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The problem asks to find the values of \(\theta\) for \(0^\circ \leq \theta < 360^\circ\) given that:

\[
\csc \theta = -2.627 \quad \text{and} \quad \cos \theta > 0
\]

### Step-by-step solution:

1. **Understanding the given information**:
   - \(\csc \theta = -2.627\): The cosecant function is the reciprocal of the sine function, so:
     \[
     \csc \theta = \frac{1}{\sin \theta} \quad \Rightarrow \quad \sin \theta = \frac{1}{\csc \theta} = \frac{1}{-2.627} \approx -0.3807
     \]
   - \(\cos \theta > 0\): This means that \(\theta\) must be in the first or fourth quadrant (since cosine is positive in these quadrants).

2. **Find the reference angle**:
   - To find the reference angle, take the inverse sine of the absolute value of \(\sin \theta\):
     \[
     \theta_{\text{ref}} = \arcsin(0.3807) \approx 22.36^\circ
     \]

3. **Determine the possible quadrants**:
   - Since \(\sin \theta\) is negative, \(\theta\) must be in the third or fourth quadrant.
   - However, since \(\cos \theta > 0\), \(\theta\) must be in the fourth quadrant.

4. **Calculate the angle**:
   - In the fourth quadrant, the angle is given by:
     \[
     \theta = 360^\circ - \theta_{\text{ref}} = 360^\circ - 22.36^\circ \approx 337.64^\circ
     \]

### Final answer:
\[
\theta \approx 338^\circ
\]

Would you like more details or have any other questions?

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Here are five related questions to expand on this concept:

1. How would the solution change if \(\cos \theta < 0\)?
2. What are the reciprocal identities for trigonometric functions?
3. How can you solve for \(\theta\) if given \(\cot \theta\) instead of \(\csc \theta\)?
4. What is the general solution for \(\sin \theta = -0.3807\) without the constraint \(0^\circ \leq \theta < 360^\circ\)?
5. How would you convert this problem to radians?

**Tip**: When using inverse trigonometric functions, always check which quadrant the angle belongs to based on the sign of the sine, cosine, or tangent functions.

Find θ for 0° ≤ θ < 360° given that csc θ = -2.627 and cos θ > 0.

Find the angle θ for 0° ≤ θ < 360° when csc θ = -2.627 and cos θ > 0. Learn how to use trigonometric identities, including the cosecant function and reference angles, to solve the problem step by step.

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