trigo identity

sin² θ + cos² θ

1

1 + tan² θ

sec² θ

1 + cot² θ

cosec² θ

sec θ + tan θ

1 / (sec θ - tan θ)

cosec θ + cot θ

1 / (cosec θ - cot θ)

sin⁴ θ + cos⁴ θ

1 - 2 sin² θ cos² θ

sin⁶ θ + cos⁶ θ

1 - 3 sin² θ cos² θ

Relation: π Radians and Degrees

π Radians = 180°

1 Radian (in degrees)

(180/π)° ≈ 57.3°

Minutes and Seconds conversion

1° = 60', 1' = 60''

Length of Arc (l)

l = rθ (θ in radians)

Area of Sector

(1/2)r²θ or (1/2)rl (θ in radians)

Sign: Anti-Clockwise (A.C.W) vs Clockwise (C.W)

A.C.W is (+); C.W is (-)

ASTC Rule (Quadrants I-IV)

I: All +, II: Sin/Cosec +, III: Tan/Cot +, IV: Cos/Sec +

sin(-θ), cos(-θ), tan(-θ)

-sin θ, cos θ, -tan θ

Function change rule (90 ± θ) or (270 ± θ)

sin↔cos, tan↔cot, sec↔cosec

Allied angle sign rule

Sign is determined by the quadrant of the ORIGINAL function

Function rule (180 ± θ) or (360 ± θ)

Trigonometric function remains the SAME

sin(A ± B)

sin A cos B ± cos A sin 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)

(tan A ± tan B) / (1 ∓ tan A tan B)

cot(A + B)

(cot A cot B - 1) / (cot A + cot B)

cot(A - B)

(cot A cot B + 1) / (cot B - cot A)

tan(A + B + C)

(tan A + tan B + tan C - tan A tan B tan C) / (1 - (tan A tan B + tan B tan C + tan C tan A))

sin 2θ

2 sin θ cos θ OR (2 tan θ) / (1 + tan² θ)

cos 2θ (All 4 forms)

cos² θ - sin² θ; 2cos² θ - 1; 1 - 2sin² θ; (1 - tan² θ)/(1 + tan² θ)

tan 2θ

(2 tan θ) / (1 - tan² θ)

sin 3θ

3 sin θ - 4 sin³ θ

cos 3θ

4 cos³ θ - 3 cos θ

tan 3θ

(3 tan θ - tan³ θ) / (1 - 3 tan² θ)

1 + cos θ and 1 - cos θ

2 cos²(θ/2) and 2 sin²(θ/2)

sin C + sin D

2 sin((C+D)/2) cos((C-D)/2)

sin C - sin D

2 cos((C+D)/2) sin((C-D)/2)

cos C + cos D

2 cos((C+D)/2) cos((C-D)/2)

cos C - cos D

2 sin((C+D)/2) sin((D-C)/2)

2 sin A cos B

sin(A+B) + sin(A-B)

2 cos A sin B

sin(A+B) - sin(A-B)

2 cos A cos B

cos(A+B) + cos(A-B)

2 sin A sin B

cos(A-B) - cos(A+B)

sin 15° / cos 75°

(√3 - 1) / 2√2

cos 15° / sin 75°

(√3 + 1) / 2√2

tan 15° / cot 75°

2 - √3

cot 15° / tan 75°

2 + √3

tan 22.5° (tan π/8)

√2 - 1

sin 18° / cos 72°

(√5 - 1) / 4

cos 36° / sin 54°

(√5 + 1) / 4

tan(π/4 + θ)

(1 + tan θ) / (1 - tan θ)

sin(A+B) · sin(A-B)

sin² A - sin² B OR cos² B - cos² A

cos(A+B) · cos(A-B)

cos² A - sin² B OR cos² B - sin² A

sin θ · sin(60° - θ) · sin(60° + θ)

(1/4) sin 3θ

cos θ · cos(60° - θ) · cos(60° + θ)

(1/4) cos 3θ

tan θ · tan(60° - θ) · tan(60° + θ)

tan 3θ

If A+B+C = 180°: tan A + tan B + tan C

tan A tan B tan C

If A+B+C = 180°: ∑ tan(A/2) tan(B/2)

1

If A+B+C = 180°: sin 2A + sin 2B + sin 2C

4 sin A sin B sin C

If A+B+C = 180°: cos 2A + cos 2B + cos 2C

-1 - 4 cos A cos B cos C

If A+B+C = 180°: sin A + sin B + sin C

4 cos(A/2) cos(B/2) cos(C/2)

If A+B+C = 180°: cos A + cos B + cos C

1 + 4 sin(A/2) sin(B/2) sin(C/2)

cos A cos 2A cos 4A … cos(2^(n-1)A)

sin(2ⁿ A) / (2ⁿ sin A)

sin α + sin(α + β) + … + sin(α + (n-1)β)

[sin(nβ/2) / sin(β/2)] · sin[α + (n-1)β/2]

Range of y = a sin x + b cos x

[-√(a²+b²), √(a²+b²)]

AM-GM Inequality (+ve numbers)

(a+b)/2 ≥ √ab

Min value: a² tan² θ + b² cot² θ

2ab

Min value: a² sec² θ + b² cosec² θ

(a + b)²

Inequality for x ∈ (0, π/2)

tan x > x > sin x