To solve for the missing angle in the given triangle using the Law of Cosines, we start with the equation:
[tex]\[
5^2 + 13^2 - 2 \cdot 5 \cdot 13 \cdot \cos(\theta) = 12^2
\][/tex]
Let's break it down step by step:
1. Square the known sides and the length opposite the angle:
[tex]\[
5^2 = 25, \quad 13^2 = 169, \quad 12^2 = 144
\][/tex]
2. Plug these values into the equation:
[tex]\[
25 + 169 - 2 \cdot 5 \cdot 13 \cdot \cos(\theta) = 144
\][/tex]
3. Simplify the left side:
[tex]\[
194 - 2 \cdot 5 \cdot 13 \cdot \cos(\theta) = 144
\][/tex]
4. Calculate the product [tex]\(2 \cdot 5 \cdot 13\)[/tex]:
[tex]\[
2 \cdot 5 \cdot 13 = 130
\][/tex]
So, the equation now is:
[tex]\[
194 - 130 \cdot \cos(\theta) = 144
\][/tex]
5. Isolate [tex]\(\cos(\theta)\)[/tex] by subtracting 144 from both sides:
[tex]\[
194 - 144 = 130 \cdot \cos(\theta)
\][/tex]
[tex]\[
50 = 130 \cdot \cos(\theta)
\][/tex]
6. Solve for [tex]\(\cos(\theta)\)[/tex]:
[tex]\[
\cos(\theta) = \frac{50}{130} = \frac{5}{13} \approx 0.38461538461538464
\][/tex]
7. Use the inverse cosine function to find [tex]\(\theta\)[/tex]:
[tex]\[
\theta = \cos^{-1}(0.38461538461538464)
\][/tex]
Upon calculating, we get:
[tex]\[
\theta \approx 67.38013505195957^{\circ}
\][/tex]
8. Now, compare this calculated angle to the given choices:
- [tex]\(90^{\circ}\)[/tex]
- [tex]\(23^{\circ}\)[/tex]
- [tex]\(180^{\circ}\)[/tex]
- [tex]\(67^{\circ}\)[/tex]
The angle [tex]\(67.38013505195957^{\circ}\)[/tex] is closest to [tex]\(67^{\circ}\)[/tex].
Thus, the correct angle that completes the law of cosines for this triangle is:
D. [tex]\(67^{\circ}\)[/tex]
