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Mathematics LibreTexts

Fundamental Trigonometric Identities

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"arc" Identities

arctanθ=tan1θ

arcsinθ=sin1θ

arccosθ=cos1θ

Quotient and reciprocal identities

tanθ=sinθcosθ

cotθ=cosθsinθ=cscθsecθ=1tanθ

secθ=1cosθ

cscθ=1sinθ

Cofunction Function identities

sinθ=cos(π2θ)

cosθ=sin(π2θ)

Even/Odd Functions

cos(θ)=cos(θ)

sin(θ)=sin(θ)

Pythagorean identities

sin2θ+cos2θ=1

tan2θ+1=sec2θ

1+cot2θ=csc2θ

Angle sum and difference identities

sin(α+β)=sinαcosβ+sinβcosα

sin(αβ)=sinαcosβsinβcosα

cos(α+β)=cosαcosβsinαsinβ

cos(αβ)=cosαcosβ+sinαsinβ

tan(α+β)=tanα+tanβ1tanαtanβ

tan(αβ)=tanαtanβ1+tanαtanβ

Double-angle identities

sin2θ=2sinθcosθ

cos2θ=cos2θsin2θ=2cos2θ1=12sin2θ

tan2θ=2tanθ1tan2θ

Half-angle identitie s

sinθ2=±1cosθ2

cosθ2=±1+cosθ2

tanθ2=±1cosθ1+cosθ=sinθ1+cosθ=1cosθsinθ

Reduction formulas

sin2θ=1cos2θ2

cos2θ=1+cos2θ2

tan2θ=1cos2θ1+cos2θ=sin2θ1+cos2θ=1cos2θsin2θ


Fundamental Trigonometric Identities is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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