Sine and Cosine of Supplementary Angles

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Theorem

Let $\alpha$ and $\beta$ be supplementary angles.

Then:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \sin \alpha$	$=$	$\displaystyle \sin \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \cos \alpha$	$=$	$\displaystyle -\cos \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \sin \alpha$	$=$	$\displaystyle \sin \left({\pi - \beta}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of supplementary
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sin \pi \cos \beta - \cos \pi \sin \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sine and Cosine of Sum (Corollary)
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle 0 \times \cos \beta - \left({-1}\right) \times \sin \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from Shape of Sine Function and Shape of Cosine Function
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sin \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Similarly:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \cos \beta$	$=$	$\displaystyle \cos \left({\pi - \alpha}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of supplementary
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \cos \pi \cos \alpha + \sin \pi \sin \alpha$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sine and Cosine of Sum (Corollary)
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({-1}\right) \times \cos \alpha + 0 \times \sin \alpha$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from Shape of Sine Function and Shape of Cosine Function
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle -\cos \alpha$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin \alpha\)	\(=\)	\(\displaystyle \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos \alpha\)	\(=\)	\(\displaystyle -\cos \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin \alpha\)	\(=\)	\(\displaystyle \sin \left({\pi - \beta}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of supplementary
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \pi \cos \beta - \cos \pi \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum (Corollary)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 0 \times \cos \beta - \left({-1}\right) \times \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Shape of Sine Function and Shape of Cosine Function
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos \beta\)	\(=\)	\(\displaystyle \cos \left({\pi - \alpha}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of supplementary
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \pi \cos \alpha + \sin \pi \sin \alpha\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum (Corollary)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({-1}\right) \times \cos \alpha + 0 \times \sin \alpha\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Shape of Sine Function and Shape of Cosine Function
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle -\cos \alpha\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)