Find the period of $|\sin x| + |\cos x - 1|$

Question

I want to find the period of this function , I know that the period of $|\sin x| + |\cos x|$ is $π/2$ but what can a $-1$ do?

Answer 1

$$|\sin(x)|+|2 \sin^2(x/2)|$$

The first one: $\pi$

The second: $2 \pi$

The total $2 \pi$

Answer 2

$$x\mapsto |\sin(x)|$$ is $\pi-$periodic.

$$x\mapsto |\cos x-1|$$ is $2\pi-$periodic,

therefore the sum is $2\pi-$periodic.

Answer 3

$$|\sin x| + |1 -\cos x| = \left\{ \begin{array}{ll} 2 \sin(x/2)\sin(x/2 + \pi/4) & \text{ if $0 \le x \le \pi$} \\ 2 \sin(x/2)\sin(x/2 - \pi/4) & \text{ if $\pi \le x \le 2\pi$} \end{array} \right. \text{ is $2\pi$-peridoic.}$$