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