id	sid	tid	token	lemma	pos
ajrhss-2207	1	1	american	american	PROPN
ajrhss-2207	1	2	journal	journal	PROPN
ajrhss-2207	1	3	of	of	ADP
ajrhss-2207	1	4	research	research	NOUN
ajrhss-2207	1	5	in	in	ADP
ajrhss-2207	1	6	humanities	humanity	NOUN
ajrhss-2207	1	7	and	and	CCONJ
ajrhss-2207	1	8	social	social	ADJ
ajrhss-2207	1	9	sciences	science	NOUN
ajrhss-2207	1	10	issn	issn	PROPN
ajrhss-2207	1	11	(	(	PUNCT
ajrhss-2207	1	12	e	e	NOUN
ajrhss-2207	1	13	):	):	PUNCT
ajrhss-2207	1	14	2832	2832	NUM
ajrhss-2207	1	15	-	-	SYM
ajrhss-2207	1	16	8019	8019	NUM
ajrhss-2207	1	17	volume	volume	NOUN
ajrhss-2207	1	18	25	25	NUM
ajrhss-2207	1	19	,	,	PUNCT
ajrhss-2207	1	20	|	|	ADV
ajrhss-2207	1	21	june	june	PROPN
ajrhss-2207	1	22	2024	2024	NUM
ajrhss-2207	1	23	p	p	NOUN
ajrhss-2207	1	24	a	a	DET
ajrhss-2207	1	25	g	g	NOUN
ajrhss-2207	1	26	e	e	NOUN
ajrhss-2207	2	1	|	|	ADV
ajrhss-2207	2	2	76	76	NUM
ajrhss-2207	2	3	www.americanjournal.org	www.americanjournal.org	PROPN
ajrhss-2207	2	4	trigonometric	trigonometric	NOUN
ajrhss-2207	2	5	equations	equation	NOUN
ajrhss-2207	2	6	abdullayev	abdullayev	PROPN
ajrhss-2207	2	7	nurbek	nurbek	PROPN
ajrhss-2207	2	8	shukhrat	shukhrat	PROPN
ajrhss-2207	2	9	ogli	ogli	NOUN
ajrhss-2207	2	10	yoldoshev	yoldoshev	PROPN
ajrhss-2207	2	11	bekmirza	bekmirza	PROPN
ajrhss-2207	2	12	shukhrat	shukhrat	PROPN
ajrhss-2207	2	13	ogli	ogli	PROPN
ajrhss-2207	2	14	matkurbanov	matkurbanov	PROPN
ajrhss-2207	2	15	ganjaboy	ganjaboy	PROPN
ajrhss-2207	2	16	botir	botir	AUX
ajrhss-2207	2	17	ogli	ogli	VERB
ajrhss-2207	2	18	a	a	DET
ajrhss-2207	2	19	b	b	PROPN
ajrhss-2207	2	20	s	s	ADP
ajrhss-2207	2	21	t	t	PROPN
ajrhss-2207	2	22	r	r	NOUN
ajrhss-2207	2	23	a	a	DET
ajrhss-2207	2	24	c	c	NOUN
ajrhss-2207	2	25	t	t	NOUN
ajrhss-2207	2	26	k	k	X
ajrhss-2207	2	27	e	e	PROPN
ajrhss-2207	2	28	y	y	PROPN
ajrhss-2207	2	29	w	w	NOUN
ajrhss-2207	2	30	o	o	NOUN
ajrhss-2207	2	31	r	r	NOUN
ajrhss-2207	2	32	d	d	X
ajrhss-2207	2	33	s	s	NOUN
ajrhss-2207	2	34	trigonometric	trigonometric	ADJ
ajrhss-2207	2	35	equations	equation	NOUN
ajrhss-2207	2	36	are	be	AUX
ajrhss-2207	2	37	mathematical	mathematical	ADJ
ajrhss-2207	2	38	expressions	expression	NOUN
ajrhss-2207	2	39	involving	involve	VERB
ajrhss-2207	2	40	trigonometric	trigonometric	ADJ
ajrhss-2207	2	41	functions	function	NOUN
ajrhss-2207	2	42	like	like	ADP
ajrhss-2207	2	43	sine	sine	NOUN
ajrhss-2207	2	44	,	,	PUNCT
ajrhss-2207	2	45	cosine	cosine	NOUN
ajrhss-2207	2	46	,	,	PUNCT
ajrhss-2207	2	47	and	and	CCONJ
ajrhss-2207	2	48	tangent	tangent	NOUN
ajrhss-2207	2	49	.	.	PUNCT
ajrhss-2207	3	1	this	this	DET
ajrhss-2207	3	2	book	book	NOUN
ajrhss-2207	3	3	provides	provide	VERB
ajrhss-2207	3	4	a	a	DET
ajrhss-2207	3	5	comprehensive	comprehensive	ADJ
ajrhss-2207	3	6	overview	overview	NOUN
ajrhss-2207	3	7	of	of	ADP
ajrhss-2207	3	8	trigonometric	trigonometric	ADJ
ajrhss-2207	3	9	equations	equation	NOUN
ajrhss-2207	3	10	,	,	PUNCT
ajrhss-2207	3	11	covering	cover	VERB
ajrhss-2207	3	12	various	various	ADJ
ajrhss-2207	3	13	solution	solution	NOUN
ajrhss-2207	3	14	techniques	technique	NOUN
ajrhss-2207	3	15	and	and	CCONJ
ajrhss-2207	3	16	applications	application	NOUN
ajrhss-2207	3	17	in	in	ADP
ajrhss-2207	3	18	mathematics	mathematic	NOUN
ajrhss-2207	3	19	and	and	CCONJ
ajrhss-2207	3	20	other	other	ADJ
ajrhss-2207	3	21	disciplines	discipline	NOUN
ajrhss-2207	3	22	.	.	PUNCT
ajrhss-2207	4	1	it	it	PRON
ajrhss-2207	4	2	offers	offer	VERB
ajrhss-2207	4	3	clear	clear	ADJ
ajrhss-2207	4	4	explanations	explanation	NOUN
ajrhss-2207	4	5	,	,	PUNCT
ajrhss-2207	4	6	examples	example	NOUN
ajrhss-2207	4	7	,	,	PUNCT
ajrhss-2207	4	8	and	and	CCONJ
ajrhss-2207	4	9	exercises	exercise	NOUN
ajrhss-2207	4	10	to	to	PART
ajrhss-2207	4	11	help	help	VERB
ajrhss-2207	4	12	readers	reader	NOUN
ajrhss-2207	4	13	understand	understand	VERB
ajrhss-2207	4	14	and	and	CCONJ
ajrhss-2207	4	15	master	master	VERB
ajrhss-2207	4	16	the	the	DET
ajrhss-2207	4	17	concepts	concept	NOUN
ajrhss-2207	4	18	of	of	ADP
ajrhss-2207	4	19	solving	solve	VERB
ajrhss-2207	4	20	trigonometric	trigonometric	ADJ
ajrhss-2207	4	21	equations	equation	NOUN
ajrhss-2207	4	22	.	.	PUNCT
ajrhss-2207	5	1	trigonometric	trigonometric	ADJ
ajrhss-2207	5	2	equations	equation	NOUN
ajrhss-2207	5	3	,	,	PUNCT
ajrhss-2207	5	4	trigonometric	trigonometric	ADJ
ajrhss-2207	5	5	functions	function	NOUN
ajrhss-2207	5	6	,	,	PUNCT
ajrhss-2207	5	7	sine	sine	NOUN
ajrhss-2207	5	8	,	,	PUNCT
ajrhss-2207	5	9	cosine	cosine	NOUN
ajrhss-2207	5	10	,	,	PUNCT
ajrhss-2207	5	11	tangent	tangent	NOUN
ajrhss-2207	5	12	,	,	PUNCT
ajrhss-2207	5	13	inverse	inverse	ADJ
ajrhss-2207	5	14	trigonometric	trigonometric	NOUN
ajrhss-2207	5	15	functions	function	NOUN
ajrhss-2207	5	16	,	,	PUNCT
ajrhss-2207	5	17	identities	identity	NOUN
ajrhss-2207	5	18	,	,	PUNCT
ajrhss-2207	5	19	solutions	solution	NOUN
ajrhss-2207	5	20	,	,	PUNCT
