Question

which values for θ have the same reference angles?
○ $\frac{pi}{6}$, $\frac{pi}{3}$, $\frac{5pi}{6}$
○ $\frac{pi}{3}$, $\frac{5pi}{6}$, $\frac{4pi}{3}$
○ $\frac{pi}{2}$, $\frac{5pi}{4}$, $\frac{7pi}{4}$
○ $\frac{pi}{4}$, $\frac{3pi}{4}$, $\frac{7pi}{4}$

Explanation:

Step1: Recall reference - angle formula

For an angle $\theta$ in standard position, if $0\leq\theta\leq2\pi$:

• In the first - quadrant ($0\leq\theta\leq\frac{\pi}{2}$), the reference angle $\theta_{r}=\theta$.
• In the second - quadrant ($\frac{\pi}{2}<\theta\leq\pi$), the reference angle $\theta_{r}=\pi - \theta$.
• In the third - quadrant ($\pi<\theta\leq\frac{3\pi}{2}$), the reference angle $\theta_{r}=\theta-\pi$.
• In the fourth - quadrant ($\frac{3\pi}{2}<\theta\leq2\pi$), the reference angle $\theta_{r}=2\pi - \theta$.

Step2: Analyze option D

• For $\theta=\frac{\pi}{4}$, since it is in the first - quadrant, the reference angle $\theta_{r}=\frac{\pi}{4}$.
• For $\theta = \frac{3\pi}{4}$, which is in the second - quadrant, $\theta_{r}=\pi-\frac{3\pi}{4}=\frac{\pi}{4}$.
• For $\theta=\frac{7\pi}{4}$, which is in the fourth - quadrant, $\theta_{r}=2\pi-\frac{7\pi}{4}=\frac{\pi}{4}$.

Answer:

D. $\frac{\pi}{4},\frac{3\pi}{4},\frac{7\pi}{4}$