Question

let $\theta = -\frac{17pi}{4}$. complete parts (a), (b), and (c) below.
(a) sketch $\theta$ in standard position.

Explanation:

Step1: Find coterminal angle

Subtract full rotations ($2\pi = \frac{8\pi}{4}$) to simplify:
$\theta = -\frac{17\pi}{4} + 3\times\frac{8\pi}{4} = -\frac{17\pi}{4} + \frac{24\pi}{4} = \frac{7\pi}{4}$

Step2: Identify angle position

$\frac{7\pi}{4}$ is equivalent to $315^\circ$, which lies in the fourth quadrant, $\frac{\pi}{4}$ radians below the positive x-axis. The negative angle $-\frac{17\pi}{4}$ rotates clockwise 2 full circles ($-4\pi = -\frac{16\pi}{4}$) plus an additional $-\frac{\pi}{4}$, ending at the same terminal side as $\frac{7\pi}{4}$.

Step3: Sketch the angle

Draw the initial side along the positive x-axis. Rotate clockwise $4\frac{1}{4}$ full circles (or directly to the terminal side in the fourth quadrant, $\frac{\pi}{4}$ from the positive x-axis towards the negative y-axis), and mark the terminal side.

Answer:

The angle $\theta = -\frac{17\pi}{4}$ has a terminal side in the fourth quadrant, coinciding with the terminal side of $\frac{7\pi}{4}$ (315°), $\frac{\pi}{4}$ radians below the positive x-axis. The sketch has the initial side on the positive x-axis, with the terminal side pointing to the fourth quadrant at the marked position $\frac{\pi}{4}$ from the x-axis.