Sine of Half-Integer Multiple of Pi
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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.
$\blacksquare$