american_scientific_journal-4481 35 2 sin sin VERB american_scientific_journal-4481 35 3 𝑐𝑐 𝑐𝑐 NOUN american_scientific_journal-4481 35 4 proof proof NOUN american_scientific_journal-4481 35 5 . . PUNCT american_scientific_journal-4481 36 1 we we PRON american_scientific_journal-4481 36 2 already already ADV american_scientific_journal-4481 36 3 proved prove VERB american_scientific_journal-4481 36 4 in in ADP american_scientific_journal-4481 36 5 cosine cosine NOUN american_scientific_journal-4481 36 6 formula formula NOUN american_scientific_journal-4481 36 7 that that DET american_scientific_journal-4481 36 8 cos𝐴𝐴 cos𝐴𝐴 NOUN american_scientific_journal-4481 36 9 = = PUNCT american_scientific_journal-4481 37 1 cos cos PROPN american_scientific_journal-4481 37 2 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 37 3 βˆ’ βˆ’ PROPN american_scientific_journal-4481 37 4 cos𝑏𝑏 cos𝑏𝑏 NOUN american_scientific_journal-4481 37 5 cos cos SCONJ american_scientific_journal-4481 37 6 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 37 7 sin sin VERB american_scientific_journal-4481 37 8 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 37 9 sin sin NOUN american_scientific_journal-4481 37 10 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 37 11 ∴ ∴ PROPN american_scientific_journal-4481 37 12 sin2 sin2 NOUN american_scientific_journal-4481 37 13 𝐴𝐴 𝐴𝐴 PROPN american_scientific_journal-4481 37 14 = = SYM american_scientific_journal-4481 37 15 1 1 NUM american_scientific_journal-4481 37 16 βˆ’ βˆ’ PROPN american_scientific_journal-4481 37 17 cos2 cos2 NOUN american_scientific_journal-4481 37 18 𝐴𝐴 𝐴𝐴 PROPN american_scientific_journal-4481 37 19 = = SYM american_scientific_journal-4481 37 20 1 1 NUM american_scientific_journal-4481 37 21 βˆ’ βˆ’ PROPN american_scientific_journal-4481 37 22 ( ( PUNCT american_scientific_journal-4481 37 23 cos cos X american_scientific_journal-4481 37 24 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 37 25 βˆ’ βˆ’ PROPN american_scientific_journal-4481 37 26 cos𝑏𝑏 cos𝑏𝑏 PROPN american_scientific_journal-4481 37 27 cos cos PROPN american_scientific_journal-4481 37 28 𝑐𝑐)2 𝑐𝑐)2 PROPN american_scientific_journal-4481 37 29 sin2 sin2 NOUN american_scientific_journal-4481 37 30 𝑏𝑏 𝑏𝑏 NUM american_scientific_journal-4481 37 31 sin2 sin2 PROPN american_scientific_journal-4481 37 32 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 37 33 π‘œπ‘œπ‘œπ‘œ π‘œπ‘œπ‘œπ‘œ NOUN american_scientific_journal-4481 37 34 , , PUNCT american_scientific_journal-4481 37 35 sin2𝐴𝐴 sin2𝐴𝐴 X american_scientific_journal-4481 38 1 = = SYM american_scientific_journal-4481 38 2 ( ( PUNCT american_scientific_journal-4481 38 3 1 1 NUM american_scientific_journal-4481 38 4 βˆ’ βˆ’ PROPN american_scientific_journal-4481 38 5 cos2𝑏𝑏)(1 cos2𝑏𝑏)(1 PROPN american_scientific_journal-4481 38 6 βˆ’ βˆ’ NOUN american_scientific_journal-4481 38 7 cos2𝑐𝑐 cos2𝑐𝑐 NOUN american_scientific_journal-4481 38 8 ) ) PUNCT american_scientific_journal-4481 38 9 βˆ’ βˆ’ PROPN american_scientific_journal-4481 38 10 ( ( PUNCT american_scientific_journal-4481 38 11 cos2π‘Žπ‘Ž cos2π‘Žπ‘Ž NOUN american_scientific_journal-4481 38 12 + + CCONJ american_scientific_journal-4481 38 13 cos2𝑏𝑏cos2𝑐𝑐 cos2𝑏𝑏cos2𝑐𝑐 VERB american_scientific_journal-4481 38 14 βˆ’ βˆ’ PROPN american_scientific_journal-4481 38 15 2 2 NUM american_scientific_journal-4481 38 16 cosπ‘Žπ‘Ž cosπ‘Žπ‘Ž NOUN american_scientific_journal-4481 38 17 cos𝑏𝑏 cos𝑏𝑏 NOUN american_scientific_journal-4481 38 18 cos cos PROPN american_scientific_journal-4481 38 19 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 38 20 ) ) PUNCT american_scientific_journal-4481 38 21 sin2 sin2 NOUN american_scientific_journal-4481 38 22 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 38 23 sin2 sin2 PROPN american_scientific_journal-4481 38 24 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 38 25 π‘œπ‘œπ‘œπ‘œ π‘œπ‘œπ‘œπ‘œ NOUN american_scientific_journal-4481 38 26 , , PUNCT american_scientific_journal-4481 38 27 sin2𝐴𝐴 sin2𝐴𝐴 X american_scientific_journal-4481 38 28 = = SYM american_scientific_journal-4481 38 29 οΏ½ οΏ½ PROPN american_scientific_journal-4481 38 30 1βˆ’cos 1βˆ’cos PROPN american_scientific_journal-4481 38 31 2π‘π‘βˆ’cos2𝑐𝑐+cos2𝑏𝑏cos2𝑐𝑐 2π‘π‘βˆ’cos2𝑐𝑐+cos2𝑏𝑏cos2𝑐𝑐 NUM american_scientific_journal-4481 38 32 οΏ½ οΏ½ PROPN american_scientific_journal-4481 38 33 βˆ’(cos2π‘Žπ‘Ž+cos2𝑏𝑏cos2π‘π‘βˆ’2cos βˆ’(cos2π‘Žπ‘Ž+cos2𝑏𝑏cos2π‘π‘βˆ’2cos PROPN american_scientific_journal-4481 38 34 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 38 35 cos cos ADP american_scientific_journal-4481 38 36 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 38 37 cos cos PROPN american_scientific_journal-4481 38 38 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 38 39 ) ) PUNCT american_scientific_journal-4481 38 40 sin2 sin2 NOUN american_scientific_journal-4481 38 41 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 38 42 sin2 sin2 NOUN american_scientific_journal-4481 38 43 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 38 44 π‘œπ‘œπ‘œπ‘œ π‘œπ‘œπ‘œπ‘œ NOUN american_scientific_journal-4481 38 45 , , PUNCT american_scientific_journal-4481 38 46 sin𝐴𝐴 sin𝐴𝐴 PROPN american_scientific_journal-4481 38 47 = = SYM american_scientific_journal-4481 38 48 οΏ½ οΏ½ PROPN american_scientific_journal-4481 38 49 ( ( PUNCT american_scientific_journal-4481 38 50 1 1 NUM american_scientific_journal-4481 38 51 βˆ’ βˆ’ PROPN american_scientific_journal-4481 38 52 cos2π‘Žπ‘Ž cos2π‘Žπ‘Ž NOUN american_scientific_journal-4481 38 53 βˆ’ βˆ’ PROPN american_scientific_journal-4481 38 54 cos2𝑏𝑏 cos2𝑏𝑏 PROPN american_scientific_journal-4481 38 55 + + CCONJ american_scientific_journal-4481 38 56 cos2𝑐𝑐 cos2𝑐𝑐 ADJ american_scientific_journal-4481 38 57 + + CCONJ american_scientific_journal-4481 38 58 2 2 NUM american_scientific_journal-4481 38 59 cos cos NOUN american_scientific_journal-4481 38 60 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 38 61 cos cos PROPN american_scientific_journal-4481 38 62 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 38 63 cos cos PROPN american_scientific_journal-4481 38 64 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 38 65 ) ) PUNCT american_scientific_journal-4481 38 66 sin sin NOUN american_scientific_journal-4481 38 67 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 38 68 sin sin NOUN american_scientific_journal-4481 38 69 𝑐𝑐 𝑐𝑐 NOUN american_scientific_journal-4481 38 70 we we PRON american_scientific_journal-4481 38 71 have have AUX american_scientific_journal-4481 38 72 taken take VERB american_scientific_journal-4481 38 73 only only ADV american_scientific_journal-4481 38 74 positive positive ADJ american_scientific_journal-4481 38 75 sign sign NOUN american_scientific_journal-4481 38 76 , , PUNCT american_scientific_journal-4481 38 77 with with ADP american_scientific_journal-4481 38 78 the the DET american_scientific_journal-4481 38 79 radical radical ADJ american_scientific_journal-4481 38 80 as as SCONJ american_scientific_journal-4481 38 81 we we PRON american_scientific_journal-4481 38 82 know know VERB american_scientific_journal-4481 38 83 that that SCONJ american_scientific_journal-4481 38 84 the the DET american_scientific_journal-4481 38 85 angle angle NOUN american_scientific_journal-4481 38 86 and and CCONJ american_scientific_journal-4481 38 87 sides side NOUN american_scientific_journal-4481 38 88 of of ADP american_scientific_journal-4481 38 89 a a DET american_scientific_journal-4481 38 90 spherical spherical ADJ american_scientific_journal-4481 38 91 triangle triangle NOUN american_scientific_journal-4481 38 92 are be AUX american_scientific_journal-4481 38 93 each each ADV american_scientific_journal-4481 38 94 less less ADJ american_scientific_journal-4481 38 95 than than ADP american_scientific_journal-4481 38 96 two two NUM american_scientific_journal-4481 38 97 right right ADJ american_scientific_journal-4481 38 98 angles angle NOUN american_scientific_journal-4481 38 99 and and CCONJ american_scientific_journal-4481 38 100 such such ADJ american_scientific_journal-4481 38 101 sin𝐴𝐴 sin𝐴𝐴 PROPN american_scientific_journal-4481 38 102 , , PUNCT american_scientific_journal-4481 38 103 sin𝐡𝐡 sin𝐡𝐡 PROPN american_scientific_journal-4481 38 104 , , PUNCT american_scientific_journal-4481 38 105 sin𝐢𝐢 sin𝐢𝐢 PROPN american_scientific_journal-4481 38 106 are be AUX american_scientific_journal-4481 38 107 all all ADV american_scientific_journal-4481 38 108 positive positive ADJ american_scientific_journal-4481 38 109 . . PUNCT american_scientific_journal-4481 39 1 ∴ ∴ PROPN american_scientific_journal-4481 39 2 sin𝐴𝐴 sin𝐴𝐴 VERB american_scientific_journal-4481 39 3 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 39 4 = = SYM american_scientific_journal-4481 39 5 οΏ½ οΏ½ PROPN american_scientific_journal-4481 39 6 ( ( PUNCT american_scientific_journal-4481 39 7 1 1 NUM american_scientific_journal-4481 39 8 βˆ’ βˆ’ PROPN american_scientific_journal-4481 39 9 cos2π‘Žπ‘Ž cos2π‘Žπ‘Ž NOUN american_scientific_journal-4481 39 10 βˆ’ βˆ’ PROPN american_scientific_journal-4481 39 11 cos2𝑏𝑏 cos2𝑏𝑏 PROPN american_scientific_journal-4481 39 12 + + CCONJ american_scientific_journal-4481 39 13 cos2𝑐𝑐 cos2𝑐𝑐 ADJ american_scientific_journal-4481 39 14 + + CCONJ american_scientific_journal-4481 39 15 2 2 NUM american_scientific_journal-4481 39 16 cosπ‘Žπ‘Ž cosπ‘Žπ‘Ž NOUN american_scientific_journal-4481 39 17 cos𝑏𝑏 cos𝑏𝑏 NOUN american_scientific_journal-4481 39 18 cos cos PROPN american_scientific_journal-4481 39 19 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 39 20 ) ) PUNCT american_scientific_journal-4481 39 21 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 39 22 sin sin NOUN american_scientific_journal-4481 39 23 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 39 24 sin sin NOUN american_scientific_journal-4481 39 25 𝑐𝑐 𝑐𝑐 ADP american_scientific_journal-4481 39 26 the the DET american_scientific_journal-4481 39 27 symmetry symmetry NOUN american_scientific_journal-4481 39 28 of of ADP american_scientific_journal-4481 39 29 the the DET american_scientific_journal-4481 39 30 result result NOUN american_scientific_journal-4481 39 31 shows show VERB american_scientific_journal-4481 39 32 that that SCONJ american_scientific_journal-4481 39 33 ∴ ∴ PROPN american_scientific_journal-4481 39 34 sin𝐡𝐡 sin𝐡𝐡 PROPN american_scientific_journal-4481 39 35 sin sin NOUN american_scientific_journal-4481 39 36 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 39 37 = = SYM american_scientific_journal-4481 39 38 οΏ½ οΏ½ PROPN american_scientific_journal-4481 39 39 ( ( PUNCT american_scientific_journal-4481 39 40 1 1 NUM american_scientific_journal-4481 39 41 βˆ’ βˆ’ PROPN american_scientific_journal-4481 39 42 cos2π‘Žπ‘Ž cos2π‘Žπ‘Ž NOUN american_scientific_journal-4481 39 43 βˆ’ βˆ’ PROPN american_scientific_journal-4481 39 44 cos2𝑏𝑏 cos2𝑏𝑏 PROPN american_scientific_journal-4481 39 45 + + CCONJ american_scientific_journal-4481 39 46 cos2𝑐𝑐 cos2𝑐𝑐 ADJ american_scientific_journal-4481 39 47 + + CCONJ american_scientific_journal-4481 39 48 2 2 NUM american_scientific_journal-4481 39 49 cos cos NOUN american_scientific_journal-4481 39 50 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 39 51 cos cos PROPN american_scientific_journal-4481 39 52 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 39 53 cos cos PROPN american_scientific_journal-4481 39 54 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 39 55 ) ) PUNCT american_scientific_journal-4481 39 56 sin sin NOUN american_scientific_journal-4481 39 57 π‘Žπ‘Ž π‘Žπ‘Ž VERB american_scientific_journal-4481 39 58 sin sin NOUN american_scientific_journal-4481 39 59 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 39 60 sin sin NOUN american_scientific_journal-4481 39 61 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 39 62 π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž PROPN american_scientific_journal-4481 39 63 sin𝐢𝐢 sin𝐢𝐢 PROPN american_scientific_journal-4481 39 64 sin sin VERB american_scientific_journal-4481 39 65 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 39 66 = = SYM american_scientific_journal-4481 39 67 οΏ½ οΏ½ PROPN american_scientific_journal-4481 39 68 ( ( PUNCT american_scientific_journal-4481 39 69 1 1 NUM