If \(2\sin^{2}\theta = 1 + \cos \theta, 0° \leq \theta \leq 90°\), find \(\theta\).
FURTHER MATHEMATICS
WAEC 2013
If \(2\sin^{2}\theta = 1 + \cos \theta, 0° \leq \theta \leq 90°\), find \(\theta\).
- A. 30°
- B. 45°
- C. 60°
- D. 90°
Correct Answer: C. 60°
Explanation
\(2\sin^{2}\theta = 1 + \cos \theta \implies 2(1 - \cos^{2}\theta) = 1 + \cos \theta\)
\(2 - 2\cos^{2}\theta = 1 + \cos \theta\)
\(2 - 2\cos^{2}\theta - 1 - \cos \theta = 0\)
\(2\cos^{2}\theta + \cos \theta - 1 = 0\)
\(2\cos^{2}\theta + 2\cos\theta - \cos \theta - 1 = 0 \implies 2\cos \theta(\cos \theta + 1) - 1(\cos \theta + 1) = 0\)
\((2\cos \theta - 1)(\cos \theta + 1) = 0 \implies \cos \theta = \frac{1}{2} \)
\(\theta = \cos^{-1} \frac{1}{2} = 60°\)
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