Question

If $ \theta \in \left[ \frac{7\pi}{6}, \frac{4\pi}{3} \right] $, then the number of solutions of $$ \sqrt{3} \csc^2\theta - 2(\sqrt{3} - 1) \csc\theta - 4 = 0, $$ is equal to:

• A. 6
• B. 8
• C. 10
• D. 7

Hint:

In trigonometric equations, look for multiple possible values of \( \sin\theta \) and count the number of solutions within the given range.

Answer 1

The Correct Option is A

\[ \csc\theta = \frac{2(\sqrt{3}-1) \pm \sqrt{16 + 8\sqrt{3}}}{2\sqrt{3}} \] \[ = \frac{2(\sqrt{3}-1) \pm \sqrt{16 + 8\sqrt{3}}}{2\sqrt{3}} \] Thus, \[ \csc\theta = 2 \text{ or } \csc\theta = -\frac{2}{\sqrt{3}} \] Therefore, \[ \sin\theta = \frac{1}{2} \text{ or } \sin\theta = -\frac{\sqrt{3}}{2} \] Thus, \( \sin\theta = \frac{1}{2} \) has 3 solutions, and \( \sin\theta = -\frac{\sqrt{3}}{2} \) has 3 solutions in the interval \( \left[ \frac{7\pi}{6}, \frac{4\pi}{3} \right] \). 
Thus, the number of solutions is 6.