To find the angle that satisfies the law of cosines for the given triangle, we need to use the formula for the law of cosines:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Given values:
- \( a = 12 \)
- \( b = 13 \)
- \( c = 5 \)
Our equation becomes:
[tex]\[ 5^2 = 12^2 + 13^2 - 2 \cdot 12 \cdot 13 \cdot \cos(C) \][/tex]
Let’s proceed step-by-step to find \(\cos(C)\) and then determine the angle \(C\):
1. Calculate \(12^2\):
[tex]\[ 12^2 = 144 \][/tex]
2. Calculate \(13^2\):
[tex]\[ 13^2 = 169 \][/tex]
3. Calculate \(5^2\):
[tex]\[ 5^2 = 25 \][/tex]
4. Substitute these values into the equation:
[tex]\[ 25 = 144 + 169 - 2 \cdot 12 \cdot 13 \cdot \cos(C) \][/tex]
5. Simplify the right-hand side:
[tex]\[ 25 = 313 - 2 \cdot 12 \cdot 13 \cdot \cos(C) \][/tex]
6. Calculate \(2 \cdot 12 \cdot 13\):
[tex]\[ 2 \cdot 12 \cdot 13 = 312 \][/tex]
So the equation is now:
[tex]\[ 25 = 313 - 312 \cos(C) \][/tex]
7. Rearrange the equation to solve for \(\cos(C)\):
[tex]\[ 312 \cos(C) = 313 - 25 \][/tex]
[tex]\[ 312 \cos(C) = 288 \][/tex]
[tex]\[ \cos(C) = \frac{288}{312} \][/tex]
8. Simplify the fraction:
[tex]\[ \cos(C) = \frac{288}{312} = \frac{24}{26} = \frac{12}{13} \approx 0.923 \][/tex]
9. Use the inverse cosine function to find the angle \(C\):
[tex]\[ C = \cos^{-1}(0.923) \][/tex]
10. Calculate the angle \(C\):
[tex]\[ C \approx 23.07^{\circ} \][/tex]
Comparing this result with the given options:
- \(23^{\circ}\)
- \(90^{\circ}\)
- \(180^{\circ}\)
- \(67^{\circ}\)
The angle \(23^{\circ}\) (Option A) is the closest to our calculated value \(23.07^{\circ}\).
Therefore, the correct angle that completes the law of cosines for this triangle is:
[tex]\[ \boxed{23^{\circ}} \][/tex]
