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Let's analyze the problem:

You are asked to find the angle \( \theta \) for \( 0^\circ \leq \theta < 360^\circ \) given:
- \( \tan \theta = -4.251 \)
- \( \cos \theta > 0 \)

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### **Step-by-Step Solution:**

1. **Identify the correct quadrant:**
   - \( \tan \theta \) is negative, so \( \theta \) must be either in **Quadrant II** or **Quadrant IV** (where tangent is negative).
   - Since \( \cos \theta > 0 \), \( \theta \) must lie in **Quadrant IV** (since cosine is positive only in Quadrants I and IV).

2. **Find the reference angle:**
   - Use the inverse tangent function to find the **reference angle** \( \alpha \).
   \[
   \alpha = \tan^{-1}(|-4.251|) = \tan^{-1}(4.251)
   \]
   Using a calculator:
   \[
   \alpha \approx 76.76^\circ
   \]
   This is the **acute angle** between the terminal side of the angle and the x-axis.

3. **Determine the correct angle \( \theta \) in Quadrant IV:**
   - In Quadrant IV, the angle \( \theta \) is calculated as:
   \[
   \theta = 360^\circ - \alpha = 360^\circ - 76.76^\circ = 283.24^\circ
   \]

4. **Answer:**
   \[
   \boxed{\theta = 283.24^\circ}
   \]

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Would you like further explanation or a verification of the steps?

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### Related Questions:
1. How do you determine which quadrant an angle lies in based on trigonometric signs?
2. What are the properties of the tangent function in different quadrants?
3. Why does inverse tangent return only values from Quadrants I and IV?
4. How would the answer change if \( \cos \theta < 0 \)?
5. How can you convert between degrees and radians?

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**Tip:** Always double-check your quadrant when working with inverse trigonometric functions since calculators give the principal value (usually only in Quadrants I and IV).

Find θ for 0° ≤ θ < 360°. Given tan θ = -4.251 and cos θ > 0. Round to two decimal places.

Solve the trigonometry problem: Find θ when tan θ = -4.251 and cos θ > 0 for 0° ≤ θ < 360°. Follow a step-by-step solution involving inverse tangent and reference angles. Suitable for students in Grades 10-12.

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