american_scientific_journal-4481 39 70 βˆ’ βˆ’ PROPN american_scientific_journal-4481 39 71 cos2π‘Žπ‘Ž cos2π‘Žπ‘Ž NOUN american_scientific_journal-4481 39 72 βˆ’ βˆ’ PROPN american_scientific_journal-4481 39 73 cos2𝑏𝑏 cos2𝑏𝑏 PROPN american_scientific_journal-4481 39 74 + + CCONJ american_scientific_journal-4481 39 75 cos2𝑐𝑐 cos2𝑐𝑐 ADJ american_scientific_journal-4481 39 76 + + CCONJ american_scientific_journal-4481 39 77 2 2 NUM american_scientific_journal-4481 39 78 cos cos NOUN american_scientific_journal-4481 39 79 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 39 80 cos cos PROPN american_scientific_journal-4481 39 81 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 39 82 cos cos PROPN american_scientific_journal-4481 39 83 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 39 84 ) ) PUNCT american_scientific_journal-4481 39 85 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 39 86 sin sin NOUN american_scientific_journal-4481 39 87 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 39 88 sin sin NOUN american_scientific_journal-4481 39 89 𝑐𝑐 𝑐𝑐 PRON american_scientific_journal-4481 40 1 so so ADV american_scientific_journal-4481 40 2 we we PRON american_scientific_journal-4481 40 3 have have VERB american_scientific_journal-4481 40 4 american american ADJ american_scientific_journal-4481 40 5 scientific scientific ADJ american_scientific_journal-4481 40 6 research research NOUN american_scientific_journal-4481 40 7 journal journal NOUN american_scientific_journal-4481 40 8 for for ADP american_scientific_journal-4481 40 9 engineering engineering NOUN american_scientific_journal-4481 40 10 , , PUNCT american_scientific_journal-4481 40 11 technology technology NOUN american_scientific_journal-4481 40 12 , , PUNCT american_scientific_journal-4481 40 13 and and CCONJ american_scientific_journal-4481 40 14 sciences science NOUN american_scientific_journal-4481 40 15 ( ( PUNCT american_scientific_journal-4481 40 16 asrjets asrjet NOUN american_scientific_journal-4481 40 17 ) ) PUNCT american_scientific_journal-4481 40 18 ( ( PUNCT american_scientific_journal-4481 40 19 2018 2018 NUM american_scientific_journal-4481 40 20 ) ) PUNCT american_scientific_journal-4481 40 21 volume volume NOUN american_scientific_journal-4481 40 22 49 49 NUM american_scientific_journal-4481 40 23 , , PUNCT american_scientific_journal-4481 40 24 no no DET american_scientific_journal-4481 40 25 1 1 NUM american_scientific_journal-4481 40 26 , , PUNCT american_scientific_journal-4481 40 27 pp pp ADV american_scientific_journal-4481 40 28 51 51 NUM american_scientific_journal-4481 40 29 - - SYM american_scientific_journal-4481 40 30 60 60 NUM american_scientific_journal-4481 40 31 55 55 NUM american_scientific_journal-4481 40 32 sin𝐴𝐴 sin𝐴𝐴 ADJ american_scientific_journal-4481 40 33 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 40 34 = = SYM american_scientific_journal-4481 40 35 sin𝐡𝐡 sin𝐡𝐡 PROPN american_scientific_journal-4481 40 36 sin sin NOUN american_scientific_journal-4481 40 37 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 40 38 = = PUNCT american_scientific_journal-4481 40 39 sin𝐢𝐢 sin𝐢𝐢 AUX american_scientific_journal-4481 40 40 sin sin VERB american_scientific_journal-4481 40 41 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 40 42 = = SYM american_scientific_journal-4481 40 43 οΏ½ οΏ½ PROPN american_scientific_journal-4481 40 44 ( ( PUNCT american_scientific_journal-4481 40 45 1 1 NUM american_scientific_journal-4481 40 46 βˆ’ βˆ’ PROPN american_scientific_journal-4481 40 47 cos2π‘Žπ‘Ž cos2π‘Žπ‘Ž NOUN american_scientific_journal-4481 40 48 βˆ’ βˆ’ PROPN american_scientific_journal-4481 40 49 cos2𝑏𝑏 cos2𝑏𝑏 PROPN american_scientific_journal-4481 40 50 + + CCONJ american_scientific_journal-4481 40 51 cos2𝑐𝑐 cos2𝑐𝑐 ADJ american_scientific_journal-4481 40 52 + + CCONJ american_scientific_journal-4481 40 53 2 2 NUM american_scientific_journal-4481 40 54 cos cos NOUN american_scientific_journal-4481 40 55 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 40 56 cos cos PROPN american_scientific_journal-4481 40 57 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 40 58 cos cos PROPN american_scientific_journal-4481 40 59 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 40 60 ) ) PUNCT american_scientific_journal-4481 40 61 sin sin NOUN american_scientific_journal-4481 40 62 π‘Žπ‘Ž π‘Žπ‘Ž VERB american_scientific_journal-4481 40 63 sin sin NOUN american_scientific_journal-4481 40 64 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 40 65 sin sin NOUN american_scientific_journal-4481 40 66 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 40 67 ∴ ∴ PROPN american_scientific_journal-4481 40 68 sin𝐴𝐴 sin𝐴𝐴 ADJ american_scientific_journal-4481 40 69 sin sin NOUN american_scientific_journal-4481 40 70 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 40 71 = = SYM american_scientific_journal-4481 40 72 sin𝐡𝐡 sin𝐡𝐡 NOUN american_scientific_journal-4481 40 73 sin sin NOUN american_scientific_journal-4481 40 74 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 41 1 = = PUNCT american_scientific_journal-4481 41 2 sin𝐢𝐢 sin𝐢𝐢 NOUN american_scientific_journal-4481 41 3 sin sin VERB american_scientific_journal-4481 41 4 𝑐𝑐 𝑐𝑐 PROPN american_scientific_journal-4481 41 5 figure figure NOUN american_scientific_journal-4481 41 6 4 4 NUM american_scientific_journal-4481 41 7 3 3 NUM american_scientific_journal-4481 41 8 . . PUNCT american_scientific_journal-4481 41 9 shortest short ADJ american_scientific_journal-4481 41 10 distance distance NOUN american_scientific_journal-4481 41 11 and and CCONJ american_scientific_journal-4481 41 12 direction direction NOUN american_scientific_journal-4481 41 13 between between ADP american_scientific_journal-4481 41 14 places place NOUN american_scientific_journal-4481 41 15 let let VERB american_scientific_journal-4481 41 16 nab nab NOUN american_scientific_journal-4481 41 17 be be AUX american_scientific_journal-4481 41 18 the the DET american_scientific_journal-4481 41 19 spherical spherical ADJ american_scientific_journal-4481 41 20 triangle triangle NOUN american_scientific_journal-4481 41 21 where where SCONJ american_scientific_journal-4481 41 22 a a PRON american_scientific_journal-4481 41 23 and and CCONJ american_scientific_journal-4481 41 24 b b NOUN american_scientific_journal-4481 41 25 the the DET american_scientific_journal-4481 41 26 two two NUM american_scientific_journal-4481 41 27 places place NOUN american_scientific_journal-4481 41 28 on on ADP american_scientific_journal-4481 41 29 the the DET american_scientific_journal-4481 41 30 sphere sphere NOUN american_scientific_journal-4481 41 31 of of ADP american_scientific_journal-4481 41 32 the the DET american_scientific_journal-4481 41 33 earth earth NOUN american_scientific_journal-4481 41 34 . . PUNCT american_scientific_journal-4481 42 1 𝐴𝐴 𝐴𝐴 PROPN american_scientific_journal-4481 42 2 ≑ ≑ PROPN american_scientific_journal-4481 42 3 ( ( PUNCT american_scientific_journal-4481 42 4 π‘₯π‘₯1 π‘₯π‘₯1 PROPN american_scientific_journal-4481 42 5 Β° Β° VERB american_scientific_journal-4481 42 6 𝑁𝑁 𝑁𝑁 PROPN american_scientific_journal-4481 42 7 , , PUNCT american_scientific_journal-4481 42 8 𝑦𝑦1 𝑦𝑦1 PROPN american_scientific_journal-4481 42 9 Β° Β° PROPN american_scientific_journal-4481 42 10 𝐷𝐷 𝐷𝐷 PROPN american_scientific_journal-4481 42 11 ) ) PUNCT american_scientific_journal-4481 42 12 𝐡𝐡 𝐡𝐡 PROPN american_scientific_journal-4481 42 13 ≑ ≑ PROPN american_scientific_journal-4481 42 14 ( ( PUNCT american_scientific_journal-4481 42 15 π‘₯π‘₯2 π‘₯π‘₯2 NOUN american_scientific_journal-4481 42 16 Β° Β° ADP american_scientific_journal-4481 42 17 𝑁𝑁 𝑁𝑁 PROPN american_scientific_journal-4481 42 18 , , PUNCT american_scientific_journal-4481 42 19 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-4481 42 20 Β° Β° PROPN american_scientific_journal-4481 42 21 𝐷𝐷 𝐷𝐷 PROPN american_scientific_journal-4481 42 22 ) ) PUNCT american_scientific_journal-4481 42 23 . . PUNCT american_scientific_journal-4481 43 1 n n PRON american_scientific_journal-4481 43 2 be be AUX american_scientific_journal-4481 43 3 the the DET american_scientific_journal-4481 43 4 north north NOUN american_scientific_journal-4481 43 5 pole pole NOUN american_scientific_journal-4481 43 6 . . PUNCT american_scientific_journal-4481 44 1 β€œ " PUNCT american_scientific_journal-4481 44 2 o o NOUN american_scientific_journal-4481 44 3 ” " PUNCT american_scientific_journal-4481 44 4 be be AUX american_scientific_journal-4481 44 5 the the DET american_scientific_journal-4481 44 6 center center NOUN american_scientific_journal-4481 44 7 of of ADP american_scientific_journal-4481 44 8 the the DET american_scientific_journal-4481 44 9 sphere sphere NOUN american_scientific_journal-4481 44 10 of of ADP american_scientific_journal-4481 44 11 the the DET american_scientific_journal-4481 44 12 earth earth NOUN american_scientific_journal-4481 44 13 . . PUNCT american_scientific_journal-4481 45 1 let let VERB american_scientific_journal-4481 45 2 , , PUNCT american_scientific_journal-4481 45 3 ∠𝐴𝐴𝐷𝐷𝐡𝐡 ∠𝐴𝐴𝐷𝐷𝐡𝐡 NOUN american_scientific_journal-4481 45 4 = = SYM american_scientific_journal-4481 45 5 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 45 6 , , PUNCT american_scientific_journal-4481 45 7 βˆ π΅π΅π·π·π‘π‘ βˆ π΅π΅π·π·π‘π‘ NOUN american_scientific_journal-4481 45 8 = = SYM american_scientific_journal-4481 45 9 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 45 10 , , PUNCT american_scientific_journal-4481 45 11 βˆ π΄π΄π·π·π‘π‘ βˆ π΄π΄π·π·π‘π‘ NOUN american_scientific_journal-4481 45 12 = = SYM american_scientific_journal-4481 45 13 𝑏𝑏. 𝑏𝑏. NOUN american_scientific_journal-4481 45 14 we we PRON american_scientific_journal-4481 45 15 have have VERB american_scientific_journal-4481 45 16 to to PART american_scientific_journal-4481 45 17 find find VERB american_scientific_journal-4481 45 18 out out ADP american_scientific_journal-4481 45 19 the the DET american_scientific_journal-4481 45 20 shortest short ADJ american_scientific_journal-4481 45 21 distance distance NOUN american_scientific_journal-4481 45 22 between between ADP american_scientific_journal-4481 45 23 the the DET american_scientific_journal-4481 45 24 given give VERB american_scientific_journal-4481 45 25 two two NUM american_scientific_journal-4481 45 26 places place NOUN american_scientific_journal-4481 45 27 and and CCONJ american_scientific_journal-4481 45 28 the the DET american_scientific_journal-4481 45 29 direction direction NOUN american_scientific_journal-4481 45 30 of of ADP american_scientific_journal-4481 45 31 the the DET american_scientific_journal-4481 45 32 two two NUM american_scientific_journal-4481 45 33 places place NOUN american_scientific_journal-4481 45 34 towards towards ADP american_scientific_journal-4481 45 35 north north NOUN american_scientific_journal-4481 45 36 pole pole NOUN american_scientific_journal-4481 45 37 . . PUNCT american_scientific_journal-4481 46 1 3.1 3.1 NUM american_scientific_journal-4481 46 2 . . PUNCT american_scientific_journal-4481 46 3 calculation calculation NOUN american_scientific_journal-4481 46 4 for for ADP american_scientific_journal-4481 46 5 the the DET american_scientific_journal-4481 46 6 shortest short ADJ american_scientific_journal-4481 46 7 distance distance NOUN american_scientific_journal-4481 46 8 between between ADP american_scientific_journal-4481 46 9 a a PRON american_scientific_journal-4481 46 10 and and CCONJ american_scientific_journal-4481 46 11 b b NOUN american_scientific_journal-4481 46 12 towards towards ADP american_scientific_journal-4481 46 13 north north NOUN american_scientific_journal-4481 46 14 pole pole NOUN american_scientific_journal-4481 46 15 we we PRON american_scientific_journal-4481 46 16 have have VERB american_scientific_journal-4481 46 17 from from ADP american_scientific_journal-4481 46 18 cosine cosine NOUN american_scientific_journal-4481 46 19 rule rule NOUN american_scientific_journal-4481 46 20 of of ADP american_scientific_journal-4481 46 21 spherical spherical ADJ american_scientific_journal-4481 46 22 trigonometry trigonometry NOUN american_scientific_journal-4481 46 23 cosπ‘Žπ‘Ž cosπ‘Žπ‘Ž NOUN american_scientific_journal-4481 46 24 = = SYM