TRIGONOMETRY IDENTITIES & FORMULAS

Identities

csc t = 1/sin t sec t = 1/cos t

cot t = 1/tan t tan t = sin t/cos t

cot t = cos t/sin t sin² t + cos² t = 1

1 + tan² t = sec² t 1 + cot² t = csc² t

Addition Formulas

sin(u + v) = sinu cosv + cosu sinv

cos(u + v) = cosu cosv = sinu sinv

tan(u + v) = (tanu + tanv)/(1 – tanu tanv)

Subtraction Formulas

sin(u – v) = sinu cosv = cosu sinv

cos(u – v) = cosu cosv + sinu sinv

tan(u – v) = (tanu – tanv)/(1 + tanu tanv)

Formulas for Negatives

sin(-t) = -sin t csc(-t) = -csc t

cos(-t) = cos t sec(-t) = sec t

tan(-t) = -tan t cot(-t) = -cot t

Double Angle Formulas

sin 2u = 2sinu cosu

cos 2u = cos² u – sin² u = 1 – 2sin² u = 2cos² u - 1

Half Angle Identities

sin² u = (1 – cos 2u)/2

cos² u = (1 + cos 2u)/2

tan² u = (1 – cos 2u)/(1 + cos 2u)

Half Angle Formulas

sin u/2 = + sqrt[(1 – cos u)/2]

cos u/2 = + sqrt[(1 + cos u)/2]

tan u/2 = (1 – cos u)/sin u = sin u/(1 + cos u)

Cofunction Formulas

sin(π/2 – u) = cos u csc(π/2 – u) = sec u

cos(π/2 – u) = sin u sec(π/2 – u) = csc u

tan(π/2 – u) = cot u cot(π/2 – u) = tan u

Product To Sum Formulas

sinu cosv = ½[sin(u + v) + sin(u – v)]

cosu sinv = ½[sin(u + v) – sin(u – v)]

cosu cosv = ½[cos(u + v) + cos(u – v)]

sinu sinv = ½[cos(u – v) – cos(u + v)]

Sum To Product Formulas

sinu + sinv = 2sin[(u+v)/2] cos[(u-v)/2]

sinu – sinv = 2cos[(u+v)/2] sin[(u-v)/2]

cosu + cosv = 2cos[(u+v)/2] cos[(u-v)/2]

cosu – cosv = -2sin[(u+v)/2] sin[(u-v)/2]