Answer
Exact value of $\sec \frac{17\pi}{3}$ = $2$
Work Step by Step
To find exact value of $\sec \frac{17\pi}{3}$, let's find its co-terminal angle between $0$ and $2\pi$ first-
$\frac{17\pi}{3}$ = $2\times2\pi + \frac{5\pi}{3}$
Therefore, co-terminal angle of $\frac{17\pi}{3}$ between $0$ and $2\pi$ = $\frac{5\pi}{3}$
$\sec \frac{17\pi}{3}$= $\sec \frac{5\pi}{3}$
( Trigonometric functions of co-terminal angles are same)
As $\sec \frac{17\pi}{3}$ terminates in quadrant IV, its $\sec$ will be positive.
The reference angle of $\frac{5\pi}{3}$ = $2\pi - \frac{5\pi}{3}$ = $\frac{\pi}{3}$
Therefore by reference angle theorem-
$\sec \frac{5\pi}{3}$ = $\sec\frac{\pi}{3}$ = $2$
Hence
$\sec \frac{17\pi}{3}$= $\sec \frac{5\pi}{3}$ = $2$
