Sine and Cosine are Periodic on Reals/Pi

Theorem
Let $\sin: \R \to \R$ be the real sine function, and let $\cos: \R: \to \R$ be the real cosine function.

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 proofs of Cosine is Periodic on Reals and Sine is Periodic on Reals, it follows that $\sin$ and $\cos$ are periodic on $\R$ with period $4 \eta \in \R$, where $\eta \in \R$ uniquely defined.

From the discussion in these proofs, it follows that $\pi$ is defined as $\pi := 2 \eta$.

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

Note
Given that we have defined sine and cosine in terms of a power series, it is a plausible proposition to define $\pi$ using the same language.

$\pi$ is, of course, the famous irrational constant $3.14159 \ldots$.