Question

The principal solutions of \(\text{cosec}(θ) = -2\) are...

• A. \(\frac{π}{6},\frac{5π}{6}\)
• B. \(\frac{5π}{6},\frac{7π}{6}\)
• C. \(\frac{7π}{6},\frac{11π}{6}\)
• D. \(\frac{π}{6},\frac{11π}{6}\)

Hint:

Convert to $\sin\theta=-\frac12$ and find angles in $[0,2\pi)$.

Answer 1

The Correct Option is C

Step 1: Use the unit circle
$\sin\theta=-\frac12$ means a $y$-coordinate of $-\frac12$. The points on the unit circle at that height are at $210^{\circ}$ and $330^{\circ}$.
These equal $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$. Option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C)}} \]