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.)

The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and angle \(𝜃\) opposite side \(c\), the formula is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(𝜃)
\]

We can rearrange this to solve for \(𝜃\):

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

Now, let's plug in the given values:

- \(a = 137.5\)
- \(b = 58.3\)
- \(c = 161.5\)

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

I'll calculate this.The angle \(𝜃\) is approximately \(104^\circ\), rounded to the nearest degree.

Would you like more details on the process, or do you have any other questions?

Here are five related questions you might explore:
1. How would the Law of Cosines change if the angle \(𝜃\) were between sides \(a\) and \(b\)?
2. Can you find the other two angles in the triangle using the Law of Cosines?
3. How does the Law of Sines compare to the Law of Cosines in solving triangles?
4. What are the real-life applications of the Law of Cosines?
5. How does changing one of the side lengths affect the angle opposite to that side?

**Tip**: Always double-check your calculator's mode (degrees vs. radians) when solving trigonometric problems.

Learn how to use the Law of Cosines to determine the angle 𝜃 for a triangle with sides a = 137.5, b = 58.3, and c = 161.5. This solution involves trigonometry and geometry concepts suitable for Grades 10-12, with step-by-step explanations.

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