If an angle is in the third quadrant and its reference angle is (\frac{2\pi}{11}), what is the principal angle?

Correct answer: (\frac{13\pi}{11})

The direct answer is option A: \(\frac{13\pi}{11}\). A reference angle is the small positive angle between the terminal arm and the nearest x-axis. In the third quadrant, the principal angle is found by adding π to the reference angle, because the angle has already passed the negative x-axis. Thus, \(\theta=\pi+\frac{2\pi}{11}=\frac{11\pi}{11}+\frac{2\pi}{11}=\frac{13\pi}{11}\). Option A is correct and lies between π and \(\frac{3\pi}{2}\), the third-quadrant range. Option B, \(\frac{9\pi}{11}\), is less than π, so it lies in the second quadrant. Option C, \(\frac{20\pi}{11}\), is larger than \(\frac{3\pi}{2}\), so it lies in the fourth quadrant rather than the third. Option D, \(\frac{15\pi}{11}\), corresponds to a different reference angle, \(\frac{4\pi}{11}\), not \(\frac{2\pi}{11}\). Memory cue: QIII means π plus the reference angle.