Central idea

〖cot〗^(-1) x= θ

Domain

(-∞,+∞)

Range

(0, π)

〖csc〗^(-1) x= θ

Domain

(-∞,-1]∪[1,+∞)

Range

[-π/2, π/2], θ≠0

〖sin〗^(-1) x= θ

Domain

(-∞,-1]∪[1,+∞)

Range

[0, π], θ≠π/2

〖tan〗^(-1) x= θ

Domain

(-∞,+∞)

Range

(-π/2, π/2)

〖cos〗^(-1) x= θ

Domain

[-1,1]

Range

[0, π]

〖sin〗^(-1) x= θ

Domain

[-1,1]

Range

[-π/2, π/2]

cotθ

Domain

R-nπ

Range

(-∞,+∞)

Period

π

cscθ

Domain

R-nπ

Range

(-∞,-1]∪[+1,+∞)

Period

2π

secθ

Domain

R-(2n+1)π/2

Range

(-∞,-1]∪[+1,+∞)

Period

2π

tanθ

Domain

R-(2n+1)π/2

Range

(-∞,+∞)

Period

π

cosθ

Domain

(-∞,+∞)

Range

[-1,+1]

Period

2π

sinθ

Domain

(-∞,+∞)

Range

[-1,+1]

Period

2π

Product to Sum

sin⁡α sin⁡β=1/2 [ cos⁡(α-β) - cos⁡(α+β) ]

cos⁡α cos⁡β=1/2 [ cos⁡(α-β) + cos⁡(α+β) ]

sin⁡α cos⁡β=1/2 [ sin⁡(α+β) + sin⁡(α-β) ]

cos⁡α sin⁡β=1/2 [ sin⁡(α+β) - sin⁡(α-β) ]

Sum to Product

sin⁡α ± sin⁡β=2 sin⁡1/2 (α±β) cos⁡1/2 (α∓β)

cosα + cos⁡β=2 cos⁡1/2 (α+β) cos⁡1/2 (α-β)

cosα - cos⁡β= - 2 sin⁡1/2 (α+β) sin⁡1/2 (α-β)

Even/Odd Identities

cos(-θ)=cosθ

sin(-θ)=-sinθ

tan(-θ)=-tanθ

cot(-θ)=-cotθ

csc(-θ)=-cscθ

sec(-θ)=secθ

Cofunction Identities

sin(π/2-θ)=cosθ and cos(π/2-θ)=sinθ

tan(π/2-θ)=cotθ and cot(π/2-θ)=tanθ

csc(π/2-θ)=secθ and sec(π/2-θ)=cscθ

Sum/Difference Identities

sin⁡(α±β) = sin⁡α cos⁡β ± cosα sinβ

cos(α±β) = cosα cos⁡β ∓ sinα sinβ

tan⁡(α±β) = (tanα ± tanβ)/(1 ∓ tanαtanβ)

Double Angle Identities

sin(2θ) = 2sin⁡θ cos⁡θ

cos(2⁡θ) = cos⁡^2⁡(θ) - sin^2(⁡θ)

cos(2⁡θ) = 2 cos^2 (⁡θ) - 1

cos(2⁡θ) = 1 - 2sin^2(⁡θ)

tan2⁡θ = 2tan⁡θ / 1 - tan^2(⁡θ)

Half Angle Identities

sin^2(θ) = (1 - cos2⁡θ)/2

cos^2(θ) = (1 + cos2θ)/2

tan^2(⁡θ) = (1 - cos2⁡θ) / (1 + cos2θ)

Pythagorean Identities

〖sin〗^2 θ+〖cos〗^2 θ=1^

〖sec〗^2 θ-〖tan〗^2 θ=1^

〖csc〗^2 θ-〖cot〗^2 θ=1^

Reciprocal Identities

sin⁡θ=1/csc⁡θ and csc⁡θ=1/sin⁡θ

cos⁡θ=1/sec⁡θ and sec⁡θ=1/cos⁡θ

tan⁡θ=1/cot⁡θ and cot⁡θ=1/tan⁡θ

Quotient Identities

tan⁡θ=sin⁡θ/cos⁡θ

cot⁡θ=cos⁡θ/sin⁡θ