Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For solving trigonometric equations involving \( \csc \theta \), first convert the equation to a quadratic form and solve for \( \csc \theta \), then find the corresponding angles in the given interval.

Answer 1

The Correct Option is C

We are given the equation: \[ \sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0 \] Let \( x = \csc \theta \). The equation becomes: \[ \sqrt{3} x^2 - 2 (\sqrt{3} - 1) x - 4 = 0 \] This is a quadratic equation in \( x \). Solving it using the quadratic formula: \[ x = \frac{-(-2 (\sqrt{3} - 1)) \pm \sqrt{(-2 (\sqrt{3} - 1))^2 - 4 \cdot \sqrt{3} \cdot (-4)}}}{2 \cdot \sqrt{3}} \] Simplifying the discriminant and solving for \( x \), we get two real solutions for \( x \). Thus, there are 2 possible values for \( \csc \theta \), which correspond to 3 solutions for \( \theta \) in the given range. Thus, the number of solutions is 3.