american_scientific_journal-4481 46 25 cos cos PROPN american_scientific_journal-4481 46 26 π‘Žπ‘Ž π‘Žπ‘Ž PROPN american_scientific_journal-4481 46 27 cos cos ADP american_scientific_journal-4481 46 28 𝑏𝑏 𝑏𝑏 PROPN american_scientific_journal-4481 46 29 + + CCONJ american_scientific_journal-4481 46 30 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 46 31 sin sin NOUN american_scientific_journal-4481 46 32 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 46 33 cos𝑁𝑁 cos𝑁𝑁 NOUN american_scientific_journal-4481 46 34 = = PUNCT american_scientific_journal-4481 46 35 cos(90 cos(90 NOUN american_scientific_journal-4481 46 36 Β° Β° NUM american_scientific_journal-4481 46 37 βˆ’x2 βˆ’x2 NOUN american_scientific_journal-4481 46 38 Β° Β° NOUN american_scientific_journal-4481 46 39 ) ) PUNCT american_scientific_journal-4481 46 40 cos(90 cos(90 VERB american_scientific_journal-4481 46 41 Β° Β° PUNCT american_scientific_journal-4481 46 42 βˆ’ βˆ’ PROPN american_scientific_journal-4481 46 43 x2 x2 PROPN american_scientific_journal-4481 46 44 Β° Β° PROPN american_scientific_journal-4481 46 45 ) ) PUNCT american_scientific_journal-4481 47 1 + + CCONJ american_scientific_journal-4481 47 2 sin(90 sin(90 NOUN american_scientific_journal-4481 47 3 Β° Β° PROPN american_scientific_journal-4481 47 4 βˆ’ βˆ’ PROPN american_scientific_journal-4481 47 5 x2 x2 PROPN american_scientific_journal-4481 47 6 Β° Β° PROPN american_scientific_journal-4481 47 7 ) ) PUNCT american_scientific_journal-4481 47 8 sin(90 sin(90 NOUN american_scientific_journal-4481 47 9 Β° Β° PROPN american_scientific_journal-4481 47 10 βˆ’ βˆ’ PROPN american_scientific_journal-4481 47 11 x2 x2 PROPN american_scientific_journal-4481 47 12 Β° Β° PRON american_scientific_journal-4481 47 13 ) ) PUNCT american_scientific_journal-4481 47 14 cos(y2 cos(y2 NOUN american_scientific_journal-4481 47 15 Β° Β° NOUN american_scientific_journal-4481 47 16 βˆ’ βˆ’ PROPN american_scientific_journal-4481 47 17 y1 y1 PROPN american_scientific_journal-4481 47 18 Β° Β° PROPN american_scientific_journal-4481 47 19 ) ) PUNCT american_scientific_journal-4481 47 20 = = PUNCT american_scientific_journal-4481 47 21 sin sin NOUN american_scientific_journal-4481 47 22 x2 x2 PROPN american_scientific_journal-4481 47 23 Β° Β° PROPN american_scientific_journal-4481 47 24 sin sin VERB american_scientific_journal-4481 47 25 x1 x1 PROPN american_scientific_journal-4481 47 26 Β° Β° NOUN american_scientific_journal-4481 48 1 + + CCONJ american_scientific_journal-4481 48 2 cos cos PROPN american_scientific_journal-4481 48 3 x2 x2 PROPN american_scientific_journal-4481 48 4 Β° Β° PROPN american_scientific_journal-4481 48 5 cos cos PROPN american_scientific_journal-4481 49 1 x1 x1 PROPN american_scientific_journal-4481 49 2 Β° Β° PROPN american_scientific_journal-4481 49 3 cos(y2 cos(y2 NOUN american_scientific_journal-4481 49 4 Β° Β° PROPN american_scientific_journal-4481 49 5 βˆ’ βˆ’ PROPN american_scientific_journal-4481 49 6 y1 y1 PROPN american_scientific_journal-4481 49 7 Β° Β° PRON american_scientific_journal-4481 49 8 ) ) PUNCT american_scientific_journal-4481 49 9 ∴ ∴ NOUN american_scientific_journal-4481 49 10 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 49 11 = = SYM american_scientific_journal-4481 49 12 cosβˆ’1 cosβˆ’1 PROPN american_scientific_journal-4481 49 13 { { PUNCT american_scientific_journal-4481 49 14 sin sin NOUN american_scientific_journal-4481 49 15 x2 x2 PROPN american_scientific_journal-4481 49 16 Β° Β° PROPN american_scientific_journal-4481 49 17 sin sin VERB american_scientific_journal-4481 49 18 x1 x1 PROPN american_scientific_journal-4481 49 19 Β° Β° NOUN american_scientific_journal-4481 50 1 + + CCONJ american_scientific_journal-4481 50 2 cos cos PROPN american_scientific_journal-4481 50 3 x2 x2 PROPN american_scientific_journal-4481 50 4 Β° Β° PROPN american_scientific_journal-4481 50 5 cos cos PROPN american_scientific_journal-4481 51 1 x1 x1 PROPN american_scientific_journal-4481 51 2 Β° Β° PROPN american_scientific_journal-4481 51 3 cos(y2 cos(y2 NOUN american_scientific_journal-4481 51 4 Β° Β° PROPN american_scientific_journal-4481 51 5 βˆ’ βˆ’ PROPN american_scientific_journal-4481 51 6 y1 y1 PROPN american_scientific_journal-4481 51 7 Β° Β° NUM american_scientific_journal-4481 51 8 ) ) PUNCT american_scientific_journal-4481 51 9 } } PUNCT american_scientific_journal-4481 51 10 ( ( PUNCT american_scientific_journal-4481 51 11 1 1 X american_scientific_journal-4481 51 12 ) ) PUNCT american_scientific_journal-4481 51 13 ∴ ∴ NOUN american_scientific_journal-4481 51 14 the the DET american_scientific_journal-4481 51 15 distance distance NOUN american_scientific_journal-4481 51 16 between between ADP american_scientific_journal-4481 51 17 a a PRON american_scientific_journal-4481 51 18 and and CCONJ american_scientific_journal-4481 51 19 b b NOUN american_scientific_journal-4481 51 20 = = NOUN american_scientific_journal-4481 51 21 the the DET american_scientific_journal-4481 51 22 length length NOUN american_scientific_journal-4481 51 23 of of ADP american_scientific_journal-4481 51 24 the the DET american_scientific_journal-4481 51 25 arc arc NOUN american_scientific_journal-4481 51 26 of of ADP american_scientific_journal-4481 51 27 the the DET american_scientific_journal-4481 51 28 great great ADJ american_scientific_journal-4481 51 29 circle circle NOUN american_scientific_journal-4481 51 30 passing pass VERB american_scientific_journal-4481 51 31 through through ADP american_scientific_journal-4481 51 32 a a DET american_scientific_journal-4481 51 33 and and CCONJ american_scientific_journal-4481 51 34 b b NOUN american_scientific_journal-4481 51 35 = = SYM american_scientific_journal-4481 51 36 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 51 37 = = SYM american_scientific_journal-4481 51 38 𝑅𝑅.π‘Žπ‘Ž 𝑅𝑅.π‘Žπ‘Ž NOUN american_scientific_journal-4481 51 39 ( ( PUNCT american_scientific_journal-4481 51 40 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-4481 51 41 = = NOUN american_scientific_journal-4481 51 42 radius radius NOUN american_scientific_journal-4481 51 43 of of ADP american_scientific_journal-4481 51 44 the the DET american_scientific_journal-4481 51 45 earth earth NOUN american_scientific_journal-4481 51 46 ) ) PUNCT american_scientific_journal-4481 51 47 . . PUNCT american_scientific_journal-4481 52 1 by by ADP american_scientific_journal-4481 52 2 equation equation NOUN american_scientific_journal-4481 52 3 ( ( PUNCT american_scientific_journal-4481 52 4 1 1 X american_scientific_journal-4481 52 5 ) ) PUNCT american_scientific_journal-4481 52 6 we we PRON american_scientific_journal-4481 52 7 can can AUX american_scientific_journal-4481 52 8 easily easily ADV american_scientific_journal-4481 52 9 find find VERB american_scientific_journal-4481 52 10 out out ADP american_scientific_journal-4481 52 11 the the DET american_scientific_journal-4481 52 12 value value NOUN american_scientific_journal-4481 52 13 of of ADP american_scientific_journal-4481 52 14 n n PRON american_scientific_journal-4481 52 15 for for ADP american_scientific_journal-4481 52 16 any any DET american_scientific_journal-4481 52 17 two two NUM american_scientific_journal-4481 52 18 places place NOUN american_scientific_journal-4481 52 19 and and CCONJ american_scientific_journal-4481 52 20 hence hence ADV american_scientific_journal-4481 52 21 we we PRON american_scientific_journal-4481 52 22 can can AUX american_scientific_journal-4481 52 23 find find VERB american_scientific_journal-4481 52 24 out out ADP american_scientific_journal-4481 52 25 shortest short ADJ american_scientific_journal-4481 52 26 distance distance NOUN american_scientific_journal-4481 52 27 d. d. PROPN american_scientific_journal-4481 52 28 3.2 3.2 NUM american_scientific_journal-4481 52 29 . . PUNCT american_scientific_journal-4481 53 1 calculation calculation NOUN american_scientific_journal-4481 53 2 for for ADP american_scientific_journal-4481 53 3 the the DET american_scientific_journal-4481 53 4 direction direction NOUN american_scientific_journal-4481 53 5 american american PROPN american_scientific_journal-4481 53 6 scientific scientific ADJ american_scientific_journal-4481 53 7 research research NOUN american_scientific_journal-4481 53 8 journal journal NOUN american_scientific_journal-4481 53 9 for for ADP american_scientific_journal-4481 53 10 engineering engineering NOUN american_scientific_journal-4481 53 11 , , PUNCT american_scientific_journal-4481 53 12 technology technology NOUN american_scientific_journal-4481 53 13 , , PUNCT american_scientific_journal-4481 53 14 and and CCONJ american_scientific_journal-4481 53 15 sciences science NOUN american_scientific_journal-4481 53 16 ( ( PUNCT american_scientific_journal-4481 53 17 asrjets asrjet NOUN american_scientific_journal-4481 53 18 ) ) PUNCT american_scientific_journal-4481 53 19 ( ( PUNCT american_scientific_journal-4481 53 20 2018 2018 NUM american_scientific_journal-4481 53 21 ) ) PUNCT american_scientific_journal-4481 53 22 volume volume NOUN american_scientific_journal-4481 53 23 49 49 NUM american_scientific_journal-4481 53 24 , , PUNCT american_scientific_journal-4481 53 25 no no DET american_scientific_journal-4481 53 26 1 1 NUM american_scientific_journal-4481 53 27 , , PUNCT american_scientific_journal-4481 53 28 pp pp ADV american_scientific_journal-4481 53 29 51 51 NUM american_scientific_journal-4481 53 30 - - SYM american_scientific_journal-4481 53 31 60 60 NUM american_scientific_journal-4481 53 32 56 56 NUM american_scientific_journal-4481 53 33 from from ADP american_scientific_journal-4481 53 34 spherical spherical ADJ american_scientific_journal-4481 53 35 triangle triangle NOUN american_scientific_journal-4481 53 36 using use VERB american_scientific_journal-4481 53 37 sine sine ADJ american_scientific_journal-4481 53 38 rule rule NOUN american_scientific_journal-4481 53 39 sin𝐴𝐴 sin𝐴𝐴 ADJ american_scientific_journal-4481 53 40 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 53 41 = = SYM american_scientific_journal-4481 53 42 sin𝐡𝐡 sin𝐡𝐡 PROPN american_scientific_journal-4481 53 43 sin sin NOUN american_scientific_journal-4481 53 44 𝑏𝑏 𝑏𝑏 NOUN american_scientific_journal-4481 53 45 = = PUNCT american_scientific_journal-4481 53 46 sin𝑁𝑁 sin𝑁𝑁 NOUN american_scientific_journal-4481 53 47 sin sin NOUN american_scientific_journal-4481 53 48 π‘Žπ‘Ž π‘Žπ‘Ž VERB american_scientific_journal-4481 53 49 or or CCONJ american_scientific_journal-4481 53 50 , , PUNCT american_scientific_journal-4481 53 51 sin𝐴𝐴 sin𝐴𝐴 ADJ american_scientific_journal-4481 53 52 = = NOUN american_scientific_journal-4481 53 53 sinπ‘Žπ‘Ž sinπ‘Žπ‘Ž NOUN american_scientific_journal-4481 53 54 sin𝑁𝑁 sin𝑁𝑁 ADJ american_scientific_journal-4481 53 55 sin sin NOUN american_scientific_journal-4481 53 56 π‘Žπ‘Ž π‘Žπ‘Ž VERB american_scientific_journal-4481 53 57 or or CCONJ american_scientific_journal-4481 53 58 , , PUNCT american_scientific_journal-4481 53 59 sin𝐴𝐴 sin𝐴𝐴 PROPN american_scientific_journal-4481 53 60 = = SYM american_scientific_journal-4481 53 61 sin(90 sin(90 NOUN american_scientific_journal-4481 53 62 Β° Β° PROPN american_scientific_journal-4481 53 63 βˆ’ βˆ’ PROPN american_scientific_journal-4481 53 64 x2 x2 PROPN american_scientific_journal-4481 53 65 Β° Β° PROPN american_scientific_journal-4481 53 66 ) ) PUNCT american_scientific_journal-4481 53 67 sin(y2 sin(y2 NOUN american_scientific_journal-4481 53 68 Β° Β° PROPN american_scientific_journal-4481 53 69 βˆ’ βˆ’ PROPN american_scientific_journal-4481 53 70 y1 y1 PROPN american_scientific_journal-4481 53 71 Β° Β° PRON american_scientific_journal-4481 53 72 ) ) PUNCT american_scientific_journal-4481 53 73 sin sin NOUN american_scientific_journal-4481 53 74 π‘Žπ‘Ž π‘Žπ‘Ž VERB american_scientific_journal-4481 53 75 or or CCONJ american_scientific_journal-4481 53 76 , , PUNCT american_scientific_journal-4481 53 77 sin𝐴𝐴 sin𝐴𝐴 PROPN american_scientific_journal-4481 53 78 = = PROPN american_scientific_journal-4481 53 79 cos cos PROPN american_scientific_journal-4481 53 80 x2 x2 PROPN american_scientific_journal-4481 53 81 Β° Β° PROPN american_scientific_journal-4481 53 82 sin(y2 sin(y2 NOUN american_scientific_journal-4481 54 1 Β° Β° PROPN american_scientific_journal-4481 54 2 