Question

sketch $\theta =-\frac{4pi}{3}$ in standard position. then show its reference angle.

Explanation:

Step1: Analyze the given angle

The given angle is $\theta = -\frac{4\pi}{3}$. A negative - angle means we rotate clockwise. First, find a positive coterminal angle. Add $2\pi$ to $-\frac{4\pi}{3}$: $-\frac{4\pi}{3}+2\pi=-\frac{4\pi}{3}+\frac{6\pi}{3}=\frac{2\pi}{3}$.

Step2: Determine the quadrant

The angle $\frac{2\pi}{3}$ is in the second - quadrant since $\frac{\pi}{2}<\frac{2\pi}{3}<\pi$.

Step3: Calculate the reference angle

For an angle $\theta$ in the second - quadrant, the reference angle $\theta_{r}$ is given by $\theta_{r}=\pi-\theta$. Substitute $\theta = \frac{2\pi}{3}$ into the formula: $\theta_{r}=\pi-\frac{2\pi}{3}=\frac{\pi}{3}$.

Answer:

The reference angle is $\frac{\pi}{3}$.