Right Triangular Ratios
Triangular Angles
\(sin\,α=\frac{a}{c}\)
\(cos\,α=\frac{b}{c}\)
\(tan\,α=\frac{a}{b}\)
\(csc\,α=\frac{c}{a}\)
\(sec\,α=\frac{c}{b}\)
\(cot\,α=\frac{b}{a}\)
\(sin\,β=\frac{b}{c}\)
\(cos\,β=\frac{a}{c}\)
\(tan\,β=\frac{b}{a}\)
\(csc\,β=\frac{c}{b}\)
\(sec\,β=\frac{c}{a}\)
\(cot\,β=\frac{a}{b}\)
Cofunction Theorem (Triangular)
\(sin\,α=cos\,β\)
\(cos\,α=sin\,β\)
\(tan\,α=cot\,β\)
\(csc\,α=sec\,β\)
\(sec\,α=csc\,β\)
\(cot\,α=tan\,β\)
Circular Ratios
Circular Angle
\(sin\,θ=\frac{y}{r}\)
\(cos\,θ=\frac{x}{r}\)
\(tan\,θ=\frac{y}{x}\)
\(csc\,θ=\frac{r}{y}\)
\(sec\,θ=\frac{r}{x}\)
\(cot\,θ=\frac{x}{y}\)
Cofunction Theorem (Circular)
\(sin\,θ=cos\,(\frac{π}{2}-θ)\)
\(cos\,θ=sin\,(\frac{π}{2}-θ)\)
\(tan\,θ=cot\,(\frac{π}{2}-θ)\)
\(csc\,θ=sec\,(\frac{π}{2}-θ)\)
\(sec\,θ=csc\,(\frac{π}{2}-θ)\)
\(cot\,θ=tan\,(\frac{π}{2}-θ)\)
Reciprocal Identities
\(sin\,θ=\frac{1}{csc\,θ}\)
\(cos\,θ=\frac{1}{sec\,θ}\)
\(tan\,θ=\frac{1}{cot\,θ}\)
\(csc\,θ=\frac{1}{sin\,θ}\)
\(sec\,θ=\frac{1}{cos\,θ}\)
\(cot\,θ=\frac{1}{tan\,θ}\)
Ratio Identities
\(sin\,θ=cos\,θ\,tan\,θ=\frac{cos\,θ}{cot\,θ}\)
\(cos\,θ=sin\,θ\,cot\,θ=\frac{sin\,θ}{tan\,θ}\)
\(tan\,θ=sin\,θ\,sec\,θ=\frac{sin\,θ}{cos\,θ}\)
\(csc\,θ=sec\,θ\,cot\,θ=\frac{cot\,θ}{cos\,θ}\)
\(sec\,θ=csc\,θ\,tan\,θ=\frac{tan\,θ}{sin\,θ}\)
\(cot\,θ=cos\,θ\,csc\,θ=\frac{cos\,θ}{sin\,θ}\)
Pythagorean Identities
\(sin^2\,θ+cos^2\,θ=1\)
\(sin^2\,θ=1-cos^2\,θ\)
\(cos^2\,θ=1-sin^2\,θ\)
\(tan^2\,θ=sec^2\,θ-1\)
\(cot^2\,θ=csc^2\,θ-1\)
\(sec^2\,θ=1+tan^2\,θ\)
\(csc^2\,θ=1+cot^2\,θ\)
Function Conversions
Sum and Difference Formulas
Inverse Identities
Miscellaneous Identities
The Law of Sines
The Law of Cosines
The Law of the Cotangent
Reciprocal Identities
Double-Angle Formulas
Half-Angle Formulas
Product to Sum Formulas
Sum to Product Formulas
Reduction Formulas