βˆ’ βˆ’ PROPN american_scientific_journal-4481 54 3 y1 y1 PROPN american_scientific_journal-4481 54 4 Β° Β° PROPN american_scientific_journal-4481 54 5 ) ) PUNCT american_scientific_journal-4481 54 6 sin sin NOUN american_scientific_journal-4481 54 7 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 54 8 ( ( PUNCT american_scientific_journal-4481 54 9 2 2 NUM american_scientific_journal-4481 54 10 ) ) PUNCT american_scientific_journal-4481 54 11 similarly similarly ADV american_scientific_journal-4481 54 12 , , PUNCT american_scientific_journal-4481 54 13 sin𝐡𝐡 sin𝐡𝐡 PROPN american_scientific_journal-4481 54 14 = = SYM american_scientific_journal-4481 54 15 cos cos PROPN american_scientific_journal-4481 54 16 x1 x1 PROPN american_scientific_journal-4481 54 17 Β° Β° PROPN american_scientific_journal-4481 54 18 sin(y2 sin(y2 NOUN american_scientific_journal-4481 54 19 Β° Β° PROPN american_scientific_journal-4481 54 20 βˆ’ βˆ’ PROPN american_scientific_journal-4481 54 21 y1 y1 PROPN american_scientific_journal-4481 54 22 Β° Β° PROPN american_scientific_journal-4481 54 23 ) ) PUNCT american_scientific_journal-4481 54 24 sin sin NOUN american_scientific_journal-4481 54 25 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 54 26 ( ( PUNCT american_scientific_journal-4481 54 27 3 3 NUM american_scientific_journal-4481 54 28 ) ) PUNCT american_scientific_journal-4481 54 29 by by ADP american_scientific_journal-4481 54 30 , , PUNCT american_scientific_journal-4481 54 31 these these DET american_scientific_journal-4481 54 32 two two NUM american_scientific_journal-4481 54 33 equations equation NOUN american_scientific_journal-4481 54 34 we we PRON american_scientific_journal-4481 54 35 can can AUX american_scientific_journal-4481 54 36 easily easily ADV american_scientific_journal-4481 54 37 find find VERB american_scientific_journal-4481 54 38 out out ADP american_scientific_journal-4481 54 39 the the DET american_scientific_journal-4481 54 40 direction direction NOUN american_scientific_journal-4481 54 41 of of ADP american_scientific_journal-4481 54 42 any any DET american_scientific_journal-4481 54 43 place place NOUN american_scientific_journal-4481 54 44 with with ADP american_scientific_journal-4481 54 45 respect respect NOUN american_scientific_journal-4481 54 46 to to ADP american_scientific_journal-4481 54 47 another another DET american_scientific_journal-4481 54 48 place place NOUN american_scientific_journal-4481 54 49 towards towards ADP american_scientific_journal-4481 54 50 north north NOUN american_scientific_journal-4481 54 51 pole pole NOUN american_scientific_journal-4481 54 52 . . PUNCT american_scientific_journal-4481 55 1 4 4 X american_scientific_journal-4481 55 2 . . X american_scientific_journal-4481 55 3 examples example NOUN american_scientific_journal-4481 55 4 4.1 4.1 NUM american_scientific_journal-4481 55 5 . . PUNCT american_scientific_journal-4481 55 6 example example NOUN american_scientific_journal-4481 55 7 to to PART american_scientific_journal-4481 55 8 find find VERB american_scientific_journal-4481 55 9 the the DET american_scientific_journal-4481 55 10 shortest short ADJ american_scientific_journal-4481 55 11 distance distance NOUN american_scientific_journal-4481 55 12 and and CCONJ american_scientific_journal-4481 55 13 direction direction NOUN american_scientific_journal-4481 55 14 from from ADP american_scientific_journal-4481 55 15 one one NUM american_scientific_journal-4481 55 16 towards towards ADP american_scientific_journal-4481 55 17 the the DET american_scientific_journal-4481 55 18 other other ADJ american_scientific_journal-4481 55 19 between between ADP american_scientific_journal-4481 55 20 london london PROPN american_scientific_journal-4481 55 21 ( ( PUNCT american_scientific_journal-4481 55 22 u.k u.k PROPN american_scientific_journal-4481 55 23 ) ) PUNCT american_scientific_journal-4481 55 24 and and CCONJ american_scientific_journal-4481 55 25 bangkok bangkok PROPN american_scientific_journal-4481 55 26 ( ( PUNCT american_scientific_journal-4481 55 27 thailand thailand PROPN american_scientific_journal-4481 55 28 ) ) PUNCT american_scientific_journal-4481 55 29 towards towards ADP american_scientific_journal-4481 55 30 north north NOUN american_scientific_journal-4481 55 31 pole pole NOUN american_scientific_journal-4481 55 32 . . PUNCT american_scientific_journal-4481 56 1 here here ADV american_scientific_journal-4481 56 2 , , PUNCT american_scientific_journal-4481 56 3 𝐴𝐴 𝐴𝐴 PROPN american_scientific_journal-4481 56 4 ≑ ≑ PROPN american_scientific_journal-4481 56 5 πΏπΏπ‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘œπ‘œπ‘Žπ‘Ž πΏπΏπ‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘œπ‘œπ‘Žπ‘Ž PROPN american_scientific_journal-4481 56 6 ; ; PUNCT american_scientific_journal-4481 56 7 π‘₯π‘₯1 π‘₯π‘₯1 NOUN american_scientific_journal-4481 56 8 = = SYM american_scientific_journal-4481 56 9 51.30 51.30 NUM american_scientific_journal-4481 56 10 Β° Β° NUM american_scientific_journal-4481 56 11 𝑁𝑁 𝑁𝑁 PROPN american_scientific_journal-4481 56 12 𝑦𝑦1 𝑦𝑦1 NOUN american_scientific_journal-4481 56 13 = = SYM american_scientific_journal-4481 56 14 0.10 0.10 NUM american_scientific_journal-4481 56 15 Β° Β° PRON american_scientific_journal-4481 56 16 π‘Šπ‘Š π‘Šπ‘Š PROPN american_scientific_journal-4481 56 17 𝐡𝐡 𝐡𝐡 PROPN american_scientific_journal-4481 56 18 ≑ ≑ PROPN american_scientific_journal-4481 56 19 π΅π΅π‘Žπ‘Žπ‘Žπ‘Žπ΅π΅π΅π΅π‘œπ‘œπ΅π΅ π΅π΅π‘Žπ‘Žπ‘Žπ‘Žπ΅π΅π΅π΅π‘œπ‘œπ΅π΅ PROPN american_scientific_journal-4481 56 20 ; ; PUNCT american_scientific_journal-4481 56 21 π‘₯π‘₯2 π‘₯π‘₯2 ADP american_scientific_journal-4481 56 22 = = PUNCT american_scientific_journal-4481 56 23 13.44 13.44 NUM american_scientific_journal-4481 56 24 Β° Β° NUM american_scientific_journal-4481 56 25 𝑁𝑁 𝑁𝑁 PROPN american_scientific_journal-4481 56 26 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-4481 56 27 = = NOUN american_scientific_journal-4481 56 28 100.30 100.30 NUM american_scientific_journal-4481 56 29 Β° Β° PROPN american_scientific_journal-4481 56 30 𝐷𝐷 𝐷𝐷 PROPN american_scientific_journal-4481 56 31 n n NOUN american_scientific_journal-4481 56 32 = = SYM american_scientific_journal-4481 56 33 y2 y2 PROPN american_scientific_journal-4481 56 34 Β° Β° PROPN american_scientific_journal-4481 56 35 βˆ’ βˆ’ PROPN american_scientific_journal-4481 56 36 y1 y1 NOUN american_scientific_journal-4481 56 37 Β° Β° NOUN american_scientific_journal-4481 56 38 = = PUNCT american_scientific_journal-4481 57 1 100.30 100.30 NUM american_scientific_journal-4481 57 2 Β° Β° NOUN american_scientific_journal-4481 57 3 βˆ’ βˆ’ PROPN american_scientific_journal-4481 57 4 ( ( PUNCT american_scientific_journal-4481 57 5 βˆ’0.10 βˆ’0.10 NOUN american_scientific_journal-4481 57 6 Β° Β° NUM american_scientific_journal-4481 57 7 ) ) PUNCT american_scientific_journal-4481 57 8 = = PUNCT american_scientific_journal-4481 58 1 100.30 100.30 NUM american_scientific_journal-4481 58 2 Β° Β° NOUN american_scientific_journal-4481 59 1 + + NOUN american_scientific_journal-4481 60 1 0.10 0.10 NUM american_scientific_journal-4481 60 2 Β° Β° NOUN american_scientific_journal-4481 60 3 = = PUNCT american_scientific_journal-4481 61 1 100.40 100.40 NUM american_scientific_journal-4481 61 2 Β° Β° NOUN american_scientific_journal-4481 61 3 we we PRON american_scientific_journal-4481 61 4 know know VERB american_scientific_journal-4481 61 5 from from ADP american_scientific_journal-4481 61 6 spherical spherical ADJ american_scientific_journal-4481 61 7 trigonometry trigonometry NOUN american_scientific_journal-4481 61 8 , , PUNCT american_scientific_journal-4481 61 9 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 61 10 = = SYM american_scientific_journal-4481 61 11 cosβˆ’1 cosβˆ’1 PROPN american_scientific_journal-4481 61 12 { { PUNCT american_scientific_journal-4481 61 13 sin sin NOUN american_scientific_journal-4481 61 14 x2 x2 PROPN american_scientific_journal-4481 61 15 Β° Β° PROPN american_scientific_journal-4481 61 16 sin sin VERB american_scientific_journal-4481 62 1 x1 x1 PROPN american_scientific_journal-4481 62 2 Β° Β° NOUN american_scientific_journal-4481 62 3 + + CCONJ american_scientific_journal-4481 62 4 cos cos PROPN american_scientific_journal-4481 62 5 x2 x2 PROPN american_scientific_journal-4481 62 6 Β° Β° PROPN american_scientific_journal-4481 62 7 cos cos PROPN american_scientific_journal-4481 63 1 x1 x1 PROPN american_scientific_journal-4481 63 2 Β° Β° PROPN american_scientific_journal-4481 63 3 cos(y2 cos(y2 NOUN american_scientific_journal-4481 63 4 Β° Β° PROPN american_scientific_journal-4481 63 5 βˆ’ βˆ’ PROPN american_scientific_journal-4481 63 6 y1 y1 PROPN american_scientific_journal-4481 63 7 Β° Β° NUM american_scientific_journal-4481 63 8 ) ) PUNCT american_scientific_journal-4481 63 9 } } PUNCT american_scientific_journal-4481 64 1 = = SYM american_scientific_journal-4481 64 2 cosβˆ’1 cosβˆ’1 NOUN american_scientific_journal-4481 64 3 { { PUNCT american_scientific_journal-4481 64 4 sin sin NOUN american_scientific_journal-4481 64 5 51.30 51.30 NUM american_scientific_journal-4481 64 6 Β° Β° NUM american_scientific_journal-4481 64 7 sin sin NOUN american_scientific_journal-4481 64 8 13.44 13.44 NUM american_scientific_journal-4481 64 9 Β° Β° NOUN american_scientific_journal-4481 65 1 + + CCONJ american_scientific_journal-4481 65 2 cos cos PROPN american_scientific_journal-4481 65 3 51.30 51.30 NUM american_scientific_journal-4481 65 4 Β° Β° NUM american_scientific_journal-4481 65 5 cos cos ADP american_scientific_journal-4481 65 6 13.44 13.44 NUM american_scientific_journal-4481 65 7 Β° Β° NOUN american_scientific_journal-4481 65 8 cos cos ADP american_scientific_journal-4481 65 9 100.40 100.40 NUM american_scientific_journal-4481 65 10 } } PUNCT american_scientific_journal-4481 65 11 american american ADJ american_scientific_journal-4481 65 12 scientific scientific ADJ american_scientific_journal-4481 65 13 research research NOUN american_scientific_journal-4481 65 14 journal journal NOUN american_scientific_journal-4481 65 15 for for ADP american_scientific_journal-4481 65 16 engineering engineering NOUN american_scientific_journal-4481 65 17 , , PUNCT american_scientific_journal-4481 65 18 technology technology NOUN american_scientific_journal-4481 65 19 , , PUNCT american_scientific_journal-4481 65 20 and and CCONJ american_scientific_journal-4481 65 21 sciences science NOUN american_scientific_journal-4481 65 22 ( ( PUNCT american_scientific_journal-4481 65 23 asrjets asrjet NOUN american_scientific_journal-4481 65 24 ) ) PUNCT american_scientific_journal-4481 65 25 ( ( PUNCT american_scientific_journal-4481 65 26 2018 2018 NUM american_scientific_journal-4481 65 27 ) ) PUNCT american_scientific_journal-4481 65 28 volume volume NOUN american_scientific_journal-4481 65 29 49 49 NUM american_scientific_journal-4481 65 30 , , PUNCT american_scientific_journal-4481 65 31 no no DET american_scientific_journal-4481 65 32 1 1 NUM american_scientific_journal-4481 65 33 , , PUNCT american_scientific_journal-4481 65 34 pp pp ADV american_scientific_journal-4481 65 35 51 51 NUM american_scientific_journal-4481 65 36 - - SYM american_scientific_journal-4481 65 37 60 60 NUM american_scientific_journal-4481 65 38 57 57 NUM american_scientific_journal-4481 65 39 = = NOUN american_scientific_journal-4481 65 40 cosβˆ’1(0.1813931 cosβˆ’1(0.1813931 DET american_scientific_journal-4481 65 41 βˆ’ βˆ’ NOUN american_scientific_journal-4481 65 42 0.1097772 0.1097772 NUM american_scientific_journal-4481 65 43 ) ) PUNCT american_scientific_journal-4481 65 44 = = SYM american_scientific_journal-4481 65 45 cosβˆ’1(0.0716159 cosβˆ’1(0.0716159 X american_scientific_journal-4481 65 46 ) ) PUNCT american_scientific_journal-4481 65 47 = = SYM american_scientific_journal-4481 65 48 85.8931980 85.8931980 NUM american_scientific_journal-4481 65 49 Β° Β° VERB american_scientific_journal-4481 65 50 = = SYM american_scientific_journal-4481 65 51 85.8931980 85.8931980 NUM american_scientific_journal-4481 65 52 Γ— Γ— NOUN american_scientific_journal-4481 65 53 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4481 65 54 180 180 NUM american_scientific_journal-4481 65 55 π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4481 