Which values for \( \theta \) have the same reference angles? \( \frac{\pi}{6}, \frac{\pi}{3}, \frac{5 \pi}{6} \) \( \frac{\pi}{3}, \frac{5 \pi}{6}, \frac{4 \pi}{3} \) \( \frac{\pi}{2}, \frac{5 \pi}{4}, \frac{7 \pi}{4} \) \( \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4} \)

To find which values of \( \theta \) have the same reference angle, we first need to identify the reference angle for each value in radians. The reference angle is the acute angle made with the x-axis. 1. For \( \frac{\pi}{6} \) (30 degrees), the reference angle is \( \frac{\pi}{6} \). 2. For \( \frac{\pi}{3} \) (60 degrees), the reference angle is \( \frac{\pi}{3} \). 3. For \( \frac{5 \pi}{6} \) (150 degrees), the reference angle is the supplementary angle: \( \pi - \frac{5 \pi}{6} = \frac{\pi}{6} \). Now for the second set: 1. \( \frac{\pi}{3} \) has a reference angle of \( \frac{\pi}{3} \). 2. \( \frac{5 \pi}{6} \) has a reference angle of \( \frac{\pi}{6} \). 3. \( \frac{4 \pi}{3} \) has a reference angle of \( \frac{4 \pi}{3} - \pi = \frac{\pi}{3} \). Next set: 1. \( \frac{\pi}{2} \) (90 degrees) has a reference angle of \( \frac{\pi}{2} \). 2. \( \frac{5 \pi}{4} \) (225 degrees) has a reference angle of \( \frac{5 \pi}{4} - \pi = \frac{\pi}{4} \). 3. \( \frac{7 \pi}{4} \) (315 degrees) has a reference angle of \( 2\pi - \frac{7 \pi}{4} = \frac{\pi}{4} \). Lastly: 1. \( \frac{\pi}{4} \) has a reference angle of \( \frac{\pi}{4} \). 2. \( \frac{3 \pi}{4} \) has a reference angle of \( \pi - \frac{3 \pi}{4} = \frac{\pi}{4} \). 3. \( \frac{7 \pi}{4} \) again has a reference angle of \( \frac{\pi}{4} \). From these calculations, we find: - \( \frac{\pi}{6} \) and \( \frac{5 \pi}{6} \) have the same reference angle \( \frac{\pi}{6} \). - \( \frac{\pi}{3} \) and \( \frac{4 \pi}{3} \) share the reference angle \( \frac{\pi}{3} \). - \( \frac{5 \pi}{4} \) and \( \frac{7 \pi}{4} \) both share the reference angle \( \frac{\pi}{4} \). - \( \frac{\pi}{4} \), \( \frac{3 \pi}{4} \), and \( \frac{7 \pi}{4} \) also share the reference angle \( \frac{\pi}{4} \). Thus, the angles \( \frac{\pi}{6} \) and \( \frac{5 \pi}{6} \) both share the same reference angle. The pairs that share the same reference angles are: - \(\frac{\pi}{6}\) and \( \frac{5 \pi}{6} \) - \(\frac{\pi}{3}\) and \( \frac{4 \pi}{3}\) - \(\frac{5 \pi}{4}\) and \(\frac{7 \pi}{4}\) - \(\frac{\pi}{4}\), \( \frac{3 \pi}{4}\), and \( \frac{7 \pi}{4}\) all have the same reference angle of \( \frac{\pi}{4} \). So to summarize, the reference angle pairs are: 1. \( \frac{\pi}{6}, \frac{5 \pi}{6} \) (Reference Angle \(= \frac{\pi}{6}\)). 2. \( \frac{\pi}{3}, \frac{4 \pi}{3} \) (Reference Angle \(= \frac{\pi}{3}\)). 3. \( \frac{5 \pi}{4}, \frac{7 \pi}{4} \) (Reference Angle \(= \frac{\pi}{4}\)). 4. \( \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{7 \pi}{4}\) (Reference Angle \(= \frac{\pi}{4}\)).