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If \(2\sin^{2} \theta = 1 + \cos \theta, 0° \leq \theta ...

If 2\sin^{2} \theta = 1 + \cos \theta, 0° \leq \theta \leq 90°, find the value of \theta

A.

90°

B.

60°

C.

45°

D.

30°

Correct answer is B

2\sin^{2} \theta = 1 + \cos \theta

2 ( 1 - \cos^{2} \theta) = 1 + \cos \theta

2 - 2\cos^{2} \theta = 1 + \cos \theta

0 = 1 - 2 + \cos \theta + 2\cos^{2} \theta

2\cos^{2} \theta + \cos \theta - 1 = 0

Factorizing, we have

(\cos \theta + 1)(2\cos \theta - 1) = 0

Note: In the range, 0° \leq \theta \leq 90°, all trig functions are positive, so we consider

2\cos \theta = 1 \implies \cos \theta = \frac{1}{2}

\theta = 60°.