why inverse trigonometric function DNE

Question

Determine the exact value

$\arccos\left[\sec\left(\dfrac{7\pi}{6}\right)\right]$ and $\text{arcsec}\left[\sin\left(\dfrac{13\pi}{6}\right)\right]$

Why does the exact value of these two questions not exist?

Answer 1

Just working with the first one.

$\sec(7\pi/6) =\frac1{\cos(7\pi/6)} =-\frac{2}{\sqrt(3)} < -1 $ so $\cos(\sec(7\pi/6)) $ does not exist (unless you go into the complex domain).

In general, since $-1 \le \cos(x) \le 1$, we have $\sec(x) \ge 1$ or $\sec(x) \le -1$.

The only times when $\arccos(\sec(x)) $ exists in the reals are when $\sec(x) =\pm 1 $, or $x = n\pi $.