Quarter-Turn Identities and Periodicity of Sine and Cosine

In the setting of The Real Numbers: Standing Notation and Background, let $\cos$ and $\sin$ be the cosine and sine functions from $\mathbb{R}$ to $\mathbb{R}$, let $\pi$ be the real number of The Number Pi (i), and let $\mathbb{Z}$ be the set of integers. The number $2=1+1$ is positive, hence invertible, by claim 8 of Elementary Order Arithmetic in an Ordered Field, and we write $\tfrac{\pi}{2}=\pi\cdot2^{-1}$. Then the following hold for every $x\in\mathbb{R}$.

1. (Quarter turn)(i)

$$\cos\bigl(x+\tfrac{\pi}{2}\bigr)=-\sin x, \qquad \sin\bigl(x+\tfrac{\pi}{2}\bigr)=\cos x .$$

2. (Half turn)(ii)

$$\cos(x+\pi)=-\cos x, \qquad \sin(x+\pi)=-\sin x .$$

3. (Period)(iii)

$$\cos(x+2\pi)=\cos x, \qquad \sin(x+2\pi)=\sin x .$$

4. (Integer multiples of the period)(iv) For every $m\in\mathbb{Z}$,

$$\cos(x+2\pi m)=\cos x, \qquad \sin(x+2\pi m)=\sin x ;$$

in particular $\cos(2\pi m)=1$ and $\sin(2\pi m)=0$.

5. (Positivity of the sine on a half period)(v) If $0<x<\pi$, then $0<\sin x$.