To find the angle \( A \) in the right triangle using the Law of Cosines, we can plug the provided values into the formula:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos A \]
Given:
- \( a = 12 \)
- \( b = 35 \)
- \( c = 37 \)
First, we calculate \( a^2 \), \( b^2 \), and \( c^2 \):
\[ a^2 = 12^2 = 144 \]
\[ b^2 = 35^2 = 1225 \]
\[ c^2 = 37^2 = 1369 \]
Now, substituting into the Law of Cosines equation:
\[ 144 = 1225 + 1369 - 2 \cdot 35 \cdot 37 \cdot \cos A \]
Calculating \( 1225 + 1369 \):
\[ 1225 + 1369 = 2594 \]
Now, substitute that into the equation:
\[ 144 = 2594 - 2 \cdot 35 \cdot 37 \cdot \cos A \]
Next, we simplify to isolate \( \cos A \):
Subtract \( 2594 \) from both sides:
\[ 144 - 2594 = -2 \cdot 35 \cdot