Sine of Multiple of Pi Plus and Half

Theorem

Let $x \in \R$ be a real number.

Let $\sin x$ be the sine of $x$.

Then:

$\forall n \in \Z: \sin \left({n + \dfrac 1 2}\right) \pi = \left({-1}\right)^n$

or alternatively:

	$\displaystyle $	$\displaystyle \forall m \in \Z:$	$\displaystyle $	$\displaystyle \sin \left({2 m + \dfrac 1 2}\right) \pi$	$=$	$\displaystyle $	$\displaystyle 1$	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle \forall m \in \Z:$	$\displaystyle $	$\displaystyle \sin \left({2 m - \dfrac 1 2}\right) \pi$	$=$	$\displaystyle $	$\displaystyle -1$	$\displaystyle $	$\displaystyle $

From the discussion of Sine and Cosine are Periodic on Reals, we have that:

$\sin \left({x + \dfrac \pi 2}\right) = \cos x$

The result then follows directly from the Cosine of Multiple of Pi.

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \forall m \in \Z:\)	\(\displaystyle \)	\(\displaystyle \sin \left({2 m + \dfrac 1 2}\right) \pi\)	\(=\)	\(\displaystyle \)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \forall m \in \Z:\)	\(\displaystyle \)	\(\displaystyle \sin \left({2 m - \dfrac 1 2}\right) \pi\)	\(=\)	\(\displaystyle \)	\(\displaystyle -1\)	\(\displaystyle \)	\(\displaystyle \)