# Trigonometric Functions of Complementary Angles

## Theorem

### Sine of Complement equals Cosine

$\map \sin {\dfrac \pi 2 - \theta} = \cos \theta$

### Cosine of Complement equals Sine

$\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$

### Tangent of Complement equals Cotangent

$\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ for $\theta \ne n \pi$

where $\tan$ and $\cot$ are tangent and cotangent respectively.

That is, the cotangent of an angle is the tangent of its complement.

This relation is defined wherever $\sin \theta \ne 0$.

### Cotangent of Complement equals Tangent

$\map \cot {\dfrac \pi 2 - \theta} = \tan \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$

where $\cot$ and $\tan$ are cotangent and tangent respectively.

That is, the tangent of an angle is the cotangent of its complement.

This relation is defined wherever $\cos \theta \ne 0$.

### Secant of Complement equals Cosecant

$\map \sec {\dfrac \pi 2 - \theta} = \csc \theta$ for $\theta \ne n \pi$

where $\sec$ and $\csc$ are secant and cosecant respectively.

That is, the cosecant of an angle is the secant of its complement.

This relation is defined wherever $\sin \theta \ne 0$.

### Cosecant of Complement equals Secant

$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$

where $\csc$ and $\sec$ are cosecant and secant respectively.

That is, the secant of an angle is the cosecant of its complement.

This relation is defined wherever $\cos \theta \ne 0$.