Question

If $ \theta \in [-2\pi,\ 2\pi] $, then the number of solutions of $$ 2\sqrt{2} \cos^2\theta + (2 - \sqrt{6}) \cos\theta - \sqrt{3} = 0 $$ is:

• A. 12
• B. 6
• C. 8
• D. 10

Hint:

Transform trig equations into algebraic form using identities to find roots easily.

Answer 1

The Correct Option is C

Let \( x = \cos\theta \). Then the equation becomes: \[ 2\sqrt{2}x^2 + (2 - \sqrt{6})x - \sqrt{3} = 0 \] Solve the quadratic: \[ (2x - \sqrt{3})(\sqrt{2}x + 1) = 0 \Rightarrow x = \frac{\sqrt{3}}{2},\ -\frac{1}{\sqrt{2}} \] Each valid cosine value gives 4 solutions in \( [-2\pi,\ 2\pi] \), so total = 8 solutions.