Sine and Cosine are Periodic on Reals/Sine

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Theorem

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


SineCos.png


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.

From Primitive of Cosine Function:

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

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 $L$.

$\blacksquare$


Sources