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, we have 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.

Proof 2
By Cosine of Zero is One:
 * $\cos 0 = 1$

By Cosine of 2 is Strictly Negative:
 * $\cos 2 < 0$

Thus by Intermediate Value Theorem/Corollary there exists an $h \in \openint 0 2$ such that:
 * $\cos h = 0$

By Sine of Sum for all $x \in \R$:

By Cosine of Sum for all $x \in \R$:

By Sum of Squares of Sine and Cosine:

Thus for all $x \in \R$:

In particular, $\cos$ is periodic.

By Nonconstant Periodic Function with no Period is Discontinuous Everywhere, $\cos$ has a period $p \in \R_{>0}$.

In view of $\paren 1$ and $\sin h \ne 0$, the periodic elements of $\sin$ are exactly those of $\cos$.

Thus $p$ is also the period of $\sin$.

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 $3.14159 \ldots$.