65 56 = = SYM american_scientific_journal-4481 65 57 1.4991191π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž 1.4991191π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž NUM american_scientific_journal-4481 65 58 calculation calculation NOUN american_scientific_journal-4481 65 59 for for ADP american_scientific_journal-4481 65 60 the the DET american_scientific_journal-4481 65 61 shortest short ADJ american_scientific_journal-4481 65 62 distance distance NOUN american_scientific_journal-4481 65 63 : : PUNCT american_scientific_journal-4481 65 64 radius radius NOUN american_scientific_journal-4481 65 65 of of ADP american_scientific_journal-4481 65 66 the the DET american_scientific_journal-4481 65 67 earth earth NOUN american_scientific_journal-4481 65 68 , , PUNCT american_scientific_journal-4481 65 69 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-4481 65 70 = = SYM american_scientific_journal-4481 65 71 6378.388 6378.388 NUM american_scientific_journal-4481 65 72 π΅π΅π‘˜π‘˜π‘˜π‘˜ π΅π΅π‘˜π‘˜π‘˜π‘˜ PROPN american_scientific_journal-4481 65 73 ∴ ∴ PROPN american_scientific_journal-4481 65 74 shortest short ADJ american_scientific_journal-4481 65 75 distance distance NOUN american_scientific_journal-4481 65 76 between between ADP american_scientific_journal-4481 65 77 πΏπΏπ‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘œπ‘œπ‘Žπ‘Ž πΏπΏπ‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘œπ‘œπ‘Žπ‘Ž PROPN american_scientific_journal-4481 65 78 and and CCONJ american_scientific_journal-4481 65 79 π΅π΅π‘Žπ‘Žπ‘Žπ‘Žπ΅π΅π΅π΅π‘œπ‘œπ΅π΅ π΅π΅π‘Žπ‘Žπ‘Žπ‘Žπ΅π΅π΅π΅π‘œπ‘œπ΅π΅ NOUN american_scientific_journal-4481 65 80 is be AUX american_scientific_journal-4481 65 81 , , PUNCT american_scientific_journal-4481 65 82 π‘Žπ‘Ž π‘Žπ‘Ž ADJ american_scientific_journal-4481 65 83 = = SYM american_scientific_journal-4481 65 84 𝑅𝑅 𝑅𝑅 ADJ american_scientific_journal-4481 65 85 β‹… β‹… NOUN american_scientific_journal-4481 65 86 π‘Žπ‘Ž π‘Žπ‘Ž X american_scientific_journal-4481 65 87 = = SYM american_scientific_journal-4481 65 88 6378.388 6378.388 NUM american_scientific_journal-4481 65 89 Γ— Γ— NOUN american_scientific_journal-4481 65 90 1.4991191 1.4991191 NUM american_scientific_journal-4481 65 91 = = SYM american_scientific_journal-4481 65 92 9561.96 9561.96 NUM american_scientific_journal-4481 65 93 π΅π΅π‘˜π‘˜π‘˜π‘˜ π΅π΅π‘˜π‘˜π‘˜π‘˜ PROPN american_scientific_journal-4481 65 94 so so ADV american_scientific_journal-4481 65 95 , , PUNCT american_scientific_journal-4481 65 96 the the DET american_scientific_journal-4481 65 97 distance distance NOUN american_scientific_journal-4481 65 98 between between ADP american_scientific_journal-4481 65 99 london london PROPN american_scientific_journal-4481 65 100 and and CCONJ american_scientific_journal-4481 65 101 bangkok bangkok PROPN american_scientific_journal-4481 65 102 is be AUX american_scientific_journal-4481 65 103 9561.96 9561.96 NUM american_scientific_journal-4481 65 104 π΅π΅π‘˜π‘˜π‘˜π‘˜. π΅π΅π‘˜π‘˜π‘˜π‘˜. ADJ american_scientific_journal-4481 65 105 calculation calculation NOUN american_scientific_journal-4481 65 106 for for ADP american_scientific_journal-4481 65 107 the the DET american_scientific_journal-4481 65 108 direction direction NOUN american_scientific_journal-4481 65 109 of of ADP american_scientific_journal-4481 65 110 london london PROPN american_scientific_journal-4481 65 111 and and CCONJ american_scientific_journal-4481 65 112 bangkok bangkok PROPN american_scientific_journal-4481 65 113 towards towards ADP american_scientific_journal-4481 65 114 north north PROPN american_scientific_journal-4481 65 115 pole pole NOUN american_scientific_journal-4481 65 116 : : PUNCT american_scientific_journal-4481 65 117 from from ADP american_scientific_journal-4481 65 118 the the DET american_scientific_journal-4481 65 119 sine sine ADJ american_scientific_journal-4481 65 120 rule rule NOUN american_scientific_journal-4481 65 121 on on ADP american_scientific_journal-4481 65 122 spherical spherical ADJ american_scientific_journal-4481 65 123 triangle triangle NOUN american_scientific_journal-4481 65 124 we we PRON american_scientific_journal-4481 65 125 have have VERB american_scientific_journal-4481 65 126 , , PUNCT american_scientific_journal-4481 65 127 sin𝐴𝐴 sin𝐴𝐴 PROPN american_scientific_journal-4481 65 128 = = PUNCT american_scientific_journal-4481 65 129 cos cos PROPN american_scientific_journal-4481 65 130 x2 x2 PROPN american_scientific_journal-4481 65 131 Β° Β° PROPN american_scientific_journal-4481 65 132 sin(y2 sin(y2 NOUN american_scientific_journal-4481 65 133 Β° Β° PROPN american_scientific_journal-4481 65 134 βˆ’ βˆ’ PROPN american_scientific_journal-4481 65 135 y1 y1 PROPN american_scientific_journal-4481 65 136 Β° Β° PROPN american_scientific_journal-4481 65 137 ) ) PUNCT american_scientific_journal-4481 65 138 sin sin NOUN american_scientific_journal-4481 65 139 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 65 140 = = SYM american_scientific_journal-4481 65 141 cos cos ADP american_scientific_journal-4481 65 142 13.44 13.44 NUM american_scientific_journal-4481 65 143 Β° Β° PRON american_scientific_journal-4481 65 144 sin sin NOUN american_scientific_journal-4481 65 145 100.40 100.40 NUM american_scientific_journal-4481 65 146 Β° Β° NOUN american_scientific_journal-4481 65 147 sin sin NOUN american_scientific_journal-4481 65 148 85.8931980 85.8931980 NUM american_scientific_journal-4481 65 149 Β° Β° NOUN american_scientific_journal-4481 66 1 = = SYM american_scientific_journal-4481 66 2 0.9590979 0.9590979 NUM american_scientific_journal-4481 66 3 ∴ ∴ PROPN american_scientific_journal-4481 66 4 ∠𝐴𝐴 ∠𝐴𝐴 PROPN american_scientific_journal-4481 66 5 = = SYM american_scientific_journal-4481 66 6 sinβˆ’1(0.9590979 sinβˆ’1(0.9590979 PROPN american_scientific_journal-4481 66 7 ) ) PUNCT american_scientific_journal-4481 66 8 = = PUNCT american_scientific_journal-4481 67 1 73.56 73.56 NUM american_scientific_journal-4481 67 2 Β° Β° PRON american_scientific_journal-4481 67 3 again again ADV american_scientific_journal-4481 67 4 , , PUNCT american_scientific_journal-4481 67 5 sin𝐡𝐡 sin𝐡𝐡 NOUN american_scientific_journal-4481 67 6 = = SYM american_scientific_journal-4481 67 7 cos cos PROPN american_scientific_journal-4481 67 8 x1 x1 PROPN american_scientific_journal-4481 67 9 Β° Β° PROPN american_scientific_journal-4481 67 10 sin(y2 sin(y2 NOUN american_scientific_journal-4481 67 11 Β° Β° PROPN american_scientific_journal-4481 67 12 βˆ’y1 βˆ’y1 NOUN american_scientific_journal-4481 67 13 Β° Β° NOUN american_scientific_journal-4481 67 14 ) ) PUNCT american_scientific_journal-4481 67 15 sin sin NOUN american_scientific_journal-4481 67 16 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4481 67 17 = = PUNCT american_scientific_journal-4481 67 18 cos cos PROPN american_scientific_journal-4481 67 19 51.30 51.30 NUM american_scientific_journal-4481 67 20 Β° Β° PROPN american_scientific_journal-4481 67 21 sin sin NOUN american_scientific_journal-4481 67 22 100.40 100.40 NUM american_scientific_journal-4481 67 23 Β° Β° ADP american_scientific_journal-4481 67 24 sin85.8931980 sin85.8931980 NOUN american_scientific_journal-4481 67 25 Β° Β° PUNCT american_scientific_journal-4481 68 1 = = PUNCT american_scientific_journal-4481 68 2 0.6165539 0.6165539 NUM american_scientific_journal-4481 68 3 american american ADJ american_scientific_journal-4481 68 4 scientific scientific ADJ american_scientific_journal-4481 68 5 research research NOUN american_scientific_journal-4481 68 6 journal journal NOUN american_scientific_journal-4481 68 7 for for ADP american_scientific_journal-4481 68 8 engineering engineering NOUN american_scientific_journal-4481 68 9 , , PUNCT american_scientific_journal-4481 68 10 technology technology NOUN american_scientific_journal-4481 68 11 , , PUNCT american_scientific_journal-4481 68 12 and and CCONJ american_scientific_journal-4481 68 13 sciences science NOUN american_scientific_journal-4481 68 14 ( ( PUNCT american_scientific_journal-4481 68 15 asrjets asrjet NOUN american_scientific_journal-4481 68 16 ) ) PUNCT american_scientific_journal-4481 68 17 ( ( PUNCT american_scientific_journal-4481 68 18 2018 2018 NUM american_scientific_journal-4481 68 19 ) ) PUNCT american_scientific_journal-4481 68 20 volume volume NOUN american_scientific_journal-4481 68 21 49 49 NUM american_scientific_journal-4481 68 22 , , PUNCT american_scientific_journal-4481 68 23 no no DET american_scientific_journal-4481 68 24 1 1 NUM american_scientific_journal-4481 68 25 , , PUNCT american_scientific_journal-4481 68 26 pp pp ADV american_scientific_journal-4481 68 27 51 51 NUM american_scientific_journal-4481 68 28 - - SYM american_scientific_journal-4481 68 29 60 60 NUM american_scientific_journal-4481 68 30 58 58 NUM american_scientific_journal-4481 68 31 ∴ ∴ NOUN american_scientific_journal-4481 68 32 ∠𝐡𝐡 ∠𝐡𝐡 NOUN american_scientific_journal-4481 68 33 = = SYM american_scientific_journal-4481 68 34 sinβˆ’1(0.6165539 sinβˆ’1(0.6165539 PROPN american_scientific_journal-4481 68 35 ) ) PUNCT american_scientific_journal-4481 68 36 = = SYM american_scientific_journal-4481 69 1 38.07 38.07 NUM american_scientific_journal-4481 69 2 Β° Β° ADP american_scientific_journal-4481 69 3 the the DET american_scientific_journal-4481 69 4 direction direction NOUN american_scientific_journal-4481 69 5 of of ADP american_scientific_journal-4481 69 6 bangkok bangkok PROPN american_scientific_journal-4481 69 7 with with ADP american_scientific_journal-4481 69 8 respect respect NOUN american_scientific_journal-4481 69 9 to to ADP american_scientific_journal-4481 69 10 london london PROPN american_scientific_journal-4481 69 11 towards towards ADP american_scientific_journal-4481 69 12 graphical graphical ADJ american_scientific_journal-4481 69 13 north north NOUN american_scientific_journal-4481 69 14 pole pole NOUN american_scientific_journal-4481 69 15 is be AUX american_scientific_journal-4481 69 16 ∠𝐴𝐴 ∠𝐴𝐴 PROPN american_scientific_journal-4481 69 17 = = PROPN american_scientific_journal-4481 69 18 73.56 73.56 NUM american_scientific_journal-4481 69 19 Β° Β° ADP american_scientific_journal-4481 69 20 the the DET american_scientific_journal-4481 69 21 direction direction NOUN american_scientific_journal-4481 69 22 of of ADP american_scientific_journal-4481 69 23 london london PROPN american_scientific_journal-4481 69 24 with with ADP american_scientific_journal-4481 69 25 respect respect NOUN american_scientific_journal-4481 69 26 to to ADP american_scientific_journal-4481 69 27 bangkok bangkok PROPN american_scientific_journal-4481 69 28 towards towards ADP american_scientific_journal-4481 69 29 graphical graphical ADJ american_scientific_journal-4481 69 30 north north NOUN american_scientific_journal-4481 69 31 pole pole NOUN american_scientific_journal-4481 69 32 is be AUX american_scientific_journal-4481 69 33 ∠𝐡𝐡 ∠𝐡𝐡 NOUN american_scientific_journal-4481 69 34 = = PUNCT american_scientific_journal-4481 69 35 38.07 38.07 NUM american_scientific_journal-4481 69 36 Β° Β° NOUN american_scientific_journal-4481 69 37 4.2 4.2 NUM american_scientific_journal-4481 69 38 . . PUNCT american_scientific_journal-4481 69 39 example example NOUN american_scientific_journal-4481 69 40 to to PART american_scientific_journal-4481 69 41 find find VERB american_scientific_journal-4481 69 42 the the DET american_scientific_journal-4481 69 43 shortest short ADJ american_scientific_journal-4481 69 44 distance distance NOUN american_scientific_journal-4481 69 45 and and CCONJ american_scientific_journal-4481 69 46 direction direction NOUN american_scientific_journal-4481 69 47 for for ADP american_scientific_journal-4481 69 48 dhaka dhaka PROPN american_scientific_journal-4481 69 49 and and CCONJ american_scientific_journal-4481 69 50 munshiganj munshiganj ADJ american_scientific_journal-4481 69 51 towards towards ADP american_scientific_journal-4481 69 52 north north NOUN american_scientific_journal-4481 69 53 pole pole NOUN american_scientific_journal-4481 69 54 . . PUNCT american_scientific_journal-4481 70 1 here here ADV american_scientific_journal-4481 70 2 , , PUNCT american_scientific_journal-4481 70 3 𝐴𝐴 𝐴𝐴 PROPN american_scientific_journal-4481 70 4 ≑ ≑ PROPN