If \(2\sin^{2} \theta = 1 + \cos \theta, 0° \leq \theta \leq 90°\), find the value of \(\theta\).
- 90°
-
60°
- 45°
- 30°
The correct answer is: B
Explanation
\(2\sin^{2} \theta = 1 + \cos \theta\)
\(2 ( 1 - \cos^{2} \theta) = 1 + \cos \theta\)
\(2 - 2\cos^{2} \theta = 1 + \cos \theta\)
\(0 = 1 - 2 + \cos \theta + 2\cos^{2} \theta\)
\(2\cos^{2} \theta + \cos \theta - 1 = 0\)
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
\(2\cos \theta = 1 \implies \cos \theta = \frac{1}{2}\)
\(\theta = 60°\).