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: Convert
\(\text{cosec}\,\theta=-2\Rightarrow\sin\theta=-\frac12\).

Step 2: Find the quadrants
Sine is negative in the third and fourth quadrants. The reference angle is \(\frac{\pi}{6}\).

Step 3: Principal solutions
Third quadrant: \(\pi+\frac{\pi}{6}=\frac{7\pi}{6}\). Fourth quadrant: \(2\pi-\frac{\pi}{6}=\frac{11\pi}{6}\).
So the answer is \(\frac{7\pi}{6},\frac{11\pi}{6}\).

Final Answer:
The principal solutions are \(\frac{7\pi}{6},\frac{11\pi}{6}\), option (C). \[ \boxed{\text{(C)}} \]