# Sine of Complement equals Cosine

## Theorem

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

where $\sin$ and $\cos$ are sine and cosine respectively.

That is, the cosine of an angle is the sine of its complement.

## Proof 1

 $\ds \map \sin {\frac \pi 2 - \theta}$ $=$ $\ds \sin \frac \pi 2 \cos \theta - \cos \frac \pi 2 \sin \theta$ Sine of Difference $\ds$ $=$ $\ds 1 \times \cos \theta - 0 \times \sin \theta$ Sine of Right Angle and Cosine of Right Angle $\ds$ $=$ $\ds \cos \theta$

$\blacksquare$

## Proof 2

 $\ds \map \sin {\frac \pi 2 - \theta}$ $=$ $\ds -\map \sin {\theta - \frac \pi 2}$ Sine Function is Odd $\ds$ $=$ $\ds \map \cos {\theta - \frac \pi 2 + \frac \pi 2}$ Cosine of Angle plus Right Angle $\ds$ $=$ $\ds \cos \theta$

$\blacksquare$

## Proof 3

 $\ds \map \sin {\dfrac \pi 2 - \theta}$ $=$ $\ds \map \Im {e^{i \paren {\frac \pi 2 - \theta} } }$ Euler's Formula $\ds$ $=$ $\ds \map \Im {e^{i \frac \pi 2} e^{-i \theta} }$ Exponent Combination Laws $\ds$ $=$ $\ds \map \Im {\paren {\cos \dfrac \pi 2+i \sin \dfrac \pi 2} e^{-i \theta} }$ Euler's Formula $\ds$ $=$ $\ds \map \Im {i e^{-i \theta} }$ Cosine of Right Angle, Sine of Right Angle $\ds$ $=$ $\ds \map \Re {e^{-i \theta} }$ $\ds$ $=$ $\ds \map \cos {-\theta}$ Euler's Formula $\ds$ $=$ $\ds \cos \theta$ Cosine Function is Even

$\blacksquare$

## Proof 4

Let $\angle xOP$ and $\angle QOy$ be complementary.

Then:

$\angle xOP = \angle QOy$

Hence:

the projection of $OP$ on the $x$-axis

equals:

the projection of $OQ$ on the $y$-axis.

Hence the result.

$\blacksquare$

## Examples

### Sine of $2 \theta - 90 \degrees$

$\map \sin {2 \theta - 90 \degrees} = -\cos 2 \theta$