Trigonometric identities involving sine and cosine

The fundamental identity

cos²(θ)+sin²(θ) = 1

Symmetry identities

cos(–θ) = cos(θ)

sin(–θ) = –sin(θ)

cos(π+θ) = –cos(θ)

sin(π+θ) = –sin(θ)

cos(π–θ) = –cos(θ)

sin(π–θ) = sin(θ)

cos(π/2 + θ) = –sin(θ)

sin(π/2 + θ) = cos(θ)

cos(π/2 – θ) = sin(θ)

sin(π/2 – θ) = cos(θ)

Addition formulas

cos(α+β) = cos(α)cos(β)–sin(α)sin(β)

cos(α–β) = cos(α)cos(β)+sin(α)sin(β)

sin(α+β) = sin(α)cos(β)+cos(α)sin(β)

sin(α–β) = sin(α)cos(β)–cos(α)sin(β)

Double-angle formulas

cos(2θ) = cos²(θ)–sin²(θ) = 1–2sin²(θ) = 2cos²(θ)–1

sin(2θ) = 2sin(θ)cos(θ)

Sum and difference formulas

cos(α)+cos(β) = 2cos(½(α+β))cos(½(α–β))

cos(α)–cos(β) = 2sin(½(α+β))sin(½(β–α))

sin(α)+sin(β) = 2sin(½(α+β))cos(½(α–β))

sin(α)–sin(β) = 2cos(½(α+β))sin(½(α–β))

Product formulas

cos(α)cos(β) = ½(cos(α–β)+cos(α+β))

sin(α)sin(β) = ½(cos(α–β)–cos(α+β))

sin(α)cos(β) = ½(sin(α–β)+sin(α+β))