american_scientific_journal-4481 70 5 π·π·β„Žπ‘Žπ‘Žπ΅π΅π‘Žπ‘Ž π·π·β„Žπ‘Žπ‘Žπ΅π΅π‘Žπ‘Ž PROPN american_scientific_journal-4481 70 6 ; ; PUNCT american_scientific_journal-4481 70 7 π‘₯π‘₯1 π‘₯π‘₯1 NOUN american_scientific_journal-4481 70 8 = = SYM american_scientific_journal-4481 70 9 23.42 23.42 NUM american_scientific_journal-4481 70 10 Β° Β° ADP american_scientific_journal-4481 70 11 𝑁𝑁 𝑁𝑁 PROPN american_scientific_journal-4481 70 12 𝑦𝑦1 𝑦𝑦1 NOUN american_scientific_journal-4481 70 13 = = SYM american_scientific_journal-4481 70 14 90.22 90.22 NUM american_scientific_journal-4481 70 15 Β° Β° ADP american_scientific_journal-4481 70 16 𝐷𝐷 𝐷𝐷 PROPN american_scientific_journal-4481 70 17 𝐡𝐡 𝐡𝐡 PROPN american_scientific_journal-4481 70 18 ≑ ≑ PROPN american_scientific_journal-4481 70 19 π‘€π‘€π‘€π‘€π‘Žπ‘Žπ‘˜π‘˜β„Žπ‘Ÿπ‘Ÿπ΅π΅π‘Žπ‘Žπ‘Žπ‘Žπ΅π΅ π‘€π‘€π‘€π‘€π‘Žπ‘Žπ‘˜π‘˜β„Žπ‘Ÿπ‘Ÿπ΅π΅π‘Žπ‘Žπ‘Žπ‘Žπ΅π΅ NOUN american_scientific_journal-4481 70 20 ; ; PUNCT american_scientific_journal-4481 70 21 π‘₯π‘₯2 π‘₯π‘₯2 ADV american_scientific_journal-4481 70 22 = = VERB american_scientific_journal-4481 71 1 23.32 23.32 NUM american_scientific_journal-4481 71 2 Β° Β° ADP american_scientific_journal-4481 71 3 𝑁𝑁 𝑁𝑁 PROPN american_scientific_journal-4481 71 4 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-4481 71 5 = = SYM american_scientific_journal-4481 71 6 90.32 90.32 NUM american_scientific_journal-4481 71 7 Β° Β° NUM american_scientific_journal-4481 71 8 𝐷𝐷 𝐷𝐷 PROPN american_scientific_journal-4481 71 9 n n NOUN american_scientific_journal-4481 71 10 = = SYM american_scientific_journal-4481 71 11 y2 y2 PROPN american_scientific_journal-4481 71 12 Β° Β° PROPN american_scientific_journal-4481 71 13 βˆ’ βˆ’ PROPN american_scientific_journal-4481 71 14 y1 y1 PROPN american_scientific_journal-4481 71 15 Β° Β° NOUN american_scientific_journal-4481 71 16 = = PUNCT american_scientific_journal-4481 72 1 90.32 90.32 NUM american_scientific_journal-4481 72 2 Β° Β° NUM american_scientific_journal-4481 72 3 βˆ’ βˆ’ PROPN american_scientific_journal-4481 72 4 90.22 90.22 NUM american_scientific_journal-4481 72 5 Β° Β° NOUN american_scientific_journal-4481 72 6 = = PUNCT american_scientific_journal-4481 73 1 0.10 0.10 NUM american_scientific_journal-4481 73 2 Β° Β° NOUN american_scientific_journal-4481 73 3 we we PRON american_scientific_journal-4481 73 4 know know VERB american_scientific_journal-4481 73 5 from from ADP american_scientific_journal-4481 73 6 spherical spherical ADJ american_scientific_journal-4481 73 7 trigonometry trigonometry NOUN american_scientific_journal-4481 73 8 , , PUNCT american_scientific_journal-4481 73 9 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 73 10 = = SYM american_scientific_journal-4481 73 11 cosβˆ’1 cosβˆ’1 PROPN american_scientific_journal-4481 73 12 { { PUNCT american_scientific_journal-4481 73 13 sin sin NOUN american_scientific_journal-4481 73 14 x2 x2 PROPN american_scientific_journal-4481 73 15 Β° Β° PROPN american_scientific_journal-4481 73 16 sin sin VERB american_scientific_journal-4481 73 17 x1 x1 PROPN american_scientific_journal-4481 73 18 Β° Β° NOUN american_scientific_journal-4481 74 1 + + CCONJ american_scientific_journal-4481 74 2 cos cos PROPN american_scientific_journal-4481 74 3 x2 x2 PROPN american_scientific_journal-4481 74 4 Β° Β° PROPN american_scientific_journal-4481 74 5 cos cos PROPN american_scientific_journal-4481 75 1 x1 x1 PROPN american_scientific_journal-4481 75 2 Β° Β° PROPN american_scientific_journal-4481 75 3 cos(y2 cos(y2 NOUN american_scientific_journal-4481 75 4 Β° Β° PROPN american_scientific_journal-4481 75 5 βˆ’ βˆ’ PROPN american_scientific_journal-4481 75 6 y1 y1 PROPN american_scientific_journal-4481 75 7 Β° Β° NUM american_scientific_journal-4481 75 8 ) ) PUNCT american_scientific_journal-4481 75 9 } } PUNCT american_scientific_journal-4481 75 10 = = SYM american_scientific_journal-4481 75 11 cosβˆ’1 cosβˆ’1 PROPN american_scientific_journal-4481 75 12 { { PUNCT american_scientific_journal-4481 75 13 sin sin NOUN american_scientific_journal-4481 75 14 23.42 23.42 NUM american_scientific_journal-4481 75 15 Β° Β° PRON american_scientific_journal-4481 75 16 sin sin NOUN american_scientific_journal-4481 75 17 23.32 23.32 NUM american_scientific_journal-4481 75 18 Β° Β° NOUN american_scientific_journal-4481 76 1 + + CCONJ american_scientific_journal-4481 76 2 cos cos PROPN american_scientific_journal-4481 76 3 23.42 23.42 NUM american_scientific_journal-4481 76 4 Β° Β° NOUN american_scientific_journal-4481 76 5 cos cos ADP american_scientific_journal-4481 76 6 23.32 23.32 NUM american_scientific_journal-4481 76 7 Β° Β° NUM american_scientific_journal-4481 76 8 cos(90.32 cos(90.32 PROPN american_scientific_journal-4481 76 9 Β° Β° PROPN american_scientific_journal-4481 76 10 βˆ’ βˆ’ PROPN american_scientific_journal-4481 76 11 90.22 90.22 NUM american_scientific_journal-4481 76 12 Β° Β° NUM american_scientific_journal-4481 76 13 ) ) PUNCT american_scientific_journal-4481 76 14 } } PUNCT american_scientific_journal-4481 76 15 = = SYM american_scientific_journal-4481 76 16 cosβˆ’1(0.157344 cosβˆ’1(0.157344 X american_scientific_journal-4481 76 17 βˆ’ βˆ’ NOUN american_scientific_journal-4481 76 18 0.842653 0.842653 NUM american_scientific_journal-4481 76 19 ) ) PUNCT american_scientific_journal-4481 76 20 = = SYM american_scientific_journal-4481 76 21 cosβˆ’1(0.999997 cosβˆ’1(0.999997 X american_scientific_journal-4481 76 22 ) ) PUNCT american_scientific_journal-4481 76 23 = = PUNCT american_scientific_journal-4481 77 1 0.1403345 0.1403345 NUM american_scientific_journal-4481 77 2 Β° Β° X american_scientific_journal-4481 78 1 = = NOUN american_scientific_journal-4481 78 2 0.1403345 0.1403345 NUM american_scientific_journal-4481 78 3 Γ— Γ— NOUN american_scientific_journal-4481 78 4 πœ‹πœ‹ πœ‹πœ‹ ADP american_scientific_journal-4481 78 5 180 180 NUM american_scientific_journal-4481 78 6 π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4481 78 7 = = SYM american_scientific_journal-4481 78 8 0.002449 0.002449 NUM american_scientific_journal-4481 78 9 π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž π‘œπ‘œπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ÿπ‘Ÿπ‘Žπ‘Žπ‘Žπ‘Ž NOUN american_scientific_journal-4481 78 10 calculation calculation NOUN american_scientific_journal-4481 78 11 for for ADP american_scientific_journal-4481 78 12 the the DET american_scientific_journal-4481 78 13 shortest short ADJ american_scientific_journal-4481 78 14 distance distance NOUN american_scientific_journal-4481 78 15 : : PUNCT american_scientific_journal-4481 78 16 american american ADJ american_scientific_journal-4481 78 17 scientific scientific ADJ american_scientific_journal-4481 78 18 research research PROPN american_scientific_journal-4481 78 19 journal journal NOUN american_scientific_journal-4481 78 20 for for ADP american_scientific_journal-4481 78 21 engineering engineering NOUN american_scientific_journal-4481 78 22 , , PUNCT american_scientific_journal-4481 78 23 technology technology NOUN american_scientific_journal-4481 78 24 , , PUNCT american_scientific_journal-4481 78 25 and and CCONJ american_scientific_journal-4481 78 26 sciences science NOUN american_scientific_journal-4481 78 27 ( ( PUNCT american_scientific_journal-4481 78 28 asrjets asrjet NOUN american_scientific_journal-4481 78 29 ) ) PUNCT american_scientific_journal-4481 78 30 ( ( PUNCT american_scientific_journal-4481 78 31 2018 2018 NUM american_scientific_journal-4481 78 32 ) ) PUNCT american_scientific_journal-4481 78 33 volume volume NOUN american_scientific_journal-4481 78 34 49 49 NUM american_scientific_journal-4481 78 35 , , PUNCT american_scientific_journal-4481 78 36 no no DET american_scientific_journal-4481 78 37 1 1 NUM american_scientific_journal-4481 78 38 , , PUNCT american_scientific_journal-4481 78 39 pp pp ADV american_scientific_journal-4481 78 40 51 51 NUM american_scientific_journal-4481 78 41 - - SYM american_scientific_journal-4481 78 42 60 60 NUM american_scientific_journal-4481 78 43 59 59 NUM american_scientific_journal-4481 78 44 radius radius NOUN american_scientific_journal-4481 78 45 of of ADP american_scientific_journal-4481 78 46 the the DET american_scientific_journal-4481 78 47 earth earth NOUN american_scientific_journal-4481 79 1 [ [ X american_scientific_journal-4481 79 2 3 3 NUM american_scientific_journal-4481 79 3 ] ] PUNCT american_scientific_journal-4481 79 4 , , PUNCT american_scientific_journal-4481 79 5 𝑅𝑅 𝑅𝑅 PROPN american_scientific_journal-4481 79 6 = = SYM american_scientific_journal-4481 79 7 6378.388 6378.388 NUM american_scientific_journal-4481 80 1 π΅π΅π‘˜π‘˜π‘˜π‘˜ π΅π΅π‘˜π‘˜π‘˜π‘˜ PROPN american_scientific_journal-4481 80 2 ∴ ∴ PROPN american_scientific_journal-4481 80 3 shortest short ADJ american_scientific_journal-4481 80 4 distance distance NOUN american_scientific_journal-4481 80 5 between between ADP american_scientific_journal-4481 80 6 dhaka dhaka PROPN american_scientific_journal-4481 80 7 and and CCONJ american_scientific_journal-4481 80 8 munshiganj munshiganj ADJ american_scientific_journal-4481 80 9 is be AUX american_scientific_journal-4481 80 10 , , PUNCT american_scientific_journal-4481 80 11 π‘Žπ‘Ž π‘Žπ‘Ž ADJ american_scientific_journal-4481 80 12 = = SYM american_scientific_journal-4481 80 13 𝑅𝑅 𝑅𝑅 ADJ american_scientific_journal-4481 80 14 β‹… β‹… NOUN american_scientific_journal-4481 80 15 π‘Žπ‘Ž π‘Žπ‘Ž X american_scientific_journal-4481 80 16 = = SYM american_scientific_journal-4481 80 17 6378.388 6378.388 NUM american_scientific_journal-4481 80 18 Γ— Γ— NOUN american_scientific_journal-4481 80 19 0.002449 0.002449 NUM american_scientific_journal-4481 80 20 = = SYM american_scientific_journal-4481 80 21 15.6237 15.6237 NUM american_scientific_journal-4481 80 22 π΅π΅π‘˜π‘˜π‘˜π‘˜ π΅π΅π‘˜π‘˜π‘˜π‘˜ PROPN american_scientific_journal-4481 80 23 so so ADV american_scientific_journal-4481 80 24 , , PUNCT american_scientific_journal-4481 80 25 the the DET american_scientific_journal-4481 80 26 distance distance NOUN american_scientific_journal-4481 80 27 between between ADP american_scientific_journal-4481 80 28 dhaka dhaka PROPN american_scientific_journal-4481 80 29 and and CCONJ american_scientific_journal-4481 80 30 munshiganj munshiganj ADJ american_scientific_journal-4481 80 31 is be AUX american_scientific_journal-4481 80 32 15.6237 15.6237 NUM american_scientific_journal-4481 80 33 π΅π΅π‘˜π‘˜π‘˜π‘˜. π΅π΅π‘˜π‘˜π‘˜π‘˜. ADJ american_scientific_journal-4481 80 34 calculation calculation NOUN american_scientific_journal-4481 80 35 for for ADP american_scientific_journal-4481 80 36 the the DET american_scientific_journal-4481 80 37 direction direction NOUN american_scientific_journal-4481 80 38 of of ADP american_scientific_journal-4481 80 39 dhaka dhaka PROPN american_scientific_journal-4481 80 40 and and CCONJ american_scientific_journal-4481 80 41 munshiganj munshiganj ADJ american_scientific_journal-4481 80 42 towards towards ADP american_scientific_journal-4481 80 43 north north NOUN american_scientific_journal-4481 80 44 pole pole NOUN american_scientific_journal-4481 80 45 : : PUNCT american_scientific_journal-4481 80 46 from from ADP american_scientific_journal-4481 80 47 the the DET american_scientific_journal-4481 80 48 sine sine ADJ american_scientific_journal-4481 80 49 rule rule NOUN american_scientific_journal-4481 80 50 on on ADP american_scientific_journal-4481 80 51 spherical spherical ADJ american_scientific_journal-4481 80 52 triangle triangle NOUN american_scientific_journal-4481 80 53 we we PRON american_scientific_journal-4481 80 54 have have VERB american_scientific_journal-4481 80 55 , , PUNCT american_scientific_journal-4481 80 56 sin𝐴𝐴 sin𝐴𝐴 PROPN american_scientific_journal-4481 80 57 = = PUNCT american_scientific_journal-4481 80 58 cos cos PROPN american_scientific_journal-4481 80 59 x2 x2 PROPN american_scientific_journal-4481 80 60 Β° Β° PROPN american_scientific_journal-4481 80 61 sin(y2 sin(y2 NOUN american_scientific_journal-4481 80 62 Β° Β° PROPN american_scientific_journal-4481 80 63 βˆ’ βˆ’ PROPN american_scientific_journal-4481 80 64 y1 y1 PROPN american_scientific_journal-4481 80 65 Β° Β° PROPN american_scientific_journal-4481 80 66 ) ) PUNCT american_scientific_journal-4481 80 67 