$\sin(z)=0$ periodic of $2k\pi$ or $k\pi$?

Question

is the result of $\sin(z)$$=$ $0$

$k\pi $ , or $2k\pi$

?

and it's the same for $\cos(z)$ $=0$

$k \frac{\pi}{2}$

?

Answer 1

It is $$\sin(x)=0$$ if $$x=k\pi$$ where $k$ is an integer number. And $$2k\pi$$ is the period of the function.

Answer 2

$\sin z = 0 \iff z = n \pi \, (n \in \mathbb{Z})$

Answer 3

Even in complex numbers, those are only solutions for $\sin(z)=0$. $$\sin(z) := \frac{e^{iz}-e^{-iz}}{2i} = 0 \iff e^{iz}=e^{-iz} \iff \exists_{k\in\mathbb{Z}} \,iz+2ik\pi=-iz \iff \exists_{k\in\mathbb{Z}} \,z=k\pi$$ Try the same for cosine.