Question

{-1} (cos(\frac{7\pi }{4} ))" alt="cos^{-1} (cos(\frac{7\pi }{4} ))" align="absmiddle" class="latex-formula">

Answer 1

For starters,

$\cos\dfrac{7\pi}4=\cos\left(2\pi-\dfrac\pi4\right)=\cos\left(-\dfrac\pi4\right)=\cos\dfrac\pi4=\dfrac1{\sqrt2}$

The second equality follows from the fact that $\cos x$ has a period of $2\pi$, and the equality after that follows from the parity of $\cos x$ (i.e. $\cos(-x)=\cos x$).

Then

$\cos^{-1}\left(\cos\dfrac{7\pi}4\right)=\cos^{-1}\left(\cos\dfrac\pi4\right)=\cos^{-1}\dfrac1{\sqrt2}=\boxed{\dfrac\pi4}$