Sine of Half-Integer Multiple of Pi

Theorem

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

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

Then:

$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$

or:

\(\ds \forall m \in \Z: \, \)

\(\ds \map \sin {2 m + \dfrac 1 2} \pi\)

\(=\)

\(\ds 1\)

\(\ds \forall m \in \Z: \, \)

\(\ds \map \sin {2 m - \dfrac 1 2} \pi\)

\(=\)

\(\ds -1\)

Proof

From the discussion of Sine and Cosine are Periodic on Reals:

$\map \sin {x + \dfrac \pi 2} = \cos x$

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

$\blacksquare$