sin sin NOUN american_scientific_journal-4481 80 68 π‘Žπ‘Ž π‘Žπ‘Ž NOUN american_scientific_journal-4481 80 69 = = SYM american_scientific_journal-4481 80 70 cos cos ADP american_scientific_journal-4481 80 71 23.32 23.32 NUM american_scientific_journal-4481 80 72 Β° Β° NOUN american_scientific_journal-4481 80 73 sin(0.10 sin(0.10 NOUN american_scientific_journal-4481 80 74 Β° Β° NUM american_scientific_journal-4481 80 75 ) ) PUNCT american_scientific_journal-4481 80 76 sin sin NOUN american_scientific_journal-4481 80 77 0.140345 0.140345 NUM american_scientific_journal-4481 80 78 Β° Β° NOUN american_scientific_journal-4481 80 79 = = SYM american_scientific_journal-4481 80 80 0.654553 0.654553 NUM american_scientific_journal-4481 80 81 ∴ ∴ PROPN american_scientific_journal-4481 80 82 ∠𝐴𝐴 ∠𝐴𝐴 PROPN american_scientific_journal-4481 80 83 = = SYM american_scientific_journal-4481 80 84 sinβˆ’1(0.654553 sinβˆ’1(0.654553 PROPN american_scientific_journal-4481 80 85 ) ) PUNCT american_scientific_journal-4481 80 86 = = PUNCT american_scientific_journal-4481 81 1 40.89 40.89 NUM american_scientific_journal-4481 81 2 Β° Β° PRON american_scientific_journal-4481 81 3 again again ADV american_scientific_journal-4481 81 4 , , PUNCT american_scientific_journal-4481 81 5 sin𝐡𝐡 sin𝐡𝐡 NOUN american_scientific_journal-4481 81 6 = = SYM american_scientific_journal-4481 81 7 cos cos PROPN american_scientific_journal-4481 81 8 x1 x1 PROPN american_scientific_journal-4481 81 9 Β° Β° PROPN american_scientific_journal-4481 81 10 sin(y2 sin(y2 NOUN american_scientific_journal-4481 81 11 Β° Β° PROPN american_scientific_journal-4481 81 12 βˆ’y1 βˆ’y1 NOUN american_scientific_journal-4481 81 13 Β° Β° NOUN american_scientific_journal-4481 81 14 ) ) PUNCT american_scientific_journal-4481 81 15 sin sin NOUN american_scientific_journal-4481 81 16 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-4481 81 17 = = PUNCT american_scientific_journal-4481 81 18 cos cos ADP american_scientific_journal-4481 81 19 23.42 23.42 NUM american_scientific_journal-4481 81 20 Β° Β° NOUN american_scientific_journal-4481 81 21 sin(0.10 sin(0.10 NOUN american_scientific_journal-4481 81 22 Β° Β° NUM american_scientific_journal-4481 81 23 ) ) PUNCT american_scientific_journal-4481 81 24 sin sin NOUN american_scientific_journal-4481 81 25 0.140345 0.140345 NUM american_scientific_journal-4481 81 26 Β° Β° NOUN american_scientific_journal-4481 81 27 = = PUNCT american_scientific_journal-4481 81 28 0.653829 0.653829 NUM american_scientific_journal-4481 81 29 ∴ ∴ PROPN american_scientific_journal-4481 81 30 ∠𝐡𝐡 ∠𝐡𝐡 X american_scientific_journal-4481 81 31 = = SYM american_scientific_journal-4481 81 32 sinβˆ’1(0.653829 sinβˆ’1(0.653829 X american_scientific_journal-4481 81 33 ) ) PUNCT american_scientific_journal-4481 81 34 = = PUNCT american_scientific_journal-4481 82 1 40.83 40.83 NUM american_scientific_journal-4481 82 2 Β° Β° ADP american_scientific_journal-4481 82 3 the the DET american_scientific_journal-4481 82 4 direction direction NOUN american_scientific_journal-4481 82 5 of of ADP american_scientific_journal-4481 82 6 munshiganj munshiganj ADJ american_scientific_journal-4481 82 7 with with ADP american_scientific_journal-4481 82 8 respect respect NOUN american_scientific_journal-4481 82 9 to to ADP american_scientific_journal-4481 82 10 dhaka dhaka PROPN american_scientific_journal-4481 82 11 towards towards ADP american_scientific_journal-4481 82 12 graphical graphical ADJ american_scientific_journal-4481 82 13 north north NOUN american_scientific_journal-4481 82 14 pole pole NOUN american_scientific_journal-4481 82 15 is be AUX american_scientific_journal-4481 82 16 ∠𝐴𝐴 ∠𝐴𝐴 PROPN american_scientific_journal-4481 82 17 = = PROPN american_scientific_journal-4481 82 18 40.89 40.89 NUM american_scientific_journal-4481 82 19 Β° Β° ADP american_scientific_journal-4481 82 20 the the DET american_scientific_journal-4481 82 21 direction direction NOUN american_scientific_journal-4481 82 22 of of ADP american_scientific_journal-4481 82 23 dhaka dhaka PROPN american_scientific_journal-4481 82 24 with with ADP american_scientific_journal-4481 82 25 respect respect NOUN american_scientific_journal-4481 82 26 to to ADP american_scientific_journal-4481 82 27 munshiganj munshiganj ADJ american_scientific_journal-4481 82 28 towards towards ADP american_scientific_journal-4481 82 29 graphical graphical ADJ american_scientific_journal-4481 82 30 north north NOUN american_scientific_journal-4481 82 31 pole pole NOUN american_scientific_journal-4481 82 32 is be AUX american_scientific_journal-4481 82 33 ∠𝐡𝐡 ∠𝐡𝐡 NOUN american_scientific_journal-4481 82 34 = = PUNCT american_scientific_journal-4481 82 35 40.83 40.83 NUM american_scientific_journal-4481 82 36 Β° Β° ADP american_scientific_journal-4481 82 37 american american PROPN american_scientific_journal-4481 82 38 scientific scientific ADJ american_scientific_journal-4481 82 39 research research NOUN american_scientific_journal-4481 82 40 journal journal NOUN american_scientific_journal-4481 82 41 for for ADP american_scientific_journal-4481 82 42 engineering engineering NOUN american_scientific_journal-4481 82 43 , , PUNCT american_scientific_journal-4481 82 44 technology technology NOUN american_scientific_journal-4481 82 45 , , PUNCT american_scientific_journal-4481 82 46 and and CCONJ american_scientific_journal-4481 82 47 sciences science NOUN american_scientific_journal-4481 82 48 ( ( PUNCT american_scientific_journal-4481 82 49 asrjets asrjet NOUN american_scientific_journal-4481 82 50 ) ) PUNCT american_scientific_journal-4481 82 51 ( ( PUNCT american_scientific_journal-4481 82 52 2018 2018 NUM american_scientific_journal-4481 82 53 ) ) PUNCT american_scientific_journal-4481 82 54 volume volume NOUN american_scientific_journal-4481 82 55 49 49 NUM american_scientific_journal-4481 82 56 , , PUNCT american_scientific_journal-4481 82 57 no no DET american_scientific_journal-4481 82 58 1 1 NUM american_scientific_journal-4481 82 59 , , PUNCT american_scientific_journal-4481 82 60 pp pp ADV american_scientific_journal-4481 82 61 51 51 NUM american_scientific_journal-4481 82 62 - - SYM american_scientific_journal-4481 82 63 60 60 NUM american_scientific_journal-4481 82 64 60 60 NUM american_scientific_journal-4481 82 65 5 5 NUM american_scientific_journal-4481 82 66 . . PUNCT american_scientific_journal-4481 83 1 conclusion conclusion NOUN american_scientific_journal-4481 83 2 differential differential NOUN american_scientific_journal-4481 83 3 geometry geometry NOUN american_scientific_journal-4481 83 4 is be AUX american_scientific_journal-4481 83 5 related relate VERB american_scientific_journal-4481 83 6 to to ADP american_scientific_journal-4481 83 7 astronomy astronomy NOUN american_scientific_journal-4481 83 8 and and CCONJ american_scientific_journal-4481 83 9 geographical geographical ADJ american_scientific_journal-4481 83 10 problems problem NOUN american_scientific_journal-4481 83 11 . . PUNCT american_scientific_journal-4481 84 1 some some DET american_scientific_journal-4481 84 2 practical practical ADJ american_scientific_journal-4481 84 3 examples example NOUN american_scientific_journal-4481 84 4 are be AUX american_scientific_journal-4481 84 5 discussed discuss VERB american_scientific_journal-4481 84 6 in in ADP american_scientific_journal-4481 84 7 this this DET american_scientific_journal-4481 84 8 paper paper NOUN american_scientific_journal-4481 84 9 . . PUNCT american_scientific_journal-4481 85 1 the the DET american_scientific_journal-4481 85 2 aim aim NOUN american_scientific_journal-4481 85 3 of of ADP american_scientific_journal-4481 85 4 this this DET american_scientific_journal-4481 85 5 paper paper NOUN american_scientific_journal-4481 85 6 is be AUX american_scientific_journal-4481 85 7 to to PART american_scientific_journal-4481 85 8 find find VERB american_scientific_journal-4481 85 9 a a DET american_scientific_journal-4481 85 10 mathematical mathematical ADJ american_scientific_journal-4481 85 11 method method NOUN american_scientific_journal-4481 85 12 of of ADP american_scientific_journal-4481 85 13 finding find VERB american_scientific_journal-4481 85 14 distance distance NOUN american_scientific_journal-4481 85 15 and and CCONJ american_scientific_journal-4481 85 16 direction direction NOUN american_scientific_journal-4481 85 17 between between ADP american_scientific_journal-4481 85 18 two two NUM american_scientific_journal-4481 85 19 places place NOUN american_scientific_journal-4481 85 20 on on ADP american_scientific_journal-4481 85 21 the the DET american_scientific_journal-4481 85 22 surfaces surface NOUN american_scientific_journal-4481 85 23 of of ADP american_scientific_journal-4481 85 24 the the DET american_scientific_journal-4481 85 25 earth earth NOUN american_scientific_journal-4481 85 26 with with ADP american_scientific_journal-4481 85 27 respect respect NOUN american_scientific_journal-4481 85 28 to to ADP american_scientific_journal-4481 85 29 the the DET american_scientific_journal-4481 85 30 geographical geographical ADJ american_scientific_journal-4481 85 31 north north NOUN american_scientific_journal-4481 85 32 pole pole NOUN american_scientific_journal-4481 85 33 or or CCONJ american_scientific_journal-4481 85 34 south south ADJ american_scientific_journal-4481 85 35 pole pole NOUN american_scientific_journal-4481 85 36 . . PUNCT american_scientific_journal-4481 86 1 since since SCONJ american_scientific_journal-4481 86 2 the the DET american_scientific_journal-4481 86 3 radius radius NOUN american_scientific_journal-4481 86 4 of of ADP american_scientific_journal-4481 86 5 the the DET american_scientific_journal-4481 86 6 earth earth NOUN american_scientific_journal-4481 86 7 is be AUX american_scientific_journal-4481 86 8 not not PART american_scientific_journal-4481 86 9 perfect perfect ADJ american_scientific_journal-4481 86 10 , , PUNCT american_scientific_journal-4481 86 11 so so SCONJ american_scientific_journal-4481 86 12 the the DET american_scientific_journal-4481 86 13 distance distance NOUN american_scientific_journal-4481 86 14 and and CCONJ american_scientific_journal-4481 86 15 directions direction NOUN american_scientific_journal-4481 86 16 found find VERB american_scientific_journal-4481 86 17 here here ADV american_scientific_journal-4481 86 18 are be AUX american_scientific_journal-4481 86 19 approximate approximate ADJ american_scientific_journal-4481 86 20 . . PUNCT american_scientific_journal-4481 87 1 references reference NOUN american_scientific_journal-4481 87 2 [ [ X american_scientific_journal-4481 87 3 1 1 NUM american_scientific_journal-4481 87 4 ] ] PUNCT american_scientific_journal-4481 87 5 . . PUNCT american_scientific_journal-4481 88 1 bartholomew bartholomew PROPN american_scientific_journal-4481 88 2 , , PUNCT american_scientific_journal-4481 88 3 j. j. PROPN american_scientific_journal-4481 88 4 times times PROPN american_scientific_journal-4481 88 5 atlas atlas PROPN american_scientific_journal-4481 88 6 of of ADP american_scientific_journal-4481 88 7 the the DET american_scientific_journal-4481 88 8 world world NOUN american_scientific_journal-4481 88 9 vol vol PROPN american_scientific_journal-4481 88 10 - - PUNCT american_scientific_journal-4481 88 11 ii ii NOUN american_scientific_journal-4481 88 12 . . PUNCT american_scientific_journal-4481 89 1 london london PROPN american_scientific_journal-4481 89 2 : : PUNCT american_scientific_journal-4481 89 3 time time NOUN american_scientific_journal-4481 89 4 publishing publishing PROPN american_scientific_journal-4481 89 5 company company PROPN american_scientific_journal-4481 89 6 ltd ltd PROPN american_scientific_journal-4481 89 7 , , PUNCT american_scientific_journal-4481 89 8 1959 1959 NUM american_scientific_journal-4481 89 9 . . PUNCT american_scientific_journal-4481 90 1 [ [ X american_scientific_journal-4481 90 2 2 2 NUM american_scientific_journal-4481 90 3 ] ] PUNCT american_scientific_journal-4481 90 4 . . PUNCT american_scientific_journal-4481 91 1 dr dr PROPN american_scientific_journal-4481 91 2 . . PROPN american_scientific_journal-4481 91 3 md md PROPN american_scientific_journal-4481 91 4 . . PUNCT american_scientific_journal-4481 92 1 showkat showkat PROPN