ajrhss-2207	5	21	applications	application	NOUN
ajrhss-2207	5	22	,	,	PUNCT
ajrhss-2207	5	23	graphical	graphical	ADJ
ajrhss-2207	5	24	methods	method	NOUN
ajrhss-2207	5	25	.	.	PUNCT
ajrhss-2207	6	1	introduction	introduction	NOUN
ajrhss-2207	6	2	a	a	DET
ajrhss-2207	6	3	trigonometric	trigonometric	ADJ
ajrhss-2207	6	4	equation	equation	NOUN
ajrhss-2207	6	5	is	be	AUX
ajrhss-2207	6	6	an	an	DET
ajrhss-2207	6	7	equation	equation	NOUN
ajrhss-2207	6	8	that	that	PRON
ajrhss-2207	6	9	involves	involve	VERB
ajrhss-2207	6	10	one	one	NUM
ajrhss-2207	6	11	or	or	CCONJ
ajrhss-2207	6	12	more	more	ADJ
ajrhss-2207	6	13	trigonometric	trigonometric	ADJ
ajrhss-2207	6	14	functions	function	NOUN
ajrhss-2207	6	15	,	,	PUNCT
ajrhss-2207	6	16	such	such	ADJ
ajrhss-2207	6	17	as	as	ADP
ajrhss-2207	6	18	sine	sine	NOUN
ajrhss-2207	6	19	,	,	PUNCT
ajrhss-2207	6	20	cosine	cosine	NOUN
ajrhss-2207	6	21	,	,	PUNCT
ajrhss-2207	6	22	tangent	tangent	NOUN
ajrhss-2207	6	23	,	,	PUNCT
ajrhss-2207	6	24	cotangent	cotangent	NOUN
ajrhss-2207	6	25	,	,	PUNCT
ajrhss-2207	6	26	secant	secant	ADJ
ajrhss-2207	6	27	,	,	PUNCT
ajrhss-2207	6	28	or	or	CCONJ
ajrhss-2207	6	29	cosecant	cosecant	ADJ
ajrhss-2207	6	30	.	.	PUNCT
ajrhss-2207	7	1	these	these	DET
ajrhss-2207	7	2	equations	equation	NOUN
ajrhss-2207	7	3	can	can	AUX
ajrhss-2207	7	4	be	be	AUX
ajrhss-2207	7	5	solved	solve	VERB
ajrhss-2207	7	6	for	for	ADP
ajrhss-2207	7	7	the	the	DET
ajrhss-2207	7	8	variable	variable	NOUN
ajrhss-2207	7	9	,	,	PUNCT
ajrhss-2207	7	10	often	often	ADV
ajrhss-2207	7	11	representing	represent	VERB
ajrhss-2207	7	12	an	an	DET
ajrhss-2207	7	13	angle	angle	NOUN
ajrhss-2207	7	14	,	,	PUNCT
ajrhss-2207	7	15	within	within	ADP
ajrhss-2207	7	16	a	a	DET
ajrhss-2207	7	17	specified	specified	ADJ
ajrhss-2207	7	18	domain	domain	NOUN
ajrhss-2207	7	19	.	.	PUNCT
ajrhss-2207	8	1	here	here	ADV
ajrhss-2207	8	2	are	be	AUX
ajrhss-2207	8	3	a	a	DET
ajrhss-2207	8	4	few	few	ADJ
ajrhss-2207	8	5	examples	example	NOUN
ajrhss-2207	8	6	:	:	PUNCT
ajrhss-2207	8	7	1	1	X
ajrhss-2207	8	8	.	.	X
ajrhss-2207	8	9	basic	basic	ADJ
ajrhss-2207	8	10	trigonometric	trigonometric	ADJ
ajrhss-2207	8	11	equations	equation	NOUN
ajrhss-2207	8	12	:	:	PUNCT
ajrhss-2207	8	13	\(\sin(x	\(\sin(x	X
ajrhss-2207	8	14	)	)	PUNCT
ajrhss-2207	8	15	=	=	SYM
ajrhss-2207	8	16	0.5\	0.5\	X
ajrhss-2207	8	17	)	)	PUNCT
ajrhss-2207	8	18	\(\cos(x	\(\cos(x	NOUN
ajrhss-2207	8	19	)	)	PUNCT
ajrhss-2207	8	20	=	=	SYM
ajrhss-2207	8	21	\frac{\sqrt{3}}{2}\	\frac{\sqrt{3}}{2}\	NOUN
ajrhss-2207	8	22	)	)	PUNCT
ajrhss-2207	8	23	\(\tan(x	\(\tan(x	ADJ
ajrhss-2207	8	24	)	)	PUNCT
ajrhss-2207	8	25	=	=	SYM
ajrhss-2207	8	26	1\	1\	NUM
ajrhss-2207	8	27	)	)	PUNCT
ajrhss-2207	8	28	2	2	NUM
ajrhss-2207	8	29	.	.	PUNCT
ajrhss-2207	8	30	equations	equation	NOUN
ajrhss-2207	8	31	involving	involve	VERB
ajrhss-2207	8	32	multiple	multiple	ADJ
ajrhss-2207	8	33	trigonometric	trigonometric	ADJ
ajrhss-2207	8	34	functions	function	NOUN
ajrhss-2207	8	35	:	:	PUNCT
ajrhss-2207	8	36	\(\sin(x	\(\sin(x	X
ajrhss-2207	8	37	)	)	PUNCT
ajrhss-2207	8	38	+	+	SYM
ajrhss-2207	8	39	\cos(x	\cos(x	ADJ
ajrhss-2207	8	40	)	)	PUNCT
ajrhss-2207	8	41	=	=	SYM
ajrhss-2207	8	42	1\	1\	NUM
ajrhss-2207	8	43	)	)	PUNCT
ajrhss-2207	8	44	\(\tan(x	\(\tan(x	ADJ
ajrhss-2207	8	45	)	)	PUNCT
ajrhss-2207	8	46	\cdot	\cdot	NOUN
ajrhss-2207	8	47	\cot(x	\cot(x	NOUN
ajrhss-2207	8	48	)	)	PUNCT
ajrhss-2207	8	49	=	=	SYM
ajrhss-2207	8	50	1\	1\	NUM
ajrhss-2207	8	51	)	)	PUNCT
ajrhss-2207	8	52	3	3	NUM
ajrhss-2207	8	53	.	.	PUNCT
ajrhss-2207	8	54	equations	equation	NOUN
ajrhss-2207	8	55	with	with	ADP
ajrhss-2207	8	56	squared	square	VERB
ajrhss-2207	8	57	functions	function	NOUN
ajrhss-2207	8	58	:	:	PUNCT
ajrhss-2207	8	59	\(\sin^2(x	\(\sin^2(x	NOUN
ajrhss-2207	8	60	)	)	PUNCT
ajrhss-2207	9	1	+	+	CCONJ
ajrhss-2207	9	2	\cos^2(x	\cos^2(x	PROPN
ajrhss-2207	9	3	)	)	PUNCT
ajrhss-2207	9	4	=	=	SYM
ajrhss-2207	9	5	1\	1\	NUM
ajrhss-2207	9	6	)	)	PUNCT
ajrhss-2207	9	7	\(\sec^2(x	\(\sec^2(x	PROPN
ajrhss-2207	9	8	)	)	PUNCT
ajrhss-2207	9	9	\tan^2(x	\tan^2(x	PROPN
ajrhss-2207	9	10	)	)	PUNCT
ajrhss-2207	10	1	=	=	SYM
ajrhss-2207	10	2	1\	1\	NUM
ajrhss-2207	10	3	)	)	PUNCT
ajrhss-2207	10	4	4	4	NUM
ajrhss-2207	10	5	.	.	PUNCT
ajrhss-2207	10	6	equations	equation	NOUN
ajrhss-2207	10	7	involving	involve	VERB
ajrhss-2207	10	8	multiple	multiple	ADJ
ajrhss-2207	10	9	angles	angle	NOUN
ajrhss-2207	10	10	:	:	PUNCT
ajrhss-2207	10	11	\(\sin(2x	\(\sin(2x	PUNCT
ajrhss-2207	10	12	)	)	PUNCT
ajrhss-2207	10	13	=	=	SYM
ajrhss-2207	10	14	\cos(x)\	\cos(x)\	PROPN
ajrhss-2207	10	15	)	)	PUNCT
ajrhss-2207	10	16	\(\cos(3x	\(\cos(3x	PUNCT
ajrhss-2207	10	17	)	)	PUNCT
ajrhss-2207	10	18	=	=	SYM
ajrhss-2207	11	1	\frac{1}{2}\	\frac{1}{2}\	VERB
ajrhss-2207	11	2	)	)	PUNCT
ajrhss-2207	11	3	5	5	NUM
ajrhss-2207	11	4	.	.	PUNCT
ajrhss-2207	11	5	complex	complex	ADJ
ajrhss-2207	11	6	trigonometric	trigonometric	ADJ
