Use the Law of Cosines to determine the indicated angle 𝜃. (Assume a = 137.5, b = 58.3, and c = 161.5. Round your answer to the nearest degree.)

To determine the indicated angle \( \theta \) using the Law of Cosines, we use the formula:

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

Given the values:
- \( a = 137.5 \)
- \( b = 58.3 \)
- \( c = 161.5 \)

Substitute these into the formula:

\[
\cos(\theta) = \frac{137.5^2 + 58.3^2 - 161.5^2}{2 \times 137.5 \times 58.3}
\]

Let's calculate this step by step.

1. Calculate \( a^2 \), \( b^2 \), and \( c^2 \):
   \[
   137.5^2 = 18906.25
   \]
   \[
   58.3^2 = 3398.89
   \]
   \[
   161.5^2 = 26082.25
   \]

2. Substitute these into the equation:
   \[
   \cos(\theta) = \frac{18906.25 + 3398.89 - 26082.25}{2 \times 137.5 \times 58.3}
   \]
   \[
   \cos(\theta) = \frac{-6777.11}{16015.25}
   \]
   \[
   \cos(\theta) \approx -0.423
   \]

3. Finally, take the arccosine (inverse cosine) to find \( \theta \):
   \[
   \theta = \cos^{-1}(-0.423)
   \]
   \[
   \theta \approx 115^\circ
   \]

So, the indicated angle \( \theta \) is approximately \( 115^\circ \).

Would you like further details or have any questions? Here are some related questions for further practice:

1. How do you use the Law of Sines to solve for an unknown angle?
2. What are the conditions under which the Law of Cosines is applied?
3. How would you solve for a side of a triangle using the Law of Cosines?
4. How does the Law of Cosines relate to the Pythagorean Theorem?
5. Can the Law of Cosines be used in non-Euclidean geometry?

**Tip:** When using the Law of Cosines, always check your angle calculation by considering whether the result makes sense in the context of the triangle.

Learn how to calculate the indicated angle θ using the Law of Cosines with sides a = 137.5, b = 58.3, and c = 161.5. This trigonometry problem provides a step-by-step solution suitable for high school students, focusing on the application of the Law of Cosines.

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