We are given that \( \cos(\theta) = -0.11 \) and \( 180^\circ < \theta < 270^\circ \). 

### Steps to solve:

1. **Determine the quadrant:**
   The given range \( 180^\circ < \theta < 270^\circ \) tells us that \( \theta \) is in the third quadrant. In the third quadrant, cosine is negative, which aligns with \( \cos(\theta) = -0.11 \).

2. **Find the reference angle:**
   The reference angle \( \theta_r \) can be found using the inverse cosine of the absolute value of \( \cos(\theta) \). Thus, we compute:

   \[
   \theta_r = \cos^{-1}(0.11)
   \]

3. **Adjust for the third quadrant:**
   Since \( \theta \) is in the third quadrant, the actual angle \( \theta \) can be found by adding the reference angle to \( 180^\circ \):

   \[
   \theta = 180^\circ + \theta_r
   \]

Let's calculate \( \theta_r \) first.The reference angle \( \theta_r \) is approximately \( 83.68^\circ \). Since \( \theta \) is in the third quadrant, the actual angle is:

\[
\theta \approx 180^\circ + 83.68^\circ = 263.68^\circ
\]

Thus, \( \theta \approx 263.68^\circ \).

Would you like more details, or do you have any questions about this? Here are some related questions to deepen understanding:

1. What is the general rule for finding angles in each quadrant?
2. How can you use the unit circle to determine the cosine of an angle?
3. How would you solve for \( \theta \) if given \( \cos(\theta) = 0.11 \) instead?
4. What is the relationship between the reference angle and the actual angle in each quadrant?
5. How does the sign of trigonometric functions change in each quadrant?

**Tip:** In trigonometry, always ensure you're aware of the quadrant to determine the correct sign of the function.

cosθ = -0.11; 180 degrees < θ < 270 degrees

Learn how to find θ when cos(θ) = -0.11 in the range 180° < θ < 270°. This trigonometry problem uses concepts such as reference angles, cosine function properties, and quadrant rules. Suitable for students in Grades 10-12.

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