Trigonometric Identities and Derivatives

Definitions

sin(θ) = opposite/hypotenuse

cos(θ) = adjacent/hypotenuse

tan(θ) = opposite/adjacent

csc(θ) = hypotenuse/opposite

sec(θ) = hypotenuse/adjacent

cot(θ) = adjacent/opposite

Identities

1 = sin^2(x) + cos^2(x)

sin(2x) = 2sin(x)cos(x)

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

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

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

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

sec^2(θ) = tan^2(θ) + 1

scs^2(θ) = cot^2(θ) + 1

Derivatives

d/dθ cot(θ) = -scs^2(θ)

d/dx arctan(x) = 1/(x^2+1)

d/dx csc(x) = -csc(x)cot(x)

d/dt sec(t) = tan(x)sec(x)

d/dt tan(t) = sec^2(t)