Question

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

• A. 10
• B. 9
• C. 8
• D. 12

Hint:

When simplifying trigonometric equations, look for useful identities to reduce complexity.

Answer 1

The Correct Option is B

Starting with the given equation: \[ 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 = 0 \] Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we rewrite the equation: \[ 3 \cos^4 \theta - 5 \cos^2 \theta - 2(1 - \cos^2 \theta) + 2 = 0 \] Simplifying: \[ 3 \cos^4 \theta - 5 \cos^2 \theta - 2 + 2 \cos^2 \theta + 2 = 0 \] \[ 3 \cos^4 \theta - 3 \cos^2 \theta = 0 \] Factoring: \[ 3 \cos^2 \theta (\cos^2 \theta - 1) = 0 \] Thus, \(\cos^2 \theta = 0\) or \(\cos^2 \theta = 1\), which gives us \(\cos \theta = 0\) or \(\cos \theta = \pm 1\). For \(\cos \theta = 0\), \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\).
For \(\cos \theta = 1\), \(\theta = 0\).
For \(\cos \theta = -1\), \(\theta = \pi\).
Hence, the total number of solutions is 9.