Question

check all that apply. $\frac{pi}{6}$ is the reference angle for: a. $\frac{3pi}{6}$ b. $\frac{5pi}{6}$ c. $\frac{8pi}{6}$ d. $\frac{13pi}{6}$

Explanation:

Step1: Recall reference - angle definition

The reference angle is the acute angle between the terminal side of the given angle and the x - axis.

Step2: Analyze option A

For $\theta=\frac{3\pi}{6}=\frac{\pi}{2}$, the reference angle is $\frac{\pi}{2}$ (not $\frac{\pi}{6}$).

Step3: Analyze option B

For $\theta = \frac{5\pi}{6}$, which is in the second - quadrant. The reference angle $\theta_{r}=\pi-\frac{5\pi}{6}=\frac{\pi}{6}$.

Step4: Analyze option C

For $\theta=\frac{8\pi}{6}=\frac{4\pi}{3}$, which is in the third - quadrant. The reference angle $\theta_{r}=\frac{4\pi}{3}-\pi=\frac{\pi}{3}$ (not $\frac{\pi}{6}$).

Step5: Analyze option D

For $\theta=\frac{13\pi}{6}$, which is in the first - quadrant. $\frac{13\pi}{6}=2\pi+\frac{\pi}{6}$, and the reference angle is $\frac{\pi}{6}$.

Answer:

B. $\frac{5\pi}{6}$, D. $\frac{13\pi}{6}$