which values for $\theta$ have the same reference angles?\n$\frac{pi}{6},\frac{pi}{3},\frac{5pi}{6}$\n$\frac{…

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: Calculate reference angles for each option

Option 1:

• For $\theta_1=\frac{\pi}{6}$ (first - quadrant), $\theta_{r1}=\frac{\pi}{6}$.
• For $\theta_2 = \frac{\pi}{3}$ (first - quadrant), $\theta_{r2}=\frac{\pi}{3}$.
• For $\theta_3=\frac{5\pi}{6}$ (second - quadrant), $\theta_{r3}=\pi-\frac{5\pi}{6}=\frac{\pi}{6}$.

Option 2:

• For $\theta_1=\frac{\pi}{3}$ (first - quadrant), $\theta_{r1}=\frac{\pi}{3}$.
• For $\theta_2=\frac{5\pi}{6}$ (second - quadrant), $\theta_{r2}=\pi - \frac{5\pi}{6}=\frac{\pi}{6}$.
• For $\theta_3=\frac{4\pi}{3}$ (third - quadrant), $\theta_{r3}=\frac{4\pi}{3}-\pi=\frac{\pi}{3}$.

Option 3:

• For $\theta_1=\frac{\pi}{2}$ (on the y - axis), reference angle is not defined in the usual sense.
• For $\theta_2=\frac{5\pi}{4}$ (third - quadrant), $\theta_{r2}=\frac{5\pi}{4}-\pi=\frac{\pi}{4}$.
• For $\theta_3=\frac{7\pi}{4}$ (fourth - quadrant), $\theta_{r3}=2\pi-\frac{7\pi}{4}=\frac{\pi}{4}$.

Option 4:

• For $\theta_1=\frac{\pi}{4}$ (first - quadrant), $\theta_{r1}=\frac{\pi}{4}$.
• For $\theta_2=\frac{3\pi}{4}$ (second - quadrant), $\theta_{r2}=\pi-\frac{3\pi}{4}=\frac{\pi}{4}$.
• For $\theta_3=\frac{7\pi}{4}$ (fourth - quadrant), $\theta_{r3}=2\pi-\frac{7\pi}{4}=\frac{\pi}{4}$.

Answer:

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