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To solve for \( \theta \) in the equation:

\[
\csc \theta = 2.559
\]

we can follow these steps:

1. **Rewrite in terms of \(\sin \theta\)**: Since \(\csc \theta = \frac{1}{\sin \theta}\), we rewrite the equation as:
   \[
   \frac{1}{\sin \theta} = 2.559
   \]

2. **Solve for \(\sin \theta\)**:
   \[
   \sin \theta = \frac{1}{2.559} \approx 0.3908
   \]

3. **Find \(\theta\)**: Now we need to find the angle \(\theta\) such that \(\sin \theta \approx 0.3908\).

   Using the inverse sine function:
   \[
   \theta = \sin^{-1}(0.3908) \approx 23.07^\circ
   \]

4. **Consider the range of solutions**: Since \(\sin\) is positive in both the first and second quadrants, we have:
   \[
   \theta \approx 23.07^\circ \quad \text{and} \quad \theta \approx 180^\circ - 23.07^\circ = 156.93^\circ
   \]

### Answer
The two solutions for \(\theta\) in degrees are:
\[
\theta \approx 23.07^\circ \quad \text{or} \quad \theta \approx 156.93^\circ
\]

Would you like further details on this solution, or do you have any additional questions?

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Here are 5 questions to expand on this topic:

1. How would the solution change if \(\csc \theta\) were negative?
2. How would you find \(\theta\) if the equation were in radians instead of degrees?
3. What are the values of \(\sin \theta\) for angles in other quadrants?
4. How does knowing the range of the \(\sin^{-1}\) function help in solving this type of equation?
5. What other trigonometric functions could you use to express \(\csc \theta\)?

**Tip**: For trigonometric equations, always consider solutions in multiple quadrants unless restricted to a specific interval.

This page provides a step-by-step solution to solve the trigonometric equation csc θ = 2.559. Learn how to find the value of θ by rewriting in terms of sin θ, solving for sin θ, and considering solutions in multiple quadrants. Suitable for high school students studying trigonometry.

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