How to write "$k$ is equal to any integer" in symbols?

Question

How do you state that $k$ is equal to any integer in the following?

The solutions to this equation $$2\sin(3x)-1=0$$ are $$ \left\{ \begin{array}{ll} x=\dfrac{\pi}{18}+\dfrac{2\pi}{3}k\\[4pt] x=\dfrac{5\pi}{18}+\dfrac{2\pi}{3}k \\ \end{array} \right. $$

Answer 1

If you want to say that $k$ can be any integer and you want to use symbols, then $$\huge k\in \mathbb{Z} $$ is a standard choice. Here $\mathbb{Z}$ means the set of integers and $\in$ means "belongs to" or "in". That is $k\in \mathbb{Z}$ means that $k$ belongs to the set of integers.

Answer 2

You have the equation

$$\sin 3x=\frac12\iff \begin{cases}3x=\frac\pi6+2k\pi\\\text{or}\\3x=\frac{5\pi}6+2k\pi\end{cases}\;\;\;,\;\;\;\;k\in\Bbb Z\iff$$$${}$$

$$\iff x=\frac\pi{18}+\frac23k\pi\;,\;\;\text{or}\;\;x=\frac{5\pi}{18}+\frac23k\pi\;\;,\;\;\;k\in\Bbb Z$$

Answer 3

Hint:

Solving $$ \sin (3x)=\frac{1}{2} $$ we have: $$ x=\frac{\pi}{6}+2k\pi \quad \mbox{or}\quad x=\frac{5\pi}{6}+2k\pi $$ with $k=0,\pm 1,\pm2,\cdots$ is a an integer and the terms$+2k\pi$ represent the periodicity of the $\sin $ function . So ....