trigonometry formulas

Identities

$sin^{2}Ѳ +cos^{2}Ѳ$ = 1

1 + $tan^{2}Ѳ = sec^{2}Ѳ$

1 + $cot^{2}Ѳ = cosec^{2}Ѳ$

Trigonometry ratios

sinѲ = $\frac{OB}{AB}$

cosѲ = $\frac{OA}{AB}$

tanѲ = $\frac{OB}{OA}$

cosecѲ = $\frac{AB}{OB}$

secѲ = $\frac{AB}{OA}$

cotѲ = $\frac{OA}{OB}$

Relation between trigonometric ratios

SinѲ = $\frac{1}{cosec\Theta}$

cosecѲ = $\frac{1}{sin\Theta}$

cosѲ= $\frac{1}{sec\Theta}$

secѲ = $\frac{1}{cos\Theta}$

tanѲ = $\frac{1}{cot\Theta}$

cotѲ = $\frac{1}{tan\Theta}$

tanѲ= $\frac{sin\Theta}{cos\Theta}$

cotѲ = $\frac{cos\Theta}{sin\Theta}$

Values of trigonometric ratios

	$0^{\circ}$	$30^{\circ}$	$45^{\circ}$	$60^{\circ}$	$90^{\circ}$	$120^{\circ}$	$135^{\circ}$	$150^{\circ}$	$180^{\circ}$
Sin Ѳ	0	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	1	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	0
cos Ѳ	1	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	0	$\frac{-1}{2}$	$\frac{-1}{\sqrt{2}}$	$\frac{-\sqrt{3}}{2}$	-1
tan Ѳ	0	$\frac{1}{\sqrt{3}}$	1	$\sqrt{3}$	∞	$-\sqrt{3}$	-1	$\frac{-1}{\sqrt{3}}$	0
cot Ѳ	∞	$\sqrt{3}$	1	$\frac{1}{\sqrt{3}}$	0	$\frac{-1}{\sqrt{3}}$	-1	$-\sqrt{3}$	∞
sec Ѳ	1	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	2	∞	-2	$-\sqrt{2}$	$\frac{-2}{\sqrt{3}}$	-1
cosec Ѳ	∞	2	$\sqrt{2}$	$\frac{2}{\sqrt{3}}$	1	$\frac{2}{\sqrt{3}}$	$\sqrt{2}$	2	∞

1. sin(-Ѳ) = -sin Ѳ

con(-Ѳ) = cosѲ

tan(-Ѳ)=-tanѲ

2. sin (90- Ѳ) =cosѲ

cos(90 – Ѳ) = sinѲ

tan(90- Ѳ) = cotѲ

3. sin(90+ Ѳ) = cos Ѳ

cos(90+ Ѳ)=-sin Ѳ

tan(90+ Ѳ)=-cot Ѳ

4. sin( $\pi$ – Ѳ) = sin Ѳ

cos( $\pi$ – Ѳ) = -cos Ѳ

tan( $\pi$ – Ѳ) = -tan Ѳ

5. sin( $\pi$ + Ѳ) = -sin Ѳ

cos( $\pi$ + Ѳ) = -cos Ѳ

tan( $\pi$ + Ѳ) = tan Ѳ

6. sin( $\frac{3\pi }{2}$ + Ѳ) = -cos Ѳ

cos( $\frac{3\pi }{2}$ + Ѳ) = sin Ѳ

tan( $\frac{3\pi }{2}$ + Ѳ) = -cot Ѳ

sin(A +B) =sin A cos B + cos A sin B,

sin(A -B) = sin A cos B – cos A sib B

cos (A +B) = cos A cos B –sin A sin B,

cos(A -B) = cos A cos B + sin A sin B

tan(A +B) = $\frac{tan A +tan B}{1 - tan A tan B}$

tan(A -B) = $\frac{tan A -tan B}{1 + tan A tan B}$

$tan(\frac{\pi }{4}+A) =\frac{1+tan A}{1 - tan A} = \frac{cos A + sin A}{cos A -sin A}$

