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Trigonometric Equations
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Solve the equation (2\sin^2\theta + \sqrt{3}\sin\theta - 1 = 0) for (0 \le \theta \le 2\pi).
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A
\(\theta = \frac{\pi}{3}, \frac{5\pi}{3}\)
💡 Explanation:
Factoring the equation, we get ((2\sin\theta - 1)(\sin\theta + 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{1}{2}) or (\sin\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{3}, \frac{5\pi}{3}).