# Sine and Cosine are Periodic on Reals/Corollary

## Corollary to Sine and Cosine are Periodic on Reals

$\map \cos {x + \pi} = -\cos x$
$\map \sin {x + \pi} = -\sin x$

$\cos x$ is strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ and strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
$\sin x$ is strictly positive on the interval $\openint 0 \pi$ and strictly negative on the interval $\openint \pi {2 \pi}$

## Proof

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

Thus:

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

It follows directly that:

$\forall x \in \closedint {-\dfrac \pi 2} {\dfrac \pi 2}: \cos x \ge 0$

Hence:

$\forall x \in \closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}: \cos x \le 0$

The result for $\sin x$ follows similarly, or we can use:

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

$\blacksquare$