# Sine and Cosine are Periodic on Reals/Sine

## Theorem

The real sine function is periodic with the same period as the real cosine function.

## Proof 1

Since Real Cosine Function is Periodic, let $K$ be its period.

Then:

$\cos K = \map \cos {0 + K} = \cos 0$

Because Cosine of Zero is One:

$\cos K = 1$

Furthermore:

 $\ds \cos^2 K + \sin^2 K$ $=$ $\ds 1$ Sum of Squares of Sine and Cosine $\ds \sin^2 K$ $=$ $\ds 0$ $\cos K = 1$ $\ds \sin K$ $=$ $\ds 0$

Then, the following holds:

 $\ds \map \sin {x + K}$ $=$ $\ds \sin x \cos K + \cos x \sin K$ Sine of Sum $\ds$ $=$ $\ds \sin x \cdot 1 + \cos x \cdot 0$ $\cos K = 1$; $\sin K = 0$ $\ds$ $=$ $\ds \sin x$

Thus $\sin$ is periodic with some period $L \leq K$.

$\Box$

The following also hold:

 $\ds \sin L$ $=$ $\ds \map \sin {0 + L}$ $\ds$ $=$ $\ds \sin 0$ Period of Periodic Real Function $\ds$ $=$ $\ds 0$ Sine of Zero is Zero $\ds \cos L$ $=$ $\ds \bigvalueat { \dfrac{\d \sin x} {\d x} } {x \mathop = L}$ Derivative of Sine Function $\ds$ $=$ $\ds \lim_{h \mathop \to 0} \dfrac{\map \sin {L + h} - \map \sin L}{h}$ Derivative of Real Function at Point $\ds$ $=$ $\ds \lim_{h \mathop \to 0} \dfrac{\map \sin {0 + h} - \map \sin 0}{h}$ Period of Periodic Real Function $\ds$ $=$ $\ds \bigvalueat { \dfrac{\d \sin x} {\d x} } {x \mathop = 0}$ Derivative of Real Function at Point $\ds$ $=$ $\ds \cos 0$ Derivative of Sine Function $\ds$ $=$ $\ds 1$ Cosine of Zero is One

So we may conclude:

 $\ds \map \cos {x + L}$ $=$ $\ds \cos x \cos L - \sin x \sin L$ Cosine of Sum $\ds$ $=$ $\ds \cos x \cdot 1 - \sin x \cdot 0$ $\cos L = 1$ and $\sin L = 0$ $\ds$ $=$ $\ds \cos x$

Therefore the period of cosine is at most that of sine:

$K \leq L$

But if $K \leq L \leq K$ then:

$K = L$

$\blacksquare$

## Proof 2

Since Real Cosine Function is Periodic, let $L$ be its period.

$\ds \int \cos x \rd x = \sin x + C$

for any constant $C$.

Therefore $\sin x$ is a Primitive of $\cos x$, for the special case of $C = 0$.

From Primitive of Periodic Real Function, it follows that $\sin x$ is periodic with period $L$.

$\blacksquare$