ajrhss-2207	11	7	equations	equation	NOUN
ajrhss-2207	11	8	:	:	PUNCT
ajrhss-2207	11	9	\(\sin(x	\(\sin(x	X
ajrhss-2207	11	10	)	)	PUNCT
ajrhss-2207	11	11	\cos(x	\cos(x	ADJ
ajrhss-2207	11	12	)	)	PUNCT
ajrhss-2207	11	13	=	=	SYM
ajrhss-2207	11	14	\frac{1}{4}\	\frac{1}{4}\	NOUN
ajrhss-2207	11	15	)	)	PUNCT
ajrhss-2207	11	16	\(\cos^2(x	\(\cos^2(x	NOUN
ajrhss-2207	11	17	)	)	PUNCT
ajrhss-2207	11	18	3	3	NUM
ajrhss-2207	11	19	\sin(x	\sin(x	X
ajrhss-2207	11	20	)	)	PUNCT
ajrhss-2207	11	21	=	=	SYM
ajrhss-2207	11	22	2\	2\	X
ajrhss-2207	11	23	)	)	PUNCT
ajrhss-2207	11	24	these	these	DET
ajrhss-2207	11	25	equations	equation	NOUN
ajrhss-2207	11	26	are	be	AUX
ajrhss-2207	11	27	solved	solve	VERB
ajrhss-2207	11	28	by	by	ADP
ajrhss-2207	11	29	using	use	VERB
ajrhss-2207	11	30	various	various	ADJ
ajrhss-2207	11	31	methods	method	NOUN
ajrhss-2207	11	32	,	,	PUNCT
ajrhss-2207	11	33	such	such	ADJ
ajrhss-2207	11	34	as	as	ADP
ajrhss-2207	11	35	:	:	PUNCT
ajrhss-2207	11	36	american	american	ADJ
ajrhss-2207	11	37	journal	journal	PROPN
ajrhss-2207	11	38	of	of	ADP
ajrhss-2207	11	39	research	research	NOUN
ajrhss-2207	11	40	in	in	ADP
ajrhss-2207	11	41	humanities	humanity	NOUN
ajrhss-2207	11	42	and	and	CCONJ
ajrhss-2207	11	43	social	social	ADJ
ajrhss-2207	11	44	sciences	science	NOUN
ajrhss-2207	11	45	volume	volume	NOUN
ajrhss-2207	11	46	25	25	NUM
ajrhss-2207	11	47	june	june	PROPN
ajrhss-2207	11	48	2024	2024	NUM
ajrhss-2207	11	49	p	p	NOUN
ajrhss-2207	11	50	a	a	DET
ajrhss-2207	11	51	g	g	NOUN
ajrhss-2207	11	52	e	e	NOUN
ajrhss-2207	11	53	|	|	ADV
ajrhss-2207	11	54	77	77	NUM
ajrhss-2207	11	55	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2207	11	56	using	use	VERB
ajrhss-2207	11	57	trigonometric	trigonometric	ADJ
ajrhss-2207	11	58	identities	identity	NOUN
ajrhss-2207	11	59	:	:	PUNCT
ajrhss-2207	11	60	simplifying	simplify	VERB
ajrhss-2207	11	61	the	the	DET
ajrhss-2207	11	62	equation	equation	NOUN
ajrhss-2207	11	63	using	use	VERB
ajrhss-2207	11	64	identities	identity	NOUN
ajrhss-2207	11	65	like	like	ADP
ajrhss-2207	11	66	\(\sin^2(x	\(\sin^2(x	NOUN
ajrhss-2207	11	67	)	)	PUNCT
ajrhss-2207	12	1	+	+	CCONJ
ajrhss-2207	12	2	\cos^2(x	\cos^2(x	PROPN
ajrhss-2207	12	3	)	)	PUNCT
ajrhss-2207	12	4	=	=	SYM
ajrhss-2207	12	5	1\	1\	NUM
ajrhss-2207	12	6	)	)	PUNCT
ajrhss-2207	12	7	or	or	CCONJ
ajrhss-2207	12	8	\(\tan(x	\(\tan(x	ADJ
ajrhss-2207	12	9	)	)	PUNCT
ajrhss-2207	12	10	=	=	SYM
ajrhss-2207	12	11	\frac{\sin(x)}{\cos(x)}\	\frac{\sin(x)}{\cos(x)}\	PROPN
ajrhss-2207	12	12	)	)	PUNCT
ajrhss-2207	12	13	.	.	PUNCT
ajrhss-2207	13	1	inverse	inverse	NOUN
ajrhss-2207	13	2	trigonometric	trigonometric	ADJ
ajrhss-2207	13	3	functions	function	NOUN
ajrhss-2207	13	4	:	:	PUNCT
ajrhss-2207	13	5	applying	apply	VERB
ajrhss-2207	13	6	the	the	DET
ajrhss-2207	13	7	inverse	inverse	NOUN
ajrhss-2207	13	8	functions	function	NOUN
ajrhss-2207	13	9	to	to	PART
ajrhss-2207	13	10	isolate	isolate	VERB
ajrhss-2207	13	11	the	the	DET
ajrhss-2207	13	12	variable	variable	NOUN
ajrhss-2207	13	13	(	(	PUNCT
ajrhss-2207	13	14	e.g.	e.g.	ADV
ajrhss-2207	13	15	,	,	PUNCT
ajrhss-2207	13	16	\(x	\(x	NOUN
ajrhss-2207	13	17	=	=	PROPN
ajrhss-2207	13	18	\arcsin(0.5)\	\arcsin(0.5)\	PROPN
ajrhss-2207	13	19	)	)	PUNCT
ajrhss-2207	13	20	)	)	PUNCT
ajrhss-2207	13	21	.	.	PUNCT
ajrhss-2207	14	1	graphical	graphical	ADJ
ajrhss-2207	14	2	methods	method	NOUN
ajrhss-2207	14	3	:	:	PUNCT
ajrhss-2207	14	4	visualizing	visualize	VERB
ajrhss-2207	14	5	the	the	DET
ajrhss-2207	14	6	functions	function	NOUN
ajrhss-2207	14	7	on	on	ADP
ajrhss-2207	14	8	a	a	DET
ajrhss-2207	14	9	graph	graph	NOUN
ajrhss-2207	14	10	to	to	PART
ajrhss-2207	14	11	find	find	VERB
ajrhss-2207	14	12	the	the	DET
ajrhss-2207	14	13	points	point	NOUN
ajrhss-2207	14	14	of	of	ADP
ajrhss-2207	14	15	intersection	intersection	NOUN
ajrhss-2207	14	16	.	.	PUNCT
ajrhss-2207	15	1	algebraic	algebraic	ADJ
ajrhss-2207	15	2	manipulation	manipulation	NOUN
ajrhss-2207	15	3	:	:	PUNCT
ajrhss-2207	15	4	rearranging	rearrange	VERB
ajrhss-2207	15	5	the	the	DET
ajrhss-2207	15	6	equation	equation	NOUN
ajrhss-2207	15	7	and	and	CCONJ
ajrhss-2207	15	8	solving	solving	NOUN
ajrhss-2207	15	9	for	for	ADP
ajrhss-2207	15	10	the	the	DET
ajrhss-2207	15	11	variable	variable	NOUN
ajrhss-2207	15	12	using	use	VERB
ajrhss-2207	15	13	algebraic	algebraic	ADJ
ajrhss-2207	15	14	techniques	technique	NOUN
ajrhss-2207	15	15	.	.	PUNCT
ajrhss-2207	16	1	the	the	DET
ajrhss-2207	16	2	goal	goal	NOUN
ajrhss-2207	16	3	is	be	AUX
ajrhss-2207	16	4	to	to	PART
ajrhss-2207	16	5	find	find	VERB
ajrhss-2207	16	6	all	all	DET
ajrhss-2207	16	7	possible	possible	ADJ
ajrhss-2207	16	8	solutions	solution	NOUN
ajrhss-2207	16	9	within	within	ADP
ajrhss-2207	16	10	the	the	DET
ajrhss-2207	16	11	given	give	VERB
ajrhss-2207	16	12	domain	domain	NOUN
ajrhss-2207	16	13	,	,	PUNCT
ajrhss-2207	16	14	often	often	ADV
