The sum of all values of \theta \in [0, 2\pi] satisfying 2 sin \theta = cos 2\theta and 2 cos \theta = 3 sin \theta is 2 2
Step-by-Step Solution
Key Concept: Apply the core result for solutions of trigonometric equations and simplify using the given constraints.
2 sin \theta = cos 2\theta 2 2 2 (3) 2 sin \theta = 1 - 2 sin \theta 2 4 sin \theta = 1 1 2 sin \theta = 4 1 sin \theta = $\pm$ 2 2 2 cos \theta = 3 sin \theta 2 2 - 2 sin \theta + 3 sin \theta - 2 = 0 (2 sin \theta - 1)(2 sin \theta - 2) = 0 1 sin \theta = 2 so common equation which satisfy both equations is sin \theta = 1 2 \pi 5\pi \theta = , (\theta \in [0, 2\pi]) 6 6 Sum = \pi
Correct Answer: 3