1. Stating the problem: We need to find the principal solution for $\csc \theta = -6$. 2. Recall the definition: $\csc \theta = \frac{1}{\sin \theta}$, so $\csc \theta = -6$ means \[ \frac{1}{\sin \theta} = -6 \] 3. Find $\sin \theta$: Multiplying both sides by $\sin \theta$ and dividing both sides by $-6$, we get \[ \sin \theta = -\frac{1}{6} \] 4. Find the angle $\theta$: We find $\theta$ such that $\sin \theta = -\frac{1}{6}$. 5. Use the inverse sine function: The reference angle is \[ \alpha = \arcsin \frac{1}{6} \] 6. Calculate $\alpha$: Numerically, $\alpha \approx 0.1674$ radians. 7. Find principal solutions: Since sine is negative in the third and fourth quadrants, principal solutions are \[ \theta = \pi + \alpha \approx 3.309 \quad \text{and} \quad \theta = 2\pi - \alpha \approx 6.116 \] 8. So, the principal solutions to $\csc \theta = -6$ on $[0, 2\pi)$ are approximately \[ \theta \approx 3.309 \text{ radians and } \theta \approx 6.116 \text{ radians}. \]