Trigonometry is the study of triangles with respect to their sides and angles and with the functions of those angles. It is extremely important in all scientific fields.

Trigonometric Relationships

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

 $$sin^2\,θ+cos^2\,θ=1$$ $$sin^2\,θ=1-cos^2\,θ$$ $$cos^2\,θ=1-sin^2\,θ$$ $$tan^2\,θ=sec^2\,θ-1$$ $$cot^2\,θ=csc^2\,θ-1$$ $$sec^2\,θ=1+tan^2\,θ$$ $$csc^2\,θ=1+cot^2\,θ$$