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IDENTITIES :

sin(theta) = a / c	csc(theta) = 1 / sin(theta) = c / a
cos(theta) = b / c	sec(theta) = 1 / cos(theta) = c / b
tan(theta) = sin(theta) / cos(theta) = a / b	cot(theta) = 1/ tan(theta) = b / a

sin(-x) = -sin(x)
csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)

sin^{^2}(x) + cos^{^2}(x) = 1	tan^{^2}(x) + 1 = sec^{^2}(x)	cot^{^2}(x) + 1 = csc^{^2}(x)
sin(x y) = sin x cos y cos x sin y
cos(x y) = cos x cosy sin x sin y

tan(x

y) = (tan x

tan y) / (1

tan x tan y)

sin(2x) = 2 sin x cos x

cos(2x) = cos^{^2}(x) - sin^{^2}(x) = 2 cos^{^2}(x) - 1 = 1 - 2 sin^{^2}(x)

tan(2x) = 2 tan(x) / (1 - tan^{^2}(x))

sin^{^2}(x) = 1/2 - 1/2 cos(2x)

cos^{^2}(x) = 1/2 + 1/2 cos(2x)

sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )

cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )

Trig Table of Common Angles
angle	0	30	45	60	90
sin^{^2}(a)	0/4	1/4	2/4	3/4	4/4
cos^{^2}(a)	4/4	3/4	2/4	1/4	0/4
tan^{^2}(a)	0/4	1/3	2/2	3/1	4/0

Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:

a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)

c^{^2} = a^{^2} + b^{^2} - 2ab cos(C)

b^{^2} = a^{^2} + c^{^2} - 2ac cos(B)

a^{^2} = b^{^2} + c^{^2} - 2bc cos(A)

(Law of Cosines)

(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)

TABLE:

PI = 3.141592... (approximately 22/7 = 3.1428)
radians = degrees x PI / 180 (deg to rad conversion)
degrees = radians x 180 / PI (rad to deg conversion)

Rad	Deg	Sin	Cos	Tan	Csc	Sec	Cot
.0000	00	.0000	1.0000	.0000	-----	1.0000	-----	90	1.5707
.0175	01	.0175	.9998	.0175	57.2987	1.0002	57.2900	89	1.5533
.0349	02	.0349	.9994	.0349	28.6537	1.0006	28.6363	88	1.5359
.0524	03	.0523	.9986	.0524	19.1073	1.0014	19.0811	87	1.5184
.0698	04	.0698	.9976	.0699	14.3356	1.0024	14.3007	86	1.5010
.0873	05	.0872	.9962	.0875	11.4737	1.0038	11.4301	85	1.4835
.1047	06	.1045	.9945	.1051	9.5668	1.0055	9.5144	84	1.4661
.1222	07	.1219	.9925	.1228	8.2055	1.0075	8.1443	83	1.4486
.1396	08	.1392	.9903	.1405	7.1853	1.0098	7.1154	82	1.4312
.1571	09	.1564	.9877	.1584	6.3925	1.0125	6.3138	81	1.4137
.1745	10	.1736	.9848	.1763	5.7588	1.0154	5.6713	80	1.3953
.1920	11	.1908	.9816	.1944	5.2408	1.0187	5.1446	79	1.3788
.2094	12	.2079	.9781	.2126	4.8097	1.0223	4.7046	78	1.3614
.2269	13	.2250	.9744	.2309	4.4454	1.0263	4.3315	77	1.3439
.2443	14	.2419	.9703	.2493	4.1336	1.0306	4.0108	76	1.3265
.2618	15	.2588	.9659	.2679	3.8637	1.0353	3.7321	75	1.3090
.2793	16	.2756	.9613	.2867	3.6280	1.0403	3.4874	74	1.2915
.2967	17	.2924	.9563	.3057	3.4203	1.0457	3.2709	73	1.2741
.3142	18	.3090	.9511	.3249	3.2361	1.0515	3.0777	72	1.2566
.3316	19	.3256	.9455	.3443	3.0716	1.0576	2.9042	71	1.2392
.3491	20	.3420	.9397	.3640	2.9238	1.0642	2.7475	70	1.2217
.3665	21	.3584	.9336	.3839	2.7904	1.0711	2.6051	69	1.2043
.3840	22	.3746	.9272	.4040	2.6695	1.0785	2.4751	68	1.1868
.4014	23	.3907	.9205	.4245	2.5593	1.0864	2.3559	67	1.1694
.4189	24	.4067	.9135	.4452	2.4586	1.0946	2.2460	66	1.1519
.4363	25	.4226	.9063	.4663	2.3662	1.1034	2.1445	65	1.1345
.4538	26	.4384	.8988	.4877	2.2812	1.1126	2.0503	64	1.1170
.4712	27	.4540	.8910	.5095	2.2027	1.1223	1.9626	63	1.0996
.4887	28	.4695	.8829	.5317	2.1301	1.1326	1.8807	62	1.0821
.5061	29	.4848	.8746	.5543	2.0627	1.1434	1.8040	61	1.0647
.5236	30	.5000	.8660	.5774	2.0000	1.1547	1.7321	60	1.0472
.5411	31	.5150	.8572	.6009	1.9416	1.1666	1.6643	59	1.0297
.5585	32	.5299	.8480	.6249	1.8871	1.1792	1.6003	58	1.0123
.5760	33	.5446	.8387	.6494	1.8361	1.1924	1.5399	57	.9948
.5934	34	.5592	.8290	.6745	1.7883	1.2062	1.4826	56	.9774
.6109	35	.5736	.8192	.7002	1.7434	1.2208	1.4281	55	.9599
.6283	36	.5878	.8090	.7265	1.7013	1.2361	1.3764	54	.9425
.6458	37	.6018	.7986	.7536	1.6616	1.2521	1.3270	53	.9250
.6632	38	.6157	.7880	.7813	1.6243	1.2690	1.2799	52	.9076
.6807	39	.6293	.7771	.8098	1.5890	1.2868	1.2349	51	.8901
.6981	40	.6428	.7660	.8391	1.5557	1.3054	1.1918	50	.8727
.7156	41	.6561	.7547	.8693	1.5243	1.3250	1.1504	49	.8552
.7330	42	.6691	.7431	.9004	1.4945	1.3456	1.1106	48	.8378
.7505	43	.6820	.7314	.9325	1.4663	1.3673	1.0724	47	.8203
.7679	44	.6947	.7193	.9657	1.4396	1.3902	1.0355	46	.8029
.7854	45	.7071	.7071	1.0000	1.4142	1.4142	1.0000	45	.7854
		COs	Sin	Cot	Sec	CSC	Tan	Deg	Rad