ajrhss-2207	16	15	considering	consider	VERB
ajrhss-2207	16	16	the	the	DET
ajrhss-2207	16	17	periodic	periodic	ADJ
ajrhss-2207	16	18	nature	nature	NOUN
ajrhss-2207	16	19	of	of	ADP
ajrhss-2207	16	20	trigonometric	trigonometric	ADJ
ajrhss-2207	16	21	functions	function	NOUN
ajrhss-2207	16	22	.	.	PUNCT
ajrhss-2207	17	1	sure	sure	ADJ
ajrhss-2207	17	2	,	,	PUNCT
ajrhss-2207	17	3	let	let	VERB
ajrhss-2207	17	4	's	us	PRON
ajrhss-2207	17	5	delve	delve	VERB
ajrhss-2207	17	6	deeper	deep	ADJ
ajrhss-2207	17	7	into	into	ADP
ajrhss-2207	17	8	trigonometric	trigonometric	ADJ
ajrhss-2207	17	9	equations	equation	NOUN
ajrhss-2207	17	10	,	,	PUNCT
ajrhss-2207	17	11	their	their	PRON
ajrhss-2207	17	12	solutions	solution	NOUN
ajrhss-2207	17	13	,	,	PUNCT
ajrhss-2207	17	14	and	and	CCONJ
ajrhss-2207	17	15	methods	method	NOUN
ajrhss-2207	17	16	for	for	ADP
ajrhss-2207	17	17	solving	solve	VERB
ajrhss-2207	17	18	them	they	PRON
ajrhss-2207	17	19	.	.	PUNCT
ajrhss-2207	18	1	types	type	NOUN
ajrhss-2207	18	2	of	of	ADP
ajrhss-2207	18	3	trigonometric	trigonometric	ADJ
ajrhss-2207	18	4	equations	equation	NOUN
ajrhss-2207	18	5	1	1	NUM
ajrhss-2207	18	6	.	.	PUNCT
ajrhss-2207	19	1	linear	linear	ADJ
ajrhss-2207	19	2	trigonometric	trigonometric	ADJ
ajrhss-2207	19	3	equations	equation	NOUN
ajrhss-2207	19	4	:	:	PUNCT
ajrhss-2207	19	5	these	these	PRON
ajrhss-2207	19	6	are	be	AUX
ajrhss-2207	19	7	equations	equation	NOUN
ajrhss-2207	19	8	where	where	SCONJ
ajrhss-2207	19	9	the	the	DET
ajrhss-2207	19	10	trigonometric	trigonometric	ADJ
ajrhss-2207	19	11	function	function	NOUN
ajrhss-2207	19	12	appears	appear	VERB
ajrhss-2207	19	13	linearly	linearly	ADV
ajrhss-2207	19	14	.	.	PUNCT
ajrhss-2207	20	1	example	example	NOUN
ajrhss-2207	20	2	:	:	PUNCT
ajrhss-2207	20	3	\(\sin(x	\(\sin(x	X
ajrhss-2207	20	4	)	)	PUNCT
ajrhss-2207	20	5	=	=	SYM
ajrhss-2207	21	1	0.5\	0.5\	NUM
ajrhss-2207	21	2	)	)	PUNCT
ajrhss-2207	21	3	2	2	NUM
ajrhss-2207	21	4	.	.	X
ajrhss-2207	21	5	quadratic	quadratic	ADJ
ajrhss-2207	21	6	trigonometric	trigonometric	ADJ
ajrhss-2207	21	7	equations	equation	NOUN
ajrhss-2207	21	8	:	:	PUNCT
ajrhss-2207	21	9	these	these	PRON
ajrhss-2207	21	10	involve	involve	VERB
ajrhss-2207	21	11	the	the	DET
ajrhss-2207	21	12	square	square	NOUN
ajrhss-2207	21	13	of	of	ADP
ajrhss-2207	21	14	a	a	DET
ajrhss-2207	21	15	trigonometric	trigonometric	ADJ
ajrhss-2207	21	16	function	function	NOUN
ajrhss-2207	21	17	.	.	PUNCT
ajrhss-2207	22	1	example	example	NOUN
ajrhss-2207	22	2	:	:	PUNCT
ajrhss-2207	22	3	\(2\cos^2(x	\(2\cos^2(x	NOUN
ajrhss-2207	22	4	)	)	PUNCT
ajrhss-2207	22	5	3\cos(x	3\cos(x	NUM
ajrhss-2207	22	6	)	)	PUNCT
ajrhss-2207	23	1	+	+	CCONJ
ajrhss-2207	23	2	1	1	NUM
ajrhss-2207	23	3	=	=	SYM
ajrhss-2207	23	4	0\	0\	PROPN
ajrhss-2207	23	5	)	)	PUNCT
ajrhss-2207	23	6	3	3	NUM
ajrhss-2207	23	7	.	.	PUNCT
ajrhss-2207	23	8	multiple	multiple	ADJ
ajrhss-2207	23	9	angle	angle	NOUN
ajrhss-2207	23	10	equations	equation	NOUN
ajrhss-2207	23	11	:	:	PUNCT
ajrhss-2207	23	12	these	these	PRON
ajrhss-2207	23	13	involve	involve	VERB
ajrhss-2207	23	14	trigonometric	trigonometric	ADJ
ajrhss-2207	23	15	functions	function	NOUN
ajrhss-2207	23	16	of	of	ADP
ajrhss-2207	23	17	multiple	multiple	ADJ
ajrhss-2207	23	18	angles	angle	NOUN
ajrhss-2207	23	19	.	.	PUNCT
ajrhss-2207	24	1	example	example	NOUN
ajrhss-2207	24	2	:	:	PUNCT
ajrhss-2207	24	3	\(\sin(2x	\(\sin(2x	PUNCT
ajrhss-2207	24	4	)	)	PUNCT
ajrhss-2207	24	5	=	=	SYM
ajrhss-2207	24	6	\cos(x)\	\cos(x)\	PROPN
ajrhss-2207	24	7	)	)	PUNCT
ajrhss-2207	24	8	4	4	NUM
ajrhss-2207	24	9	.	.	X
ajrhss-2207	24	10	sum	sum	NOUN
ajrhss-2207	24	11	and	and	CCONJ
ajrhss-2207	24	12	difference	difference	NOUN
ajrhss-2207	24	13	equations	equation	NOUN
ajrhss-2207	24	14	:	:	PUNCT
ajrhss-2207	24	15	these	these	PRON
ajrhss-2207	24	16	involve	involve	VERB
ajrhss-2207	24	17	sums	sum	NOUN
ajrhss-2207	24	18	or	or	CCONJ
ajrhss-2207	24	19	differences	difference	NOUN
ajrhss-2207	24	20	of	of	ADP
ajrhss-2207	24	21	trigonometric	trigonometric	ADJ
ajrhss-2207	24	22	functions	function	NOUN
ajrhss-2207	24	23	.	.	PUNCT
ajrhss-2207	25	1	example	example	NOUN
ajrhss-2207	25	2	:	:	PUNCT
ajrhss-2207	25	3	\(\sin(x	\(\sin(x	X
ajrhss-2207	25	4	)	)	PUNCT
ajrhss-2207	25	5	+	+	SYM
ajrhss-2207	25	6	\cos(x	\cos(x	ADJ
ajrhss-2207	25	7	)	)	PUNCT
ajrhss-2207	25	8	=	=	SYM
ajrhss-2207	25	9	\frac{\sqrt{2}}{2}\	\frac{\sqrt{2}}{2}\	X
ajrhss-2207	25	10	)	)	PUNCT
ajrhss-2207	25	11	solving	solve	VERB
ajrhss-2207	25	12	trigonometric	trigonometric	ADJ
ajrhss-2207	25	13	equations	equation	NOUN
ajrhss-2207	25	14	1	1	NUM
ajrhss-2207	25	15	.	.	PUNCT
ajrhss-2207	26	1	using	use	VERB
ajrhss-2207	26	2	identities	identity	NOUN
ajrhss-2207	26	3	:	:	PUNCT
ajrhss-2207	26	4	trigonometric	trigonometric	ADJ
ajrhss-2207	26	5	identities	identity	NOUN
ajrhss-2207	26	6	can	can	AUX
