Cosine of Complement equals Sine/Proof 3

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Theorem

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


Proof

\(\displaystyle \map \cos {\dfrac \pi 2 - \theta}\) \(=\) \(\displaystyle \frac 1 2 \paren {e^{i \paren {\frac \pi 2 - \theta} } + e^{-i \paren {\frac \pi 2 - \theta} } }\) 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

$\blacksquare$