If $\tan\left(\frac{2\pi}{3} - x\right) = \frac{\sin\frac{2\pi}{3} - \sin x}{\cos\frac{2\pi}{3} - \cos x}$ where $0 < x < \frac{3\pi}{2}$, and the values of $x$ are $x_1$ and $x_2$, then the value of $\frac{12}{\pi}|x_2 - x_1|$ is
Step-by-Step Solution
Key Concept: Convert tangent to cotangent using complementary angle identities, then solve the resulting trigonometric equation systematically.
Starting with $\tan\left(\frac{2\pi}{3} - x\right)$, we convert using sine and cosine formulas involving $\frac{\pi}{3}$ and $\frac{x}{2}$. After simplification, we obtain $\cot\left(\frac{\pi}{3} - \frac{x}{2}\right) = \tan\left(\frac{\pi}{6} + \frac{x}{2}\right)$. Solving the resulting equation gives $x = \frac{-2n\pi}{3} + \frac{\pi}{9}$, and for specific solutions, $|x_2 - x_1| = 8$.
Correct Answer: 8