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 38.41 is opposite the angle 𝜃.
The second side of length 42.35 is opposite an unlabeled angle.
The third side of length 65.01 is opposite an unlabeled angle. what is theta in degrees?

The Law of Cosines relates the sides of a triangle to the cosine of one of its angles. For a triangle with sides \(a\), \(b\), and \(c\), where \( \theta \) is the angle opposite side \(a\), the Law of Cosines states:

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

We are given:
- \(a = 38.41\) (side opposite angle \( \theta \))
- \(b = 42.35\)
- \(c = 65.01\)

To find \( \theta \), we rearrange the Law of Cosines formula to solve for \( \cos(\theta) \):

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

Substituting the values for \(a\), \(b\), and \(c\):

\[
\cos(\theta) = \frac{42.35^2 + 65.01^2 - 38.41^2}{2 \cdot 42.35 \cdot 65.01}
\]

Let's compute \( \theta \).The angle \( \theta \) is approximately \( 34.4^\circ \), rounded to one decimal place.

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How does the Law of Cosines compare to the Law of Sines?
2. Can the Law of Cosines be used to find side lengths given all three angles?
3. What are the steps for solving an obtuse triangle using the Law of Cosines?
4. How can you verify the result of an angle calculation in a triangle?
5. How do different types of triangles affect the use of the Law of Cosines?

**Tip:** When using the Law of Cosines, ensure your calculator is set to the correct mode (degrees or radians) based on the context of the problem.

Use the Law of Cosines to determine the indicated angle 𝜃 (in degrees) shown in the following figure. A triangle is given. The first side of length 38.41 is opposite the angle 𝜃. The second side of length 42.35 is opposite an unlabeled angle. The third side of length 65.01 is opposite an unlabeled angle. What is theta in degrees?

Learn how to calculate the angle θ in a triangle using the Law of Cosines, with side lengths 38.41, 42.35, and 65.01. This trigonometry problem includes a step-by-step solution and is suitable for students in Grades 10-12.

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