To use the Law of Cosines to find angle \( A \) in a triangle where \( a = 5 \), \( b = 12 \), and \( c = 13 \), we start with the formula:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos A \]
Plugging the values into the equation:
\[ 5^2 = 12^2 + 13^2 - 2 \cdot 12 \cdot 13 \cdot \cos A \]
Calculating \( a^2 \), \( b^2 \), and \( c^2 \):
\[ 25 = 144 + 169 - 2 \cdot 12 \cdot 13 \cdot \cos A \]
Now, add \( 144 \) and \( 169 \):
\[ 25 = 313 - 2 \cdot 12 \cdot 13 \cdot \cos A \]
Now, calculate \( 2 \cdot 12 \cdot 13 \):
\[ 2 \cdot 12 \cdot 13 = 312 \]
So, our equation becomes:
\[ 25 = 313 - 312 \cdot \cos A \]
Now, isolate the term with \( \cos A \):
\[ 312 \cdot \cos A = 313 - 25 \]
\[ 312 \cdot \cos A = 288 \]
Now solve for \( \cos A \):
\[ \cos A = \frac{288}{312} = \frac{12}{13} \]
Now, to find angle \( A \), we take the arccosine:
\[ A = \cos^{-1}\left(\frac{12}{13}\right) \]
Using a calculator:
\[ A \approx 22.6 \text{ degrees} \]
Rounding to the nearest whole number:
\[ A \approx 23 \text{ degrees} \]
Thus, the final answer is:
\[ A = 23 \text{ degrees} \]