Choose the correct answer from the drop-down menu. Solve \( \sin 2 \theta=\cos \theta \) on the interval \( [0,2 \pi) \). \( \theta= \) Select... \( 0, \frac{\pi}{4}, \frac{\pi}{2} \), or \( \frac{3 \pi}{4} \)

To solve the equation \( \sin 2\theta = \cos \theta \), we can use the double angle identity for sine: \( \sin 2\theta = 2\sin \theta \cos \theta \). This turns our equation into \( 2 \sin \theta \cos \theta = \cos \theta \). If we factor out \( \cos \theta \), we get: \[ \cos \theta (2 \sin \theta - 1) = 0 \] This gives us two cases to solve: 1. \( \cos \theta = 0 \) 2. \( 2 \sin \theta - 1 = 0 \) or \( \sin \theta = \frac{1}{2} \) For the first case \( \cos \theta = 0 \), this happens at \( \theta = \frac{\pi}{2} \) within the interval \( [0, 2\pi) \). For the second case \( \sin \theta = \frac{1}{2} \), this occurs at \( \theta = \frac{\pi}{6}, \frac{5\pi}{6} \). However, since we need values that are among the options: - \( \theta = 0 \) - \( \theta = \frac{\pi}{4} \) - \( \theta = \frac{\pi}{2} \) - \( \theta = \frac{3\pi}{4} \) The correct answer is \( \theta = \frac{\pi}{2} \). Now, if you check through your answer choices, it seems like the down menu doesn't have a glaringly correct option listed based on traditional solutions. Perhaps you meant the larger scope of values in separate angles, but with your request, your answer is: \( \theta= \frac{\pi}{2} \).