Question

One of the principal solutions of \( \sqrt{3} \sec x = -2 \) is equal to:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{2\pi}{3} \)
• C. \( \frac{\pi}{6} \)
• D. \( \frac{5\pi}{6} \)

Hint:

To find principal solutions for trigonometric equations, identify the reference angle and determine the quadrants where the function is positive or negative based on the equation.

Answer 1

The Correct Option is D

Step 1: The initial equation is given as: \[ \sqrt{3} \sec x = -2 \] Step 2: Isolate \( \sec x \): \[ \sec x = \frac{-2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \] Step 3: Substitute \( \sec x \) with \( \frac{1}{\cos x} \): \[ \cos x = -\frac{\sqrt{3}}{2} \] Step 4: Find the principal values of \( x \) for which \( \cos x = -\frac{\sqrt{3}}{2} \). This occurs in the second and third quadrants: \[ x = \frac{5\pi}{6}, \frac{7\pi}{6} \] Step 5: The solution \( \frac{5\pi}{6} \) is among the provided options.