How many values of \theta, where -\pi \lt \theta \lt \pi, satisfy <strong>BOTH</strong> the equations \cot~\theta=-\sqrt{3} and \csc~\theta=-2 simultaneously?
- A. 4
- B. 2
- C. 1 ✓
- D. None
Correct Answer: C. 1
Explanation
The condition \cot\theta = -\sqrt{3} implies \theta is in Quadrant II or IV. The condition \csc\theta = -2 (or \sin\theta = -1/2) implies \theta is in Quadrant III or IV. The only quadrant satisfying both is Quadrant IV. In the interval (-\pi, \pi), the single common value is \theta = -\frac{\pi}{6}. Thus, there is exactly 1 solution.