# Cosine of Complement equals Sine/Proof 3

$\map \cos {\dfrac \pi 2 - \theta} = \sin \theta$
 $\displaystyle \map \cos {\dfrac \pi 2 - \theta}$ $=$ $\displaystyle \frac 1 2 \paren {e^{i \paren {\frac \pi 2 - \theta} } + e^{-i \paren {\frac \pi 2 - \theta} } }$ Cosine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \frac 1 2 \paren {e^{i \frac \pi 2} e^{-i \theta} + e^{-i \frac \pi 2} e^{i \theta} }$ Exponential of Sum: Complex Numbers $\displaystyle$ $=$ $\displaystyle \frac 1 2 \paren {\paren {\map \cos {\frac \pi 2} + i \, \map \sin {\frac \pi 2} } e^{-i \theta} + \paren {\map \cos {-\frac \pi 2} + i \, \map \sin {-\frac \pi 2} } e^{i \theta} }$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \frac 1 2 \paren {i e^{-i \theta} - i e^{i \theta} }$ Cosine of Right Angle, Sine of Right Angle, Cosine Function is Even, Sine Function is Odd $\displaystyle$ $=$ $\displaystyle \frac 1 {2 i} \paren {e^{i \theta} - e^{-i \theta} }$ Definition of Imaginary Unit $\displaystyle$ $=$ $\displaystyle \sin \theta$ Sine Exponential Formulation
$\blacksquare$