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Trigonometric Identities

This page is meant to serve as a quick overview of the trigonometric identities.

Reciprocal Identities

cot Θ = 1 / tan Θ sec Θ = 1 / cos Θ csc Θ = 1 / sin Θ
tan Θ = 1 / cot Θ cos Θ = 1 / sec Θ sin Θ = 1 / csc Θ

Quotient Identities

tan Θ = sin Θ / cos Θ cot Θ = cos Θ / sin Θ

Pythagorean Identities

sin² Θ + cos² Θ = 1 tan² Θ + 1 = sec² Θ 1 + cot² Θ = csc² Θ

Negative-Angle Identities

sin (-Θ) = -sin Θ cos (-Θ) = -cos Θ tan (-Θ) = -tan Θ

Cofunction Identities

For any acute angle Θ in standard position, the following applies:

sin Θ = cos(90-Θ) cos Θ = sin(90-Θ) tan Θ = cot(90-Θ)
csc Θ = sec(90-Θ) sec Θ = csc(90-Θ) cot Θ = tan(90-Θ)

Sum and Difference Identities

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

Double-Angle Identities

Half-Angle Identities