What are the solutions of \( \sin 2 \theta=\sqrt{2} \cos \theta \) on the interval \( [0,2 \pi) \) ? Select 4 choice(s) \( \square \frac{\pi}{4} \) \( \square \frac{3 \pi}{2} \) \( \square \)

To solve the equation \( \sin 2\theta = \sqrt{2} \cos \theta \), we can use the double angle identity for sine, which gives us \( \sin 2\theta = 2 \sin \theta \cos \theta \). This transforms the equation into: \[ 2 \sin \theta \cos \theta = \sqrt{2} \cos \theta. \] Assuming \( \cos \theta \neq 0 \), we can divide both sides by \( \cos \theta \): \[ 2 \sin \theta = \sqrt{2}. \] This simplifies to: \[ \sin \theta = \frac{\sqrt{2}}{2}. \] The solutions to this equation are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) within the interval \( [0, 2\pi) \). Now, let’s also consider the condition \( \cos \theta = 0 \). This condition gives us: \[ \theta = \frac{\pi}{2} \text{ and } \theta = \frac{3\pi}{2}. \] Combining both scenarios, the final solutions for the equation \( \sin 2\theta = \sqrt{2} \cos \theta \) on the interval \( [0, 2\pi) \) are: - \( \frac{\pi}{4} \) - \( \frac{3\pi}{4} \) - \( \frac{\pi}{2} \) - \( \frac{3\pi}{2} \) Thus, you can select: \( \square \frac{\pi}{4} \) \( \square \frac{3 \pi}{4} \) \( \square \frac{\pi}{2} \) \( \square \frac{3 \pi}{2} \)