Sine and Cosine are Periodic on Reals/Pi

Theorem
Let the period of the real sine function $\sin$ and the real cosine function $\cos$ be $4 \eta$ for $\eta \in \R_{>0}$.

Then we define the real number $\pi$ (called pi, pronounced pie) as:


 * $\pi := 2 \eta$

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 $4 \eta$.

We then set $\pi := 2 \eta$.

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 $3.14159 \ldots$.