Sine and Cosine of Conjugate Angles

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Theorem

Let $\alpha$ and $\beta$ be conjugate 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 $	$\displaystyle \sin \alpha$	$=$	$\displaystyle \sin \left({2 \pi - \beta}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of conjugate
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sin \left({2 \pi}\right) \cos \beta - \cos \left({2 \pi}\right) \sin \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sine and Cosine of Sum (Corollary)
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle 0 \times \cos \beta - 1 \times \sin \beta$	$\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 $	$\displaystyle \cos \alpha$	$=$	$\displaystyle \cos \left({2 \pi - \beta}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of conjugate
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \cos \left({2 \pi}\right) \cos \beta + \sin \left({2 \pi}\right) \sin \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sine and Cosine of Sum (Corollary)
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle 1 \times \cos \beta + 0 \times \sin \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from Shape of Sine Function and Shape of Cosine Function
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \cos \beta$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

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

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin \alpha\)	\(=\)	\(\displaystyle \sin \left({2 \pi - \beta}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of conjugate
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \left({2 \pi}\right) \cos \beta - \cos \left({2 \pi}\right) \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum (Corollary)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 0 \times \cos \beta - 1 \times \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Shape of Sine Function and Shape of Cosine Function
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle - \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos \alpha\)	\(=\)	\(\displaystyle \cos \left({2 \pi - \beta}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of conjugate
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \left({2 \pi}\right) \cos \beta + \sin \left({2 \pi}\right) \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine of Sum (Corollary)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 1 \times \cos \beta + 0 \times \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Shape of Sine Function and Shape of Cosine Function
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)