Let's explore the problem step-by-step using the Law of Cosines:
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Given:
[tex]\[ a = 12 \][/tex]
[tex]\[ b = 13 \][/tex]
[tex]\[ c = 5 \][/tex]
We need to find the angle \( C \). According to the Law of Cosines:
[tex]\[ 5^2 = 12^2 + 13^2 - 2 \cdot 12 \cdot 13 \cdot \cos(C) \][/tex]
First, let's plug in the values:
[tex]\[ 25 = 144 + 169 - 312 \cos(C) \][/tex]
Combine the values:
[tex]\[ 25 = 313 - 312 \cos(C) \][/tex]
Rearrange 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]
[tex]\[ \cos(C) = \frac{24}{26} \][/tex]
[tex]\[ \cos(C) = \frac{12}{13} \][/tex]
Now, to find the angle \( C \), we take the arccos (inverse cosine) of \(\frac{12}{13}\):
[tex]\[ C = \arccos \left(\frac{12}{13}\right) \][/tex]
Using a calculator or tables, we find:
[tex]\[ C \approx 22.62^{\circ} \][/tex]
Among the given choices, the angle that fits closest is:
C. \( 23^{\circ} \)
Therefore, the correct answer is [tex]\( 23^{\circ} \)[/tex].
