How can I write $\frac{(4k-15)\pi}8$ as $n+2k\pi$ where $k\in\mathbb Z$ and $n\in(-\pi,\pi]$

Question

How can I write $\frac{(4k-15)\pi}8$ as $n+2k\pi$ where $k\in\mathbb Z$ and $n\in(-\pi,\pi]$

$\boxed{\bf My\,try::}$

$$\begin{align} \frac{(4k-15)\pi}{8}&=\frac{4k\pi-15\pi}{8}\\ &=\frac{4k\pi-16\pi+\pi}{8}\\ &=\frac{4(k\pi-4\pi)+\pi}{8}\\&=\frac{k\pi-4\pi}2+\pi/8 \end{align}$$

then I can't.

Thank you in advance.

Answer 1

If $k$ is even then we can write $k=2\ell$ with $\ell\in\Bbb{Z}$.

$$\frac{4k-15}{8}\pi=\frac{8\ell-15}{8}\pi=(\ell-2)\pi+\frac{\pi}{8}$$

If $k$ is odd then we can write $k=2\ell+1$ with $\ell\in\Bbb{Z}$.

$$\frac{4k-15}{8}\pi=\frac{8\ell-11}{8}\pi=(\ell-2)\pi+\frac{5}{8}\pi$$