ajrhss-2207	26	7	simplify	simplify	VERB
ajrhss-2207	26	8	complex	complex	ADJ
ajrhss-2207	26	9	equations	equation	NOUN
ajrhss-2207	26	10	.	.	PUNCT
ajrhss-2207	27	1	pythagorean	pythagorean	PROPN
ajrhss-2207	27	2	identities	identity	NOUN
ajrhss-2207	27	3	:	:	PUNCT
ajrhss-2207	28	1	\	\	PROPN
ajrhss-2207	28	2	[	[	PUNCT
ajrhss-2207	28	3	\sin^2(x	\sin^2(x	PROPN
ajrhss-2207	28	4	)	)	PUNCT
ajrhss-2207	28	5	+	+	SYM
ajrhss-2207	28	6	\cos^2(x	\cos^2(x	PROPN
ajrhss-2207	28	7	)	)	PUNCT
ajrhss-2207	28	8	=	=	PUNCT
ajrhss-2207	28	9	1\	1\	NUM
ajrhss-2207	28	10	]	]	X
ajrhss-2207	28	11	\[1	\[1	X
ajrhss-2207	28	12	+	+	X
ajrhss-2207	28	13	\tan^2(x	\tan^2(x	PROPN
ajrhss-2207	28	14	)	)	PUNCT
ajrhss-2207	28	15	=	=	SYM
ajrhss-2207	28	16	\sec^2(x)\	\sec^2(x)\	PROPN
ajrhss-2207	28	17	]	]	X
ajrhss-2207	28	18	\	\	PROPN
ajrhss-2207	28	19	[	[	PUNCT
ajrhss-2207	28	20	1	1	NUM
ajrhss-2207	28	21	+	+	NUM
ajrhss-2207	28	22	\cot^2(x	\cot^2(x	PROPN
ajrhss-2207	28	23	)	)	PUNCT
ajrhss-2207	28	24	=	=	SYM
ajrhss-2207	28	25	\csc^2(x)\	\csc^2(x)\	PROPN
ajrhss-2207	28	26	]	]	PUNCT
ajrhss-2207	28	27	angle	angle	NOUN
ajrhss-2207	28	28	sum	sum	NOUN
ajrhss-2207	28	29	and	and	CCONJ
ajrhss-2207	28	30	difference	difference	NOUN
ajrhss-2207	28	31	identities	identity	NOUN
ajrhss-2207	28	32	:	:	PUNCT
ajrhss-2207	29	1	\	\	NOUN
ajrhss-2207	29	2	[	[	PUNCT
ajrhss-2207	29	3	\sin(a	\sin(a	ADP
ajrhss-2207	29	4	\pm	\pm	PROPN
ajrhss-2207	29	5	b	b	NOUN
ajrhss-2207	29	6	)	)	PUNCT
ajrhss-2207	29	7	=	=	PUNCT
ajrhss-2207	29	8	\sin(a)\cos(b	\sin(a)\cos(b	NUM
ajrhss-2207	29	9	)	)	PUNCT
ajrhss-2207	29	10	\pm	\pm	PROPN
ajrhss-2207	29	11	\cos(a)\sin(b)\	\cos(a)\sin(b)\	PROPN
ajrhss-2207	29	12	]	]	PUNCT
ajrhss-2207	29	13	\[\cos(a	\[\cos(a	NOUN
ajrhss-2207	29	14	\pm	\pm	PROPN
ajrhss-2207	29	15	b	b	NOUN
ajrhss-2207	29	16	)	)	PUNCT
ajrhss-2207	29	17	=	=	SYM
ajrhss-2207	29	18	\cos(a)\cos(b	\cos(a)\cos(b	X
ajrhss-2207	29	19	)	)	PUNCT
ajrhss-2207	29	20	\mp	\mp	PROPN
ajrhss-2207	29	21	\sin(a)\sin(b)\	\sin(a)\sin(b)\	PROPN
ajrhss-2207	29	22	]	]	X
ajrhss-2207	29	23	double	double	ADJ
ajrhss-2207	29	24	angle	angle	NOUN
ajrhss-2207	29	25	identities	identity	NOUN
ajrhss-2207	29	26	:	:	PUNCT
ajrhss-2207	29	27	\[\sin(2x	\[\sin(2x	ADJ
ajrhss-2207	29	28	)	)	PUNCT
ajrhss-2207	29	29	=	=	SYM
ajrhss-2207	29	30	2\sin(x)\cos(x)\	2\sin(x)\cos(x)\	NUM
ajrhss-2207	29	31	]	]	PUNCT
ajrhss-2207	29	32	\[\cos(2x	\[\cos(2x	ADJ
ajrhss-2207	29	33	)	)	PUNCT
ajrhss-2207	29	34	=	=	SYM
ajrhss-2207	29	35	\cos^2(x	\cos^2(x	PROPN
ajrhss-2207	29	36	)	)	PUNCT
ajrhss-2207	29	37	\sin^2(x)\	\sin^2(x)\	PROPN
ajrhss-2207	29	38	]	]	X
ajrhss-2207	29	39	2	2	NUM
ajrhss-2207	29	40	.	.	X
ajrhss-2207	29	41	using	use	VERB
ajrhss-2207	29	42	inverse	inverse	NOUN
ajrhss-2207	29	43	trigonometric	trigonometric	ADJ
ajrhss-2207	29	44	functions	function	NOUN
ajrhss-2207	29	45	:	:	PUNCT
ajrhss-2207	29	46	to	to	PART
ajrhss-2207	29	47	solve	solve	VERB
ajrhss-2207	29	48	for	for	ADP
ajrhss-2207	29	49	\(x\	\(x\	NOUN
ajrhss-2207	29	50	)	)	PUNCT
ajrhss-2207	29	51	,	,	PUNCT
ajrhss-2207	29	52	we	we	PRON
ajrhss-2207	29	53	can	can	AUX
ajrhss-2207	29	54	use	use	VERB
ajrhss-2207	29	55	inverse	inverse	NOUN
ajrhss-2207	29	56	trigonometric	trigonometric	NOUN
ajrhss-2207	29	57	functions	function	NOUN
ajrhss-2207	29	58	.	.	PUNCT
ajrhss-2207	30	1	example	example	NOUN
ajrhss-2207	30	2	:	:	PUNCT
ajrhss-2207	30	3	\(\sin(x	\(\sin(x	X
ajrhss-2207	30	4	)	)	PUNCT
ajrhss-2207	30	5	=	=	SYM
ajrhss-2207	30	6	0.5\	0.5\	X
ajrhss-2207	30	7	)	)	PUNCT
ajrhss-2207	30	8	american	american	ADJ
ajrhss-2207	30	9	journal	journal	PROPN
ajrhss-2207	30	10	of	of	ADP
ajrhss-2207	30	11	research	research	NOUN
ajrhss-2207	30	12	in	in	ADP
ajrhss-2207	30	13	humanities	humanity	NOUN
ajrhss-2207	30	14	and	and	CCONJ
ajrhss-2207	30	15	social	social	ADJ
ajrhss-2207	30	16	sciences	science	NOUN
ajrhss-2207	30	17	volume	volume	NOUN
ajrhss-2207	30	18	25	25	NUM
ajrhss-2207	30	19	june	june	PROPN
ajrhss-2207	30	20	2024	2024	NUM
ajrhss-2207	30	21	p	p	NOUN
ajrhss-2207	30	22	a	a	DET
ajrhss-2207	30	23	g	g	NOUN
ajrhss-2207	30	24	e	e	NOUN
ajrhss-2207	30	25	|	|	ADV
ajrhss-2207	30	26	78	78	NUM
ajrhss-2207	30	27	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2207	30	28	\[x	\[x	NOUN
ajrhss-2207	30	29	=	=	SYM
ajrhss-2207	30	30	\arcsin(0.5	\arcsin(0.5	NUM
ajrhss-2207	30	31	)	)	PUNCT
ajrhss-2207	31	1	\	\	NOUN
ajrhss-2207	31	2	]	]	X
ajrhss-2207	31	3	the	the	DET
ajrhss-2207	31	4	solutions	solution	NOUN
ajrhss-2207	31	5	are:\	are:\	VERB
ajrhss-2207	31	6	[	[	PUNCT
ajrhss-2207	31	7	x	x	SYM
ajrhss-2207	31	8	=	=	SYM
ajrhss-2207	31	9	\frac{\pi}{6	\frac{\pi}{6	NOUN
ajrhss-2207	31	10	}	}	PUNCT
ajrhss-2207	31	11	+	+	NUM
ajrhss-2207	31	12	2k\pi	2k\pi	NUM
ajrhss-2207	31	13	\quad	\quad	PROPN
ajrhss-2207	31	14	\text{and	\text{and	PROPN
ajrhss-2207	31	15	}	}	PUNCT
ajrhss-2207	31	16	\quad	\quad	PROPN
