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To find the measure of angle \( C \) using the Law of Cosines, we start with the given values:

- \( a = 17 \)
- \( b = 8 \)
- \( c = 16 \)

The Law of Cosines formula is:

\[
c^2 = a^2 + b^2 - 2ab \cos C
\]

We need to solve for \( \cos C \):

\[
16^2 = 17^2 + 8^2 - 2 \cdot 17 \cdot 8 \cdot \cos C
\]

First, we calculate the squares:

\[
256 = 289 + 64 - 2 \cdot 17 \cdot 8 \cdot \cos C
\]

\[
256 = 353 - 272 \cos C
\]

Next, we isolate \( \cos C \):

\[
272 \cos C = 353 - 256
\]

\[
272 \cos C = 97
\]

\[
\cos C = \frac{97}{272}
\]

\[
\cos C \approx 0.3566
\]

Now, we find the measure of angle \( C \) by taking the inverse cosine:

\[
C = \cos^{-1}(0.3566)
\]

Using a calculator:

\[
C \approx 69.1^\circ
\]

Therefore, the measure of angle \( C \) rounded to the nearest tenth is:

\[
C \approx 69.1^\circ
\]

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How would you verify this result using the Law of Sines?
2. What are the steps to solve for other angles using the Law of Cosines?
3. How do you determine if a triangle is acute, obtuse, or right using side lengths?
4. What is the relationship between the Law of Sines and the Law of Cosines?
5. How can the Law of Cosines be applied to real-world problems?
6. How do you calculate the area of a triangle using Heron's formula with these side lengths?
7. What is the significance of the Law of Cosines in trigonometry?
8. How do you solve for an angle in a triangle given all three side lengths using a different method?

**Tip:** When using the Law of Cosines, always double-check your calculations for accuracy, especially when solving for angles, as small errors in the cosine value can lead to significant differences in the angle measure.

Learn how to find the measure of angle C in a triangle using the Law of Cosines formula. This problem involves trigonometric calculations with given side lengths and includes steps for calculating cosine values and inverse cosine. Suitable for high school students studying trigonometry.

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