Sine of Complement equals Cosine/Proof 3

Theorem

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

Proof

$\ds \map \sin {\dfrac \pi 2 - \theta}$

$=$

$\ds \map \Im {e^{i \paren {\frac \pi 2 - \theta} } }$

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

$\blacksquare$

This article, or a section of it, needs explaining.
In particular: Just had a thought -- do we need a result which says something like $\Im i z = \map \Re z$? It's not completely obvious, specially since $\Re i z = -\Im z$.
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