$tan(\frac{\pi }{4}-A) =\frac{1-tan A}{1 + tan A} = \frac{cos A - sin A}{cos A +sin A}$

cot(A +B) = $\frac{cot A cot B - 1}{cot A + cot B}$

cot(A – B) = $\frac{cot A cot B + 1}{cot B - cot A}$

sin(A + B) sin (A -B) = $sin^{2}A -sin^{2}B$

= $cos^{2}B -cos^{2}A$

Cos(A + B) cos (A – B) = $cos^{2} A -sin^{2} B$

= $cos^{2} B - sin^{2}A$

2sin A cos B = sin (A +B) +sin (A – B)

2cos A Sin B = sin (A +B) – sin (A – B)

2cos A cos B= cos (A +B) + cos (A – B)

2sin A sin B = cos (A -B) – cos (A + B)

sin C + sin D = 2 sin ( $\frac{C+D}{2}) cos(\frac{C-D}{2})$

sin C – sin D = -2 cos ( $\frac{C+D}{2})sin(\frac{C-D}{2})$

cos C + Cos D = 2 cos ( $\frac{C + D}{2}) cos(\frac{C-D}{2})$

cos C – cos D = 2 sin ( $\frac{C+D}{2})sin(\frac{D-C}{2})$

sin2A = 2 sin A cos A

cos 2A = $cos^{2} - sin^{2}A$

= $2cos^{2}A - 1$

= 1 – $2sin^{2}A$

sin2A = $\frac{2tanA}{1+tan^{a}A}$

cos2A = $\frac{1-tan^{a}A}{1+tan^{2}}$

tan 2A = $\frac{2tanA}{1-tan^{2}A}$

tan 3A = $\frac{3tanA - tan^{3}A}{1-3tan^{2}A}$

sin 3A = 3 sin A – 4 $sin^{3}A$

cos 3A = 4 $cos^{3}A$ -3 cos A

$\sqrt{1+sin2A}$ = ±(sin A +cos A)

$\sqrt{1-sin2A}$ = ±(sin A – cos A)

$\frac{sinA}{a} = \frac{sinB}{b}=\frac{sinC}{c}$

$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$

Cosine formula

cosA = $\frac{b^{2}+c^{2}-a^{2}}{2bc}$

cosB = $\frac{c^{2}+a^{2}-b^{2}}{2ca}$

cosC = $\frac{a^{2}+b^{2}-c^{2}}{2ab}$

Projection formula

a = b cos C + c cos B

b = c cos A + a cos C

c = a cos B + b cos A

tangent formula

tan( $\frac{B-C}{2}) =\frac{b-c}{b+c}cot\frac{A}{2}$

tan( $\frac{A-B}{2}) =\frac{a-b}{b+c}cot\frac{C}{2}$

tan( $\frac{C-A}{2}) =\frac{c-a}{c+a}cot\frac{B}{2}$

sin $\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}}$

sin $\frac{B}{2} = \sqrt{\frac{(s-c)(s-a)}{ ca}}$

sin $\frac{C}{2} = \sqrt{\frac{(s-a)(s-b)}{ab}}$

cos $\frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}}$

cos $\frac{B}{2} = \sqrt{\frac{s(s-c)}{ab}}$

tan $\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$

tan $\frac{B}{2} = \sqrt{\frac{(s-c)(s-a)}{s(s-b)}}$

tan $\frac{C}{2} = \sqrt{\frac{(s-a)(s-b)}{s(s-c)}}$

Where s = $\frac{1}{2}$ (a+b+c)

sin A = $\frac{2}{bc}\sqrt{s(s-a)(s-b)(s-c)} = \frac{2S}{bc}$

sin B = $\frac{2}{ca}\sqrt{s(s-a)(s-b)(s-c)} = \frac{2S}{ca}$

sin C = $\frac{2}{ab}\sqrt{s(s-a)(s-b)(s-c)} = \frac{2S}{ab}$

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