ajrhss-2207	31	17	x	x	NOUN
ajrhss-2207	31	18	=	=	SYM
ajrhss-2207	31	19	\pi	\pi	NOUN
ajrhss-2207	31	20	\frac{\pi}{6	\frac{\pi}{6	NOUN
ajrhss-2207	31	21	}	}	PUNCT
ajrhss-2207	31	22	+	+	CCONJ
ajrhss-2207	31	23	2k\pi	2k\pi	NUM
ajrhss-2207	31	24	\	\	NOUN
ajrhss-2207	31	25	]	]	PUNCT
ajrhss-2207	31	26	where	where	SCONJ
ajrhss-2207	31	27	\(k\	\(k\	NOUN
ajrhss-2207	31	28	)	)	PUNCT
ajrhss-2207	31	29	is	be	AUX
ajrhss-2207	31	30	any	any	DET
ajrhss-2207	31	31	integer	integer	NOUN
ajrhss-2207	31	32	.	.	PUNCT
ajrhss-2207	32	1	3	3	X
ajrhss-2207	32	2	.	.	X
ajrhss-2207	32	3	algebraic	algebraic	ADJ
ajrhss-2207	32	4	manipulation	manipulation	NOUN
ajrhss-2207	32	5	:	:	PUNCT
ajrhss-2207	32	6	rearranging	rearrange	VERB
ajrhss-2207	32	7	and	and	CCONJ
ajrhss-2207	32	8	factoring	factor	VERB
ajrhss-2207	32	9	the	the	DET
ajrhss-2207	32	10	equation	equation	NOUN
ajrhss-2207	32	11	.	.	PUNCT
ajrhss-2207	33	1	example	example	NOUN
ajrhss-2207	33	2	:	:	PUNCT
ajrhss-2207	33	3	solve	solve	VERB
ajrhss-2207	33	4	\(2\cos^2(x	\(2\cos^2(x	NOUN
ajrhss-2207	33	5	)	)	PUNCT
ajrhss-2207	33	6	3\cos(x	3\cos(x	NUM
ajrhss-2207	33	7	)	)	PUNCT
ajrhss-2207	34	1	+	+	CCONJ
ajrhss-2207	34	2	1	1	NUM
ajrhss-2207	34	3	=	=	SYM
ajrhss-2207	34	4	0\	0\	PROPN
ajrhss-2207	34	5	)	)	PUNCT
ajrhss-2207	34	6	.	.	PUNCT
ajrhss-2207	35	1	\	\	PROPN
ajrhss-2207	35	2	[	[	PUNCT
ajrhss-2207	35	3	\text{let	\text{let	NOUN
ajrhss-2207	35	4	}	}	PUNCT
ajrhss-2207	35	5	y	y	PROPN
ajrhss-2207	35	6	=	=	SYM
ajrhss-2207	35	7	\cos(x)\	\cos(x)\	PROPN
ajrhss-2207	35	8	]	]	X
ajrhss-2207	35	9	\[2y^2	\[2y^2	PROPN
ajrhss-2207	35	10	3y	3y	NUM
ajrhss-2207	36	1	+	+	NOUN
ajrhss-2207	36	2	1	1	NUM
ajrhss-2207	36	3	=	=	SYM
ajrhss-2207	36	4	0\	0\	PROPN
ajrhss-2207	36	5	]	]	PUNCT
ajrhss-2207	36	6	factor	factor	NOUN
ajrhss-2207	36	7	the	the	DET
ajrhss-2207	36	8	quadratic	quadratic	NOUN
ajrhss-2207	36	9	:	:	PUNCT
ajrhss-2207	36	10	\[(2y	\[(2y	PROPN
ajrhss-2207	36	11	1)(y	1)(y	NUM
ajrhss-2207	36	12	1	1	NUM
ajrhss-2207	36	13	)	)	PUNCT
ajrhss-2207	36	14	=	=	SYM
ajrhss-2207	36	15	0\	0\	PROPN
ajrhss-2207	36	16	]	]	X
ajrhss-2207	37	1	so	so	CCONJ
ajrhss-2207	37	2	,	,	PUNCT
ajrhss-2207	37	3	\(y	\(y	PRON
ajrhss-2207	37	4	=	=	SYM
ajrhss-2207	37	5	\frac{1}{2}\	\frac{1}{2}\	NOUN
ajrhss-2207	37	6	)	)	PUNCT
ajrhss-2207	37	7	or	or	CCONJ
ajrhss-2207	37	8	\(y	\(y	PRON
ajrhss-2207	37	9	=	=	SYM
ajrhss-2207	37	10	1\	1\	NUM
ajrhss-2207	37	11	)	)	PUNCT
ajrhss-2207	37	12	.	.	PUNCT
ajrhss-2207	38	1	hence	hence	ADV
ajrhss-2207	38	2	,	,	PUNCT
ajrhss-2207	38	3	\(\cos(x	\(\cos(x	X
ajrhss-2207	38	4	)	)	PUNCT
ajrhss-2207	38	5	=	=	SYM
ajrhss-2207	38	6	\frac{1}{2}\	\frac{1}{2}\	VERB
ajrhss-2207	38	7	)	)	PUNCT
ajrhss-2207	38	8	or	or	CCONJ
ajrhss-2207	38	9	\(\cos(x	\(\cos(x	NOUN
ajrhss-2207	38	10	)	)	PUNCT
ajrhss-2207	38	11	=	=	SYM
ajrhss-2207	38	12	1\	1\	NUM
ajrhss-2207	38	13	)	)	PUNCT
ajrhss-2207	38	14	.	.	PUNCT
ajrhss-2207	39	1	4	4	X
ajrhss-2207	39	2	.	.	X
ajrhss-2207	39	3	graphical	graphical	ADJ
ajrhss-2207	39	4	methods	method	NOUN
ajrhss-2207	39	5	:	:	PUNCT
ajrhss-2207	39	6	plotting	plot	VERB
ajrhss-2207	39	7	the	the	DET
ajrhss-2207	39	8	functions	function	NOUN
ajrhss-2207	39	9	to	to	PART
ajrhss-2207	39	10	find	find	VERB
ajrhss-2207	39	11	intersections	intersection	NOUN
ajrhss-2207	39	12	.	.	PUNCT
ajrhss-2207	40	1	example	example	NOUN
ajrhss-2207	40	2	:	:	PUNCT
ajrhss-2207	40	3	to	to	PART
ajrhss-2207	40	4	solve	solve	VERB
ajrhss-2207	40	5	\(\sin(x	\(\sin(x	NOUN
ajrhss-2207	40	6	)	)	PUNCT
ajrhss-2207	40	7	=	=	SYM
ajrhss-2207	40	8	0.5\	0.5\	NUM
ajrhss-2207	40	9	)	)	PUNCT
ajrhss-2207	40	10	,	,	PUNCT
ajrhss-2207	40	11	plot	plot	VERB
ajrhss-2207	40	12	\(y	\(y	PRON
ajrhss-2207	40	13	=	=	SYM
ajrhss-2207	40	14	\sin(x)\	\sin(x)\	PROPN
ajrhss-2207	40	15	)	)	PUNCT
ajrhss-2207	40	16	and	and	CCONJ
ajrhss-2207	40	17	\(y	\(y	NUM
ajrhss-2207	40	18	=	=	SYM
ajrhss-2207	40	19	0.5\	0.5\	NUM
ajrhss-2207	40	20	)	)	PUNCT
ajrhss-2207	40	21	on	on	ADP
ajrhss-2207	40	22	the	the	DET
ajrhss-2207	40	23	same	same	ADJ
ajrhss-2207	40	24	graph	graph	NOUN
ajrhss-2207	40	25	.	.	PUNCT
ajrhss-2207	41	1	the	the	DET
ajrhss-2207	41	2	\(x\)coordinates	\(x\)coordinates	PROPN
ajrhss-2207	41	3	of	of	ADP
ajrhss-2207	41	4	the	the	DET
ajrhss-2207	41	5	intersection	intersection	NOUN
ajrhss-2207	41	6	points	point	NOUN
ajrhss-2207	41	7	give	give	VERB
ajrhss-2207	41	8	the	the	DET
ajrhss-2207	41	9	solutions	solution	NOUN
ajrhss-2207	41	10	.	.	PUNCT
ajrhss-2207	42	1	general	general	ADJ
ajrhss-2207	42	2	solution	solution	NOUN
ajrhss-2207	42	3	form	form	NOUN
ajrhss-2207	42	4	trigonometric	trigonometric	NOUN
ajrhss-2207	42	5	functions	function	NOUN
ajrhss-2207	42	6	are	be	AUX
ajrhss-2207	42	7	periodic	periodic	ADJ
ajrhss-2207	42	8	,	,	PUNCT
ajrhss-2207	42	9	so	so	SCONJ
ajrhss-2207	42	10	solutions	solution	NOUN
ajrhss-2207	42	11	are	be	AUX
ajrhss-2207	42	12	often	often	ADV
