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))