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

Proof
From the proofs of Real Cosine Function is Periodic and Real Sine Function is Periodic, 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.