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The equation provided is \( \csc(\theta) = \frac{3}{2} \), and you are asked to find two possible angles \( \theta \) within the domain \( 0^\circ \leq \theta \leq 360^\circ \).

We start by recalling that the cosecant function is the reciprocal of the sine function. That is, 
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]
So, the equation becomes:
\[
\frac{1}{\sin(\theta)} = \frac{3}{2}
\]
Multiplying both sides by \( \sin(\theta) \) and solving for \( \sin(\theta) \), we get:
\[
\sin(\theta) = \frac{2}{3}
\]

Next, we find the angles \( \theta \) that satisfy this condition within the domain \( 0^\circ \leq \theta \leq 360^\circ \).

1. **First Quadrant**: Since \( \sin(\theta) = \frac{2}{3} \) is positive, there is a solution in the first quadrant. The inverse sine of \( \frac{2}{3} \) gives:
   \[
   \theta_1 = \arcsin\left(\frac{2}{3}\right) \approx 41.81^\circ
   \]

2. **Second Quadrant**: Since sine is also positive in the second quadrant, the second angle is given by:
   \[
   \theta_2 = 180^\circ - 41.81^\circ = 138.19^\circ
   \]

Thus, the two possible angles between \( 0^\circ \) and \( 360^\circ \) are approximately:
\[
\theta_1 \approx 41.81^\circ \quad \text{and} \quad \theta_2 \approx 138.19^\circ
\]

Would you like any further details or have any other questions?

Here are 5 related questions to explore:

1. How do we calculate the other quadrants where sine is negative?
2. What would the graph of \( \csc(\theta) = \frac{3}{2} \) look like?
3. How can we express solutions for this equation if the domain is extended beyond \( 360^\circ \)?
4. What is the relationship between the sine and cosecant functions?
5. How do we use radians instead of degrees for this equation?

**Tip**: When solving trigonometric equations, always consider the quadrants where the function is positive or negative. This will guide you in finding all possible solutions within a given domain.

Given the following function, choose the graph with the two possible angles with the domain 0° ≤ θ ≤ 360°. csc(θ) = 3/2.

Explore how to solve the trigonometric equation csc(θ) = 3/2 to find two angles within the domain 0° ≤ θ ≤ 360°. This problem covers key concepts in trigonometry, including the cosecant and sine functions, and is suitable for students in Grades 9-12.

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