Sine and Cosine of Complementary Angles

Theorem

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

Then:

$\sin \alpha = \cos \beta$

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

Proof 1

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin \alpha\)	\(=\)	\(\displaystyle \sin \left({\frac \pi 2 - \beta}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of complementary
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \frac \pi 2 \cos \beta - \cos \frac \pi 2 \sin \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	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
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \beta\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Similarly and alternatively:

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

$\blacksquare$

Proof 2

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin \left({\frac \pi 2 - x}\right)\)	\(=\)	\(\displaystyle - \sin \left({x - \frac \pi 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine Function is Odd
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos \left({x - \frac \pi 2 + \frac \pi 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine are Periodic on Reals
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \cos x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos \left({\frac \pi 2 - x}\right)\)	\(=\)	\(\displaystyle \cos \left({x - \frac \pi 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Cosine Function is Even
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin \left({x - \frac \pi 2 + \frac \pi 2}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sine and Cosine are Periodic on Reals
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sin x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$