Trig Table of Common Angles
angle (degrees)	0	30	45	60	90	120	135	150	180	210	225	240	270	300	315	330	360 = 0
angle (radians)	0	PI/6	PI/4	PI/3	PI/2	2/3PI	3/4PI	5/6PI	PI	7/6PI	5/4PI	4/3PI	3/2PI	5/3PI	7/4PI	11/6PI	2PI = 0
sin(a)	(0/4)	(1/4)	(2/4)	(3/4)	(4/4)	(3/4)	(2/4)	(1/4)	(0/4)	-(1/4)	-(2/4)	-(3/4)	-(4/4)	-(3/4)	-(2/4)	-(1/4)	(0/4)
COs(a)	(4/4)	(3/4)	(2/4)	(1/4)	(0/4)	-(1/4)	-(2/4)	-(3/4)	-(4/4)	-(3/4)	-(2/4)	-(1/4)	(0/4)	(1/4)	(2/4)	(3/4)	(4/4)
tan(a)	(0/4)	(1/3)	(2/2)	(3/1)	(4/0)	-(3/1)	-(2/2)	-(1/3)	-(0/4)	(1/3)	(2/2)	(3/1)	(4/0)	-(3/1)	-(2/2)	-(1/3)	(0/4)

Those with a zero in the denominator are undefined. They are included solely to demonstrate the pattern.

$unit circle picture$

HYPERBOLA:

Hyperbolic Definitions

sinh(x) = ( e^x - e^-x )/2

csch(x) = 1/sinh(x) = 2/( e^x - e^-x )

cosh(x) = ( e^x + e^-x )/2

sech(x) = 1/cosh(x) = 2/( e^x + e^-x )

tanh(x) = sinh(x)/cosh(x) = ( e^x - e^-x )/( e^x + e^-x )

coth(x) = 1/tanh(x) = ( e^x + e^-x)/( e^x - e^-x )

cosh²(x) - sinh²(x) = 1

tanh²(x) + sech²(x) = 1

coth²(x) - csch²(x) = 1

Inverse Hyperbolic Definitions

arcsinh(z) = ln( z + $sqrt$ (z² + 1) )

arccosh(z) = ln( z

$sqrt$ (z² - 1) )

arctanh(z) = 1/2 ln( (1+z)/(1-z) )

arccsch(z) = ln( (1+

(1+z²) )/z )

arcsech(z) = ln( (1

(1-z²) )/z )

arccoth(z) = 1/2 ln( (z+1)/(z-1) )

Relations to Trigonometric Functions

sinh(z) = -i sin(iz)

csch(z) = i csc(iz)

cosh(z) = cos(iz)

sech(z) = sec(iz)

tanh(z) = -i tan(iz)

coth(z) = i cot(iz)

GRAPHS:

Trig Functions: The Functions

$triangle 1$ $triangle 2$

sine(q) = opp/hyp	cosecant(q) = hyp/opp
cosine(q) = adj/hyp	secant(q) = hyp/adj
tangent(q) = opp/adj	cotangent(q) = adj/opp

The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).

It is often simpler to memorize the the trig functions in terms of only sine and cosine:

sine(q) = opp/hyp	csc(q) = 1/sin(q)
cos(q) = adj/hyp	sec(q) = 1/COs(q)
tan(q) = sin(q)/cos(q)	cot(q) = 1/tan(q)

**Inverse Functions**
arcsine(opp/hyp) = q	arccosecant(hyp/opp) = q
arccosine(adj/hyp) = q	arcsecant(hyp/adj) = q
arctangent(opp/adj) = q	arccotangent(adj/opp) = q

The functions are usually abbreviated:

arcsine (arcsin)
arccosine (arccos)
arctangent (arctan)
arccosecant (arccsc)
arcsecant (arcsec)
arccotangent (arccot).

According to the standard notation for inverse functions (f^-1), you will also often see these written as sin^-1, cos-1, tan^-1 arccsc^-1, arcsec^-1, and arccot^-1. Beware, though, there is another common notation that writes the square of the trig functions, such as (sin(x))² as sin²(x). This can be confusing, for you then might then be lead to think that sin^-1(x) = (sin(x))^-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).

Community Contributions - Articles by goIITians

Hyperbolic Definitions

Inverse Hyperbolic Definitions

Relations to Trigonometric Functions