### Instruction: Find $\sec \frac{5 \pi}{3}.$.### Response: I know that $\sec$ is the reciprocal of $\cos$, so I can rewrite this as $\frac{1}{\cos \frac{5 \pi}{3}}.$
To find $\cos \frac{5 \pi}{3}$, I can use the unit circle and the reference angle.
The angle $\frac{5 \pi}{3}$ is in the fourth quadrant, and its reference angle is $\frac{\pi}{3}.$
The cosine of an angle in the fourth quadrant is positive, and the cosine of $\frac{\pi}{3}$ is $\frac{1}{2}.$
So, $\cos \frac{5 \pi}{3} = \frac{1}{2}.$
Therefore, $\sec \frac{5 \pi}{3} = \frac{1}{\cos \frac{5 \pi}{3}} = \frac{1}{\frac{1}{2}} = 2.$