Question

sketch $\theta = -\frac{8pi}{3}$ in standard position.

Explanation:

Step1: Convert negative angle to positive

First, add $2\pi$ (a full rotation) to the negative angle to find its coterminal positive equivalent.
$\theta = -\frac{8\pi}{3} + 2\pi = -\frac{8\pi}{3} + \frac{6\pi}{3} = -\frac{2\pi}{3}$
Add $2\pi$ again to get a positive coterminal angle:
$\theta = -\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}$

Step2: Identify quadrant of the angle

$\pi < \frac{4\pi}{3} < \frac{3\pi}{2}$, so the angle lies in the third quadrant. Each tick mark on the circle represents $\frac{\pi}{6}$ (since $2\pi$ divided by 12 ticks is $\frac{\pi}{6}$). $\frac{4\pi}{3} = \frac{8\pi}{6}$, which is 8 tick marks counterclockwise from the positive x-axis, or equivalently 4 tick marks clockwise from the positive x-axis (matching the negative angle direction).

Step3: Sketch the angle in standard position

Draw a ray starting at the origin, rotating $\frac{4\pi}{3}$ counterclockwise (or $-\frac{8\pi}{3}$ clockwise) from the positive x-axis, ending at the 8th tick mark on the lower-left portion of the unit circle.

Answer:

The angle $\theta = -\frac{8\pi}{3}$ (coterminal with $\frac{4\pi}{3}$) is a ray starting at the origin, passing through the 8th counterclockwise tick mark (or 4th clockwise tick mark) on the unit circle, located in the third quadrant.