# Sine of Complement equals Cosine

## Theorem

$\sin \left({\dfrac \pi 2 - \theta}\right) = \cos \theta$

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

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

## Proof 1

 $\displaystyle \sin \left({\frac \pi 2 - \theta}\right)$ $=$ $\displaystyle \sin \frac \pi 2 \cos \theta - \cos \frac \pi 2 \sin \theta$ Sine of Difference $\displaystyle$ $=$ $\displaystyle 1 \times \cos \theta - 0 \times \sin \theta$ Sine of Right Angle and Cosine of Right Angle $\displaystyle$ $=$ $\displaystyle \cos \theta$

$\blacksquare$

## Proof 2

 $\displaystyle \sin \left({\frac \pi 2 - \theta}\right)$ $=$ $\displaystyle - \sin \left({\theta - \frac \pi 2}\right)$ Sine Function is Odd $\displaystyle$ $=$ $\displaystyle \cos \left({\theta - \frac \pi 2 + \frac \pi 2}\right)$ Cosine of Angle plus Right Angle $\displaystyle$ $=$ $\displaystyle \cos \theta$

$\blacksquare$

## Proof 3

 $\displaystyle \sin \left({\dfrac \pi 2 - \theta}\right)$ $=$ $\displaystyle \operatorname{Im} \left({e^{i \left({\frac \pi 2 - \theta}\right) } }\right)$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \operatorname{Im} \left({e^{i \frac \pi 2} e^{-i \theta} }\right)$ Exponent Combination Laws $\displaystyle$ $=$ $\displaystyle \operatorname{Im} \left({\left({\cos \dfrac \pi 2+i \sin \dfrac \pi 2}\right) e^{-i \theta} }\right)$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \operatorname{Im} \left({i e^{-i \theta} }\right)$ Cosine of Right Angle, Sine of Right Angle $\displaystyle$ $=$ $\displaystyle \operatorname{Re} \left({e^{-i \theta} }\right)$ $\displaystyle$ $=$ $\displaystyle \cos \left({-\theta}\right)$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \cos \theta$ Cosine Function is Even

$\blacksquare$