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

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

$\blacksquare$