Certainly! Let's use the Law of Cosines to find the angle that completes the given equation:
[tex]\[8^2 + 17^2 - 2 \cdot 8 \cdot 17 \cdot \cos(\theta) = 15^2\][/tex]
1. Begin by squaring the given side lengths:
[tex]\[
8^2 = 64
\][/tex]
[tex]\[
17^2 = 289
\][/tex]
[tex]\[
15^2 = 225
\][/tex]
2. Substitute these values into the equation:
[tex]\[
64 + 289 - 2 \cdot 8 \cdot 17 \cdot \cos(\theta) = 225
\][/tex]
3. Simplify the left side of the equation:
[tex]\[
353 - 272 \cdot \cos(\theta) = 225
\][/tex]
4. Isolate the cosine term by subtracting 225 from both sides:
[tex]\[
353 - 225 = 272 \cdot \cos(\theta)
\][/tex]
[tex]\[
128 = 272 \cdot \cos(\theta)
\][/tex]
5. Solve for [tex]\(\cos(\theta)\)[/tex]:
[tex]\[
\cos(\theta) = \frac{128}{272} = \frac{16}{34} = \frac{8}{17}
\][/tex]
6. Determine the angle [tex]\(\theta\)[/tex] that corresponds to [tex]\(\cos(\theta) = \frac{8}{17}\)[/tex]:
Using the inverse cosine (arccos) function:
[tex]\[
\theta \approx 61.93^{\circ}
\][/tex]
7. Compare the calculated angle with the given options to find the closest value:
- 180°
- 62°
- 28°
- 90°
The closest angle to [tex]\(61.93^{\circ}\)[/tex] is [tex]\(62^{\circ}\)[/tex].
So, the correct answer is:
B. [tex]\(62^{\circ}\)[/tex]
