To solve for the exact value of $\csc(150^\circ)$ in simplest form with a rational denominator, follow these steps:

Step 1: Relationship between cosecant and sine

$\csc \theta = \frac{1}{\sin \theta}$ Thus, $\csc(150^\circ) = \frac{1}{\sin(150^\circ)}$.

Step 2: Find $\sin(150^\circ)$

Using the unit circle or trigonometric reference angles: $\sin(150^\circ) = \sin(180^\circ - 30^\circ) = \sin(30^\circ),$ and from known values, $\sin(30^\circ) = \frac{1}{2}$.

So: $\sin(150^\circ) = \frac{1}{2}.$

Step 3: Calculate $\csc(150^\circ)$

$\csc(150^\circ) = \frac{1}{\sin(150^\circ)} = \frac{1}{\frac{1}{2}} = 2.$

Final Answer:

$\csc(150^\circ) = 2$

Would you like further details on the process, or do you have additional questions? Here are five related questions to consider:

1. How do reference angles help in trigonometric calculations?
2. Can you explain how the unit circle aids in finding trigonometric values?
3. What is the relationship between $\sin$ and $\csc$, and how do they behave for other angles?
4. How would this calculation change if the angle was in radians rather than degrees?
5. What are the other trigonometric values for $150^\circ$ (e.g., $\cos, \tan$)?

Tip: Always double-check whether the angle lies in a quadrant that changes the sign of the trigonometric function!