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The problem is asking to find the value of \(\theta\) to the nearest tenth of a degree, where:

\[
0^\circ < \theta < 360^\circ, \quad \sec(\theta) = 8.0905 \quad \text{with} \quad \theta \in \text{QIV (Quadrant IV)}.
\]

### Solution:

1. **Relationship Between \(\sec(\theta)\) and \(\cos(\theta)\)**:
   The secant function is the reciprocal of the cosine function:
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)}.
   \]
   So, to find \(\cos(\theta)\), take the reciprocal of the secant value:
   \[
   \cos(\theta) = \frac{1}{\sec(\theta)} = \frac{1}{8.0905} \approx 0.1236.
   \]

2. **Find the Reference Angle**:
   Use the inverse cosine function to find the reference angle in degrees. The reference angle is:
   \[
   \theta_{\text{ref}} = \cos^{-1}(0.1236) \approx 82.9^\circ.
   \]

3. **Determine the Angle in Quadrant IV**:
   Since \(\theta\) is in Quadrant IV (QIV), the formula to find the angle is:
   \[
   \theta = 360^\circ - \theta_{\text{ref}} = 360^\circ - 82.9^\circ = 277.1^\circ.
   \]

Thus, the value of \(\theta\) is:

\[
\boxed{277.1^\circ}.
\]

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. What is the difference between secant and cosine?
2. How do we determine which quadrant the angle is in using trigonometric values?
3. What other trigonometric functions can be used to find angles in different quadrants?
4. How would this problem change if the secant value was negative?
5. How does the unit circle help in understanding quadrants and angles?

**Tip**: When working with trigonometric functions, always pay attention to the quadrant, as it determines whether the function value is positive or negative.

Use a calculator to find θ to the nearest tenth of a degree, if 0° < θ < 360° and sec(θ) = 8.0905 with θ in QIV.

Learn how to find the angle θ to the nearest tenth of a degree when sec(θ) = 8.0905 in Quadrant IV. This trigonometry problem involves secant function properties and inverse cosine calculations, suitable for students in Grades 10-12.

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