Use the Law of Cosines to determine the indicated angle 𝜃 (in degrees) shown in the following figure. (Round your answer to one decimal place.) A triangle is given.
The first side of length 24 is opposite the angle 𝜃.
The second side of length 15 is opposite an unlabeled angle.
The third side of length 12 is opposite an unlabeled angle.

To solve for the angle \( \theta \) using the Law of Cosines, we can use the formula:

\[
\cos(\theta) = \frac{b^2 + c^2 - a^2}{2bc}
\]

Where:
- \( a \) is the side opposite the angle \( \theta \),
- \( b \) and \( c \) are the other two sides of the triangle.

Given:
- \( a = 24 \) (the side opposite angle \( \theta \)),
- \( b = 15 \),
- \( c = 12 \).

### Step-by-Step Calculation:

1. Substitute the given values into the Law of Cosines formula:

\[
\cos(\theta) = \frac{15^2 + 12^2 - 24^2}{2 \times 15 \times 12}
\]

2. Simplify:

\[
\cos(\theta) = \frac{225 + 144 - 576}{2 \times 15 \times 12}
\]
\[
\cos(\theta) = \frac{-207}{360}
\]

3. Calculate the value of \( \cos(\theta) \):

\[
\cos(\theta) = -0.575
\]

4. Use the inverse cosine function to find \( \theta \):

\[
\theta = \cos^{-1}(-0.575)
\]

Now, let's compute \( \theta \).The angle \( \theta \) is approximately \( 125.1^\circ \) (rounded to one decimal place).

Would you like more details on this solution or have any other questions?

Here are some related questions you might find interesting:
1. How does the Law of Cosines differ from the Law of Sines?
2. Can the Law of Cosines be used to find unknown sides instead of angles?
3. What are the conditions when the Law of Cosines is most useful?
4. How would the Law of Sines apply in a triangle with angle \( \theta \) and known sides?
5. How does the Law of Cosines change in right triangles?

**Tip**: The Law of Cosines can be especially helpful in obtuse triangles, where the Law of Sines might not easily apply.

Use the Law of Cosines to determine the indicated angle 𝜃 (in degrees) shown in the following figure. (Round your answer to one decimal place.) A triangle is given. The first side of length 24 is opposite the angle 𝜃. The second side of length 15 is opposite an unlabeled angle. The third side of length 12 is opposite an unlabeled angle.

Learn how to use the Law of Cosines to find angle 𝜃 in a triangle where the sides are 24, 15, and 12 units long. This step-by-step solution shows how to apply the formula cos(𝜃) = (b² + c² - a²) / (2bc) and use inverse cosine to calculate the angle. Suitable for students in Grades 10-12.

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