How to find $\arccos(\cos\frac{15\pi}{11})$? How to find $\arccos(\cos\frac{15\pi}{11})$?
I'm completely lost. $\frac{15\pi}{11}$ isn't on the unit circle so how do I find the cosine of it?

As $\cos x=\cos A\implies x=2n\pi\pm A$ where $n$ is any integer
The general value of $$\arccos\left(\cos\frac{15\pi}{11}\right)=2n\pi\pm \frac{15\pi}{11}$$
Based on the definition of principal value inverse cosine ratio, $\displaystyle0\le \arccos x\le\pi$
Taking '+'  sign, $\displaystyle0\le 2n\pi+\frac{15\pi}{11}\le\pi\implies 0\le 22n+15\le 11\implies0\le n<0$ which is impossible
Taking '-'  sign, $\displaystyle0\le 2n\pi-\frac{15\pi}{11}\le\pi\implies 0\le 22n-15\le 11\implies 1\le n\le1\implies n=1$ 
So, the principal value is $\displaystyle 2\pi-\frac{15\pi}{11}=??$

Alternatively, as we know $\cos\left(\pi\pm y\right)=-\cos y$
$$\cos\left(\frac{15\pi}{11}\right)=\cos\left(\pi+\frac{4\pi}{11}\right)=-\cos\left(\frac{4\pi}{11}\right)=\cos\left(\pi-\frac{4\pi}{11}\right)=\cos\left(\frac{7\pi}{11}\right)$$