ajrhss-2207	42	13	given	give	VERB
ajrhss-2207	42	14	in	in	ADP
ajrhss-2207	42	15	a	a	DET
ajrhss-2207	42	16	general	general	ADJ
ajrhss-2207	42	17	form	form	NOUN
ajrhss-2207	42	18	.	.	PUNCT
ajrhss-2207	43	1	for	for	ADP
ajrhss-2207	43	2	\(\sin(x	\(\sin(x	NOUN
ajrhss-2207	43	3	)	)	PUNCT
ajrhss-2207	43	4	=	=	SYM
ajrhss-2207	43	5	a\	a\	NOUN
ajrhss-2207	43	6	):	):	PUNCT
ajrhss-2207	43	7	\[x	\[x	NOUN
ajrhss-2207	43	8	=	=	SYM
ajrhss-2207	43	9	\arcsin(a	\arcsin(a	PRON
ajrhss-2207	43	10	)	)	PUNCT
ajrhss-2207	43	11	+	+	CCONJ
ajrhss-2207	43	12	2k\pi	2k\pi	NUM
ajrhss-2207	43	13	\quad	\quad	PROPN
ajrhss-2207	43	14	\text{or	\text{or	PROPN
ajrhss-2207	43	15	}	}	PUNCT
ajrhss-2207	43	16	\quad	\quad	PROPN
ajrhss-2207	43	17	x	x	NOUN
ajrhss-2207	43	18	=	=	SYM
ajrhss-2207	43	19	\pi	\pi	PROPN
ajrhss-2207	43	20	\arcsin(a	\arcsin(a	PRON
ajrhss-2207	43	21	)	)	PUNCT
ajrhss-2207	43	22	+	+	CCONJ
ajrhss-2207	43	23	2k\pi\	2k\pi\	NUM
ajrhss-2207	43	24	]	]	PUNCT
ajrhss-2207	43	25	where	where	SCONJ
ajrhss-2207	43	26	\(k\	\(k\	NOUN
ajrhss-2207	43	27	)	)	PUNCT
ajrhss-2207	43	28	is	be	AUX
ajrhss-2207	43	29	any	any	DET
ajrhss-2207	43	30	integer	integer	NOUN
ajrhss-2207	43	31	.	.	PUNCT
ajrhss-2207	44	1	for	for	ADP
ajrhss-2207	44	2	\(\cos(x	\(\cos(x	NOUN
ajrhss-2207	44	3	)	)	PUNCT
ajrhss-2207	44	4	=	=	SYM
ajrhss-2207	44	5	a\):\[x	a\):\[x	PROPN
ajrhss-2207	44	6	=	=	PUNCT
ajrhss-2207	44	7	\arccos(a	\arccos(a	NOUN
ajrhss-2207	44	8	)	)	PUNCT
ajrhss-2207	44	9	+	+	CCONJ
ajrhss-2207	44	10	2k\pi	2k\pi	NUM
ajrhss-2207	44	11	\quad	\quad	PROPN
ajrhss-2207	44	12	\text{or	\text{or	PROPN
ajrhss-2207	44	13	}	}	PUNCT
ajrhss-2207	44	14	\quad	\quad	PROPN
ajrhss-2207	44	15	x	x	SYM
ajrhss-2207	44	16	=	=	SYM
ajrhss-2207	44	17	-\arccos(a	-\arccos(a	PROPN
ajrhss-2207	44	18	)	)	PUNCT
ajrhss-2207	44	19	+	+	NUM
ajrhss-2207	44	20	2k\pi\	2k\pi\	NUM
ajrhss-2207	44	21	]	]	PUNCT
ajrhss-2207	44	22	where	where	SCONJ
ajrhss-2207	44	23	\(k\	\(k\	NOUN
ajrhss-2207	44	24	)	)	PUNCT
ajrhss-2207	44	25	is	be	AUX
ajrhss-2207	44	26	any	any	DET
ajrhss-2207	44	27	integer	integer	NOUN
ajrhss-2207	44	28	.	.	PUNCT
ajrhss-2207	45	1	for	for	ADP
ajrhss-2207	45	2	\(\tan(x	\(\tan(x	ADV
ajrhss-2207	45	3	)	)	PUNCT
ajrhss-2207	45	4	=	=	SYM
ajrhss-2207	45	5	a\	a\	NOUN
ajrhss-2207	45	6	):	):	PUNCT
ajrhss-2207	45	7	\[x	\[x	NOUN
ajrhss-2207	45	8	=	=	SYM
ajrhss-2207	45	9	\arctan(a	\arctan(a	X
ajrhss-2207	45	10	)	)	PUNCT
ajrhss-2207	46	1	+	+	CCONJ
ajrhss-2207	46	2	k\pi\	k\pi\	NOUN
ajrhss-2207	46	3	]	]	PUNCT
ajrhss-2207	46	4	where	where	SCONJ
ajrhss-2207	46	5	\(k\	\(k\	NOUN
ajrhss-2207	46	6	)	)	PUNCT
ajrhss-2207	46	7	is	be	AUX
ajrhss-2207	46	8	any	any	DET
ajrhss-2207	46	9	integer	integer	NOUN
ajrhss-2207	46	10	.	.	PUNCT
ajrhss-2207	47	1	examples	example	NOUN
ajrhss-2207	47	2	and	and	CCONJ
ajrhss-2207	47	3	solutions	solution	NOUN
ajrhss-2207	47	4	1	1	NUM
ajrhss-2207	47	5	.	.	PUNCT
ajrhss-2207	47	6	example	example	NOUN
ajrhss-2207	47	7	1	1	NUM
ajrhss-2207	47	8	:	:	PUNCT
ajrhss-2207	47	9	solve	solve	VERB
ajrhss-2207	47	10	\(\sin(x	\(\sin(x	NOUN
ajrhss-2207	47	11	)	)	PUNCT
ajrhss-2207	47	12	=	=	SYM
ajrhss-2207	47	13	0.5\	0.5\	NUM
ajrhss-2207	47	14	)	)	PUNCT
ajrhss-2207	47	15	.	.	PUNCT
ajrhss-2207	48	1	solutions	solution	NOUN
ajrhss-2207	48	2	:	:	PUNCT
ajrhss-2207	48	3	\[x	\[x	NOUN
ajrhss-2207	48	4	=	=	SYM
ajrhss-2207	48	5	\frac{\pi}{6	\frac{\pi}{6	X
ajrhss-2207	48	6	}	}	PUNCT
ajrhss-2207	48	7	+	+	NUM
ajrhss-2207	48	8	2k\pi	2k\pi	NUM
ajrhss-2207	48	9	\quad	\quad	PROPN
ajrhss-2207	48	10	\text{or	\text{or	PROPN
ajrhss-2207	48	11	}	}	PUNCT
ajrhss-2207	48	12	\quad	\quad	PROPN
ajrhss-2207	48	13	x	x	NOUN
ajrhss-2207	48	14	=	=	SYM
ajrhss-2207	48	15	\frac{5\pi}{6	\frac{5\pi}{6	PROPN
ajrhss-2207	48	16	}	}	PUNCT
ajrhss-2207	48	17	+	+	NUM
ajrhss-2207	48	18	2k\pi\	2k\pi\	NUM
ajrhss-2207	48	19	]	]	PUNCT
ajrhss-2207	48	20	where	where	SCONJ
ajrhss-2207	48	21	\(k\	\(k\	NOUN
ajrhss-2207	48	22	)	)	PUNCT
ajrhss-2207	48	23	is	be	AUX
ajrhss-2207	48	24	any	any	DET
ajrhss-2207	48	25	integer	integer	NOUN
ajrhss-2207	48	26	.	.	PUNCT
ajrhss-2207	49	1	2	2	X
ajrhss-2207	49	2	.	.	NOUN
ajrhss-2207	49	3	example	example	NOUN
ajrhss-2207	49	4	2	2	NUM
ajrhss-2207	49	5	:	:	PUNCT
ajrhss-2207	49	6	solve	solve	VERB
ajrhss-2207	49	7	\(\tan(x	\(\tan(x	ADV
ajrhss-2207	49	8	)	)	PUNCT
ajrhss-2207	49	9	=	=	PUNCT
ajrhss-2207	49	10	\sqrt{3}\	\sqrt{3}\	NOUN
ajrhss-2207	49	11	)	)	PUNCT
ajrhss-2207	49	12	.	.	PUNCT
ajrhss-2207	50	1	solutions	solution	NOUN
ajrhss-2207	50	2	:	:	PUNCT
ajrhss-2207	51	1	\	\	PROPN
ajrhss-2207	51	2	[	[	PUNCT
ajrhss-2207	51	3	x	x	SYM
ajrhss-2207	51	4	=	=	SYM
ajrhss-2207	51	5	\frac{\pi}{3	\frac{\pi}{3	NOUN
ajrhss-2207	51	6	}	}	PUNCT
ajrhss-2207	51	7	+	+	CCONJ
ajrhss-2207	51	8	k\pi\	k\pi\	NOUN
ajrhss-2207	51	9	]	]	PUNCT
ajrhss-2207	51	10	where	where	SCONJ
ajrhss-2207	51	11	\(k\	\(k\	NOUN
