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

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