Sine and Cosine are Periodic on Reals/Pi/Proof 1

Theorem

The real number $\pi$ (called pi, pronounced pie) is uniquely defined as:

$\pi := \dfrac p 2$

where $p \in \R$ is the period of $\sin$ and $\cos$.

Proof

From the Real Cosine Function is Periodic and Real Sine Function is Periodic, we have that $\cos x$ and $\sin x$ are periodic on $\R$ with the same period.

If we denote the period of $\cos x$ and $\sin x$ as $p$, it follows that $\pi = \dfrac p 2$ is uniquely defined.

$\blacksquare$