ajrhss-2207	51	12	)	)	PUNCT
ajrhss-2207	51	13	is	be	AUX
ajrhss-2207	51	14	any	any	DET
ajrhss-2207	51	15	integer	integer	NOUN
ajrhss-2207	51	16	.	.	PUNCT
ajrhss-2207	52	1	3	3	X
ajrhss-2207	52	2	.	.	NOUN
ajrhss-2207	52	3	example	example	NOUN
ajrhss-2207	52	4	3	3	NUM
ajrhss-2207	52	5	:	:	PUNCT
ajrhss-2207	52	6	solve	solve	VERB
ajrhss-2207	52	7	\(\cos(2x	\(\cos(2x	NUM
ajrhss-2207	52	8	)	)	PUNCT
ajrhss-2207	52	9	=	=	SYM
ajrhss-2207	52	10	-1\	-1\	PROPN
ajrhss-2207	52	11	)	)	PUNCT
ajrhss-2207	52	12	.	.	PUNCT
ajrhss-2207	53	1	american	american	PROPN
ajrhss-2207	53	2	journal	journal	PROPN
ajrhss-2207	53	3	of	of	ADP
ajrhss-2207	53	4	research	research	NOUN
ajrhss-2207	53	5	in	in	ADP
ajrhss-2207	53	6	humanities	humanity	NOUN
ajrhss-2207	53	7	and	and	CCONJ
ajrhss-2207	53	8	social	social	ADJ
ajrhss-2207	53	9	sciences	science	NOUN
ajrhss-2207	53	10	volume	volume	NOUN
ajrhss-2207	53	11	25	25	NUM
ajrhss-2207	53	12	june	june	PROPN
ajrhss-2207	53	13	2024	2024	NUM
ajrhss-2207	53	14	p	p	NOUN
ajrhss-2207	53	15	a	a	DET
ajrhss-2207	53	16	g	g	NOUN
ajrhss-2207	53	17	e	e	NOUN
ajrhss-2207	53	18	|	|	ADV
ajrhss-2207	53	19	79	79	NUM
ajrhss-2207	53	20	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2207	53	21	solutions	solution	NOUN
ajrhss-2207	53	22	:	:	PUNCT
ajrhss-2207	53	23	\[2x	\[2x	ADJ
ajrhss-2207	53	24	=	=	SYM
ajrhss-2207	53	25	\pi	\pi	PROPN
ajrhss-2207	53	26	+	+	CCONJ
ajrhss-2207	53	27	2k\pi\	2k\pi\	NUM
ajrhss-2207	53	28	]	]	X
ajrhss-2207	53	29	\[x	\[x	NOUN
ajrhss-2207	53	30	=	=	SYM
ajrhss-2207	53	31	\frac{\pi}{2	\frac{\pi}{2	PROPN
ajrhss-2207	53	32	}	}	PUNCT
ajrhss-2207	53	33	+	+	CCONJ
ajrhss-2207	53	34	k\pi\	k\pi\	NOUN
ajrhss-2207	53	35	]	]	PUNCT
ajrhss-2207	54	1	where	where	SCONJ
ajrhss-2207	54	2	\(k\	\(k\	NOUN
ajrhss-2207	54	3	)	)	PUNCT
ajrhss-2207	54	4	is	be	AUX
ajrhss-2207	54	5	any	any	DET
ajrhss-2207	54	6	integer	integer	NOUN
ajrhss-2207	54	7	.	.	PUNCT
ajrhss-2207	55	1	by	by	ADP
ajrhss-2207	55	2	applying	apply	VERB
ajrhss-2207	55	3	these	these	DET
ajrhss-2207	55	4	methods	method	NOUN
ajrhss-2207	55	5	and	and	CCONJ
ajrhss-2207	55	6	principles	principle	NOUN
ajrhss-2207	55	7	,	,	PUNCT
ajrhss-2207	55	8	you	you	PRON
ajrhss-2207	55	9	can	can	AUX
ajrhss-2207	55	10	solve	solve	VERB
ajrhss-2207	55	11	a	a	DET
ajrhss-2207	55	12	wide	wide	ADJ
ajrhss-2207	55	13	range	range	NOUN
ajrhss-2207	55	14	of	of	ADP
ajrhss-2207	55	15	trigonometric	trigonometric	ADJ
ajrhss-2207	55	16	equations	equation	NOUN
ajrhss-2207	55	17	.	.	PUNCT
ajrhss-2207	56	1	references	reference	NOUN
ajrhss-2207	56	2	1	1	NUM
ajrhss-2207	56	3	.	.	PUNCT
ajrhss-2207	56	4	"	"	PUNCT
ajrhss-2207	56	5	trigonometry	trigonometry	NOUN
ajrhss-2207	56	6	"	"	PUNCT
ajrhss-2207	56	7	by	by	ADP
ajrhss-2207	56	8	michael	michael	PROPN
ajrhss-2207	56	9	sullivan	sullivan	PROPN
ajrhss-2207	56	10	2	2	PROPN
ajrhss-2207	56	11	.	.	PUNCT
ajrhss-2207	57	1	"	"	PUNCT
ajrhss-2207	57	2	precalculus	precalculus	NOUN
ajrhss-2207	57	3	:	:	PUNCT
ajrhss-2207	57	4	mathematics	mathematic	NOUN
ajrhss-2207	57	5	for	for	ADP
ajrhss-2207	57	6	calculus	calculus	NOUN
ajrhss-2207	57	7	"	"	PUNCT
ajrhss-2207	57	8	by	by	ADP
ajrhss-2207	57	9	james	james	PROPN
ajrhss-2207	57	10	stewart	stewart	PROPN
ajrhss-2207	57	11	,	,	PUNCT
ajrhss-2207	57	12	lothar	lothar	PROPN
ajrhss-2207	57	13	redlin	redlin	PROPN
ajrhss-2207	57	14	,	,	PUNCT
ajrhss-2207	57	15	and	and	CCONJ
ajrhss-2207	57	16	saleem	saleem	PROPN
ajrhss-2207	57	17	watson	watson	PROPN
ajrhss-2207	57	18	3	3	NUM
ajrhss-2207	57	19	.	.	PUNCT
ajrhss-2207	58	1	"	"	PUNCT
ajrhss-2207	58	2	trigonometry	trigonometry	NOUN
ajrhss-2207	58	3	for	for	ADP
ajrhss-2207	58	4	dummies	dummy	NOUN
ajrhss-2207	58	5	"	"	PUNCT
ajrhss-2207	58	6	by	by	ADP
ajrhss-2207	58	7	mary	mary	PROPN
ajrhss-2207	58	8	jane	jane	PROPN
ajrhss-2207	58	9	sterling	sterling	PROPN
ajrhss-2207	58	10	4	4	NUM
ajrhss-2207	58	11	.	.	PUNCT
ajrhss-2207	59	1	"	"	PUNCT
ajrhss-2207	59	2	trigonometry	trigonometry	NOUN
ajrhss-2207	59	3	workbook	workbook	NOUN
ajrhss-2207	59	4	for	for	ADP
ajrhss-2207	59	5	dummies	dummy	NOUN
ajrhss-2207	59	6	"	"	PUNCT
ajrhss-2207	59	7	by	by	ADP
ajrhss-2207	59	8	mary	mary	PROPN
ajrhss-2207	59	9	jane	jane	PROPN
ajrhss-2207	59	10	sterling	sterling	PROPN
ajrhss-2207	59	11	5	5	NUM
ajrhss-2207	59	12	.	.	PUNCT
ajrhss-2207	59	13	"	"	PUNCT
ajrhss-2207	59	14	algebra	algebra	NOUN
ajrhss-2207	59	15	and	and	CCONJ
ajrhss-2207	59	16	trigonometry	trigonometry	NOUN
ajrhss-2207	59	17	"	"	PUNCT
ajrhss-2207	59	18	by	by	ADP
ajrhss-2207	59	19	ron	ron	PROPN
ajrhss-2207	59	20	larson	larson	PROPN
ajrhss-2207	59	21	6	6	NUM
ajrhss-2207	59	22	.	.	PUNCT
ajrhss-2207	59	23	"	"	PUNCT
ajrhss-2207	59	24	trigonometry	trigonometry	NOUN
ajrhss-2207	59	25	"	"	PUNCT
ajrhss-2207	59	26	by	by	ADP
ajrhss-2207	59	27	cynthia	cynthia	PROPN
ajrhss-2207	59	28	y.	y.	PROPN
ajrhss-2207	59	29	young	young	PROPN