Question

The number of elements in the set \(S = \{ \theta \in [-4\pi, 4\pi] : 3 \cos^2(2\theta) + 6 \cos^2(\theta) - 10\cos(2\theta) + 5 = 0 \}\) is______.

Answer 1

Correct Answer: 32

The correct answer is: 32

\(3 \cos^2(2\theta) + 6 \cos^2(\theta) - 10\cos(2\theta) + 5 = 0\)\(-\frac{10(1+cos2θ)}{2}\)

\(3\cos^2(2\theta) + \cos^2(\theta) = 0\)

\(\cos^2(\theta) = 0 \quad \text{or} \quad \cos^2(\theta) =\) \(-\frac{1}{3}\)

As \(\theta \in [0, \pi], \quad \cos(2\theta) = -\frac{1}{3} ⇒ 2 times\)

\(⇒\) \(\theta \in [-4\pi, 4\pi], \quad \cos(2\theta) = -\frac{1}{3}\)\(-\frac{1}{3}\) \(⇒\) 16 times

Similarly, \(\cos(2\theta) = 0\) \(⇒ 16\) times

∴ Total is 32 solutions.