american_scientific_journal-4481 92 2 ali ali PROPN american_scientific_journal-4481 92 3 . . PROPN american_scientific_journal-4481 93 1 tensor tensor NOUN american_scientific_journal-4481 93 2 analysis analysis NOUN american_scientific_journal-4481 93 3 with with ADP american_scientific_journal-4481 93 4 applications application NOUN american_scientific_journal-4481 93 5 . . PUNCT american_scientific_journal-4481 94 1 bangladesh bangladesh NOUN american_scientific_journal-4481 94 2 : : PUNCT american_scientific_journal-4481 94 3 educational educational ADJ american_scientific_journal-4481 94 4 publishers publisher NOUN american_scientific_journal-4481 94 5 , , PUNCT american_scientific_journal-4481 94 6 2006 2006 NUM american_scientific_journal-4481 94 7 . . PUNCT american_scientific_journal-4481 95 1 [ [ X american_scientific_journal-4481 95 2 3 3 NUM american_scientific_journal-4481 95 3 ] ] PUNCT american_scientific_journal-4481 95 4 . . PUNCT american_scientific_journal-4481 96 1 duncan duncan PROPN american_scientific_journal-4481 96 2 , , PUNCT american_scientific_journal-4481 96 3 j.c j.c PROPN american_scientific_journal-4481 96 4 . . PROPN american_scientific_journal-4481 96 5 astronomy astronomy PROPN american_scientific_journal-4481 96 6 . . PUNCT american_scientific_journal-4481 97 1 newyork newyork NOUN american_scientific_journal-4481 97 2 : : PUNCT american_scientific_journal-4481 97 3 hamper hamper NOUN american_scientific_journal-4481 97 4 and and CCONJ american_scientific_journal-4481 97 5 brothers brother NOUN american_scientific_journal-4481 97 6 , , PUNCT american_scientific_journal-4481 97 7 1946 1946 NUM american_scientific_journal-4481 97 8 . . PUNCT american_scientific_journal-4481 98 1 [ [ X american_scientific_journal-4481 98 2 4 4 NUM american_scientific_journal-4481 98 3 ] ] PUNCT american_scientific_journal-4481 98 4 . . PUNCT american_scientific_journal-4481 99 1 harry harry PROPN american_scientific_journal-4481 99 2 lass lass PROPN american_scientific_journal-4481 99 3 . . PUNCT american_scientific_journal-4481 99 4 vector vector NOUN american_scientific_journal-4481 99 5 and and CCONJ american_scientific_journal-4481 99 6 tensor tensor NOUN american_scientific_journal-4481 99 7 analysis analysis NOUN american_scientific_journal-4481 99 8 . . PUNCT american_scientific_journal-4481 100 1 mc mc PROPN american_scientific_journal-4481 100 2 graw graw PROPN american_scientific_journal-4481 100 3 - - PUNCT american_scientific_journal-4481 100 4 hill hill NOUN american_scientific_journal-4481 100 5 book book NOUN american_scientific_journal-4481 100 6 company company NOUN american_scientific_journal-4481 100 7 , , PUNCT american_scientific_journal-4481 100 8 1950 1950 NUM american_scientific_journal-4481 100 9 . . PUNCT american_scientific_journal-4481 101 1 [ [ X american_scientific_journal-4481 101 2 5 5 NUM american_scientific_journal-4481 101 3 ] ] PUNCT american_scientific_journal-4481 101 4 . . PUNCT american_scientific_journal-4481 101 5 lipschutz lipschutz PROPN american_scientific_journal-4481 101 6 , , PUNCT american_scientific_journal-4481 101 7 m.m m.m PROPN american_scientific_journal-4481 101 8 . . PROPN american_scientific_journal-4481 101 9 differential differential PROPN american_scientific_journal-4481 101 10 geometry geometry NOUN american_scientific_journal-4481 101 11 . . PUNCT american_scientific_journal-4481 102 1 newyork newyork PROPN american_scientific_journal-4481 102 2 , , PUNCT american_scientific_journal-4481 102 3 toronto toronto PROPN american_scientific_journal-4481 102 4 , , PUNCT american_scientific_journal-4481 102 5 sydney sydney PROPN american_scientific_journal-4481 102 6 : : PUNCT american_scientific_journal-4481 102 7 mc mc PROPN american_scientific_journal-4481 102 8 graw graw PROPN american_scientific_journal-4481 102 9 - - PUNCT american_scientific_journal-4481 102 10 hill hill NOUN american_scientific_journal-4481 102 11 international international ADJ american_scientific_journal-4481 102 12 editions edition NOUN american_scientific_journal-4481 102 13 , , PUNCT american_scientific_journal-4481 102 14 1969 1969 NUM american_scientific_journal-4481 102 15 . . PUNCT american_scientific_journal-4481 103 1 [ [ X american_scientific_journal-4481 103 2 6 6 NUM american_scientific_journal-4481 103 3 ] ] PUNCT american_scientific_journal-4481 103 4 . . PUNCT american_scientific_journal-4481 104 1 mital mital ADJ american_scientific_journal-4481 104 2 , , PUNCT american_scientific_journal-4481 104 3 s.c s.c PROPN american_scientific_journal-4481 104 4 . . PROPN american_scientific_journal-4481 104 5 and and CCONJ american_scientific_journal-4481 104 6 agarwal agarwal PROPN american_scientific_journal-4481 104 7 , , PUNCT american_scientific_journal-4481 104 8 d.c d.c PROPN american_scientific_journal-4481 104 9 . . PROPN american_scientific_journal-4481 104 10 differential differential PROPN american_scientific_journal-4481 104 11 geometry geometry NOUN american_scientific_journal-4481 104 12 . . PUNCT american_scientific_journal-4481 105 1 india india PROPN american_scientific_journal-4481 105 2 : : PUNCT american_scientific_journal-4481 105 3 krishna krishna PROPN american_scientific_journal-4481 105 4 prakashan prakashan NOUN american_scientific_journal-4481 105 5 mandir mandir NOUN american_scientific_journal-4481 105 6 , , PUNCT american_scientific_journal-4481 105 7 1994 1994 NUM american_scientific_journal-4481 105 8 . . PUNCT american_scientific_journal-4481 106 1 [ [ X american_scientific_journal-4481 106 2 7 7 NUM american_scientific_journal-4481 106 3 ] ] PUNCT american_scientific_journal-4481 106 4 . . PUNCT american_scientific_journal-4481 107 1 m.l m.l PROPN american_scientific_journal-4481 107 2 . . PROPN american_scientific_journal-4481 107 3 khanna khanna PROPN american_scientific_journal-4481 107 4 , , PUNCT american_scientific_journal-4481 107 5 differential differential NOUN american_scientific_journal-4481 107 6 geometry geometry NOUN american_scientific_journal-4481 107 7 . . PUNCT american_scientific_journal-4481 108 1 india india PROPN american_scientific_journal-4481 108 2 : : PUNCT american_scientific_journal-4481 108 3 jal jal PROPN american_scientific_journal-4481 108 4 prakash prakash PROPN american_scientific_journal-4481 108 5 nath nath PROPN american_scientific_journal-4481 108 6 and and CCONJ american_scientific_journal-4481 108 7 co co NOUN american_scientific_journal-4481 108 8 mandir mandir PROPN american_scientific_journal-4481 108 9 , , PUNCT american_scientific_journal-4481 108 10 1976 1976 NUM american_scientific_journal-4481 108 11 . . PUNCT american_scientific_journal-4481 109 1 [ [ X american_scientific_journal-4481 109 2 8 8 NUM american_scientific_journal-4481 109 3 ] ] PUNCT american_scientific_journal-4481 109 4 . . PUNCT american_scientific_journal-4481 110 1 pandey pandey PROPN american_scientific_journal-4481 110 2 , , PUNCT american_scientific_journal-4481 110 3 h.d h.d PROPN american_scientific_journal-4481 110 4 . . PROPN american_scientific_journal-4481 110 5 , , PUNCT american_scientific_journal-4481 110 6 malik malik PROPN american_scientific_journal-4481 110 7 , , PUNCT american_scientific_journal-4481 110 8 g.s g.s PROPN american_scientific_journal-4481 110 9 . . PROPN american_scientific_journal-4481 110 10 and and CCONJ american_scientific_journal-4481 110 11 gupta gupta PROPN american_scientific_journal-4481 110 12 p.p p.p PROPN american_scientific_journal-4481 110 13 . . PROPN american_scientific_journal-4481 110 14 tensors tensor NOUN american_scientific_journal-4481 110 15 and and CCONJ american_scientific_journal-4481 110 16 differential differential ADJ american_scientific_journal-4481 110 17 geometry geometry NOUN american_scientific_journal-4481 110 18 . . PUNCT american_scientific_journal-4481 111 1 pragati pragati PROPN american_scientific_journal-4481 111 2 prakashan prakashan VERB american_scientific_journal-4481 111 3 , , PUNCT american_scientific_journal-4481 111 4 2006 2006 NUM american_scientific_journal-4481 111 5 . . PUNCT american_scientific_journal-4481 112 1 [ [ X american_scientific_journal-4481 112 2 9 9 NUM american_scientific_journal-4481 112 3 ] ] PUNCT american_scientific_journal-4481 112 4 . . PUNCT american_scientific_journal-4481 113 1 struik struik PROPN american_scientific_journal-4481 113 2 , , PUNCT american_scientific_journal-4481 113 3 d.j d.j PROPN american_scientific_journal-4481 113 4 . . PROPN american_scientific_journal-4481 113 5 lectures lecture NOUN american_scientific_journal-4481 113 6 on on ADP american_scientific_journal-4481 113 7 classical classical ADJ american_scientific_journal-4481 113 8 differential differential NOUN american_scientific_journal-4481 113 9 geometry geometry NOUN american_scientific_journal-4481 113 10 . . PUNCT american_scientific_journal-4481 114 1 usa usa PROPN american_scientific_journal-4481 114 2 : : PUNCT american_scientific_journal-4481 114 3 wisley wisley PROPN american_scientific_journal-4481 114 4 publishing publishing PROPN american_scientific_journal-4481 114 5 company company PROPN american_scientific_journal-4481 114 6 , , PUNCT american_scientific_journal-4481 114 7 inc inc PROPN american_scientific_journal-4481 114 8 , , PUNCT american_scientific_journal-4481 114 9 , , PUNCT american_scientific_journal-4481 114 10 1961 1961 NUM american_scientific_journal-4481 114 11 . . PUNCT american_scientific_journal-4481 115 1 [ [ X american_scientific_journal-4481 115 2 10 10 NUM american_scientific_journal-4481 115 3 ] ] PUNCT american_scientific_journal-4481 115 4 . . PUNCT american_scientific_journal-4481 115 5 weatherburn weatherburn NOUN american_scientific_journal-4481 115 6 , , PUNCT american_scientific_journal-4481 115 7 c.e c.e PROPN american_scientific_journal-4481 115 8 . . PROPN american_scientific_journal-4481 115 9 differential differential PROPN american_scientific_journal-4481 115 10 geometry geometry NOUN american_scientific_journal-4481 115 11 of of ADP american_scientific_journal-4481 115 12 three three NUM american_scientific_journal-4481 115 13 dimention dimention NOUN american_scientific_journal-4481 115 14 , , PUNCT american_scientific_journal-4481 115 15 vol-1 vol-1 NUM american_scientific_journal-4481 115 16 . . PUNCT american_scientific_journal-4481 115 17 london london PROPN american_scientific_journal-4481 115 18 , , PUNCT american_scientific_journal-4481 115 19 newyork newyork PROPN american_scientific_journal-4481 115 20 : : PUNCT american_scientific_journal-4481 115 21 cambridge cambridge PROPN american_scientific_journal-4481 115 22 university university PROPN american_scientific_journal-4481 115 23 press press NOUN american_scientific_journal-4481 115 24 , , PUNCT american_scientific_journal-4481 115 25 1939 1939 NUM american_scientific_journal-4481 115 26 . . PUNCT american_scientific_journal-4481 116 1 [ [ X american_scientific_journal-4481 116 2 8 8 NUM american_scientific_journal-4481 116 3 ] ] PUNCT american_scientific_journal-4481 116 4 . . PUNCT american_scientific_journal-4481 117 1 pandey pandey PROPN american_scientific_journal-4481 117 2 , , PUNCT american_scientific_journal-4481 117 3 h.d h.d PROPN american_scientific_journal-4481 117 4 . . PROPN american_scientific_journal-4481 117 5 , , PUNCT american_scientific_journal-4481 117 6 malik malik PROPN american_scientific_journal-4481 117 7 , , PUNCT american_scientific_journal-4481 117 8 g.s g.s PROPN american_scientific_journal-4481 117 9 . . PROPN american_scientific_journal-4481 117 10 and and CCONJ american_scientific_journal-4481 117 11 gupta gupta PROPN american_scientific_journal-4481 117 12 p.p p.p PROPN american_scientific_journal-4481 117 13 . . PROPN american_scientific_journal-4481 117 14 tensors tensor NOUN american_scientific_journal-4481 117 15 and and CCONJ american_scientific_journal-4481 117 16 differential differential ADJ american_scientific_journal-4481 117 17 geometry geometry NOUN american_scientific_journal-4481 117 18 . . PUNCT american_scientific_journal-4481 118 1 pragati pragati PROPN american_scientific_journal-4481 118 2 prakashan prakashan VERB american_scientific_journal-4481 118 3 , , PUNCT american_scientific_journal-4481 118 4 2006 2006 NUM american_scientific_journal-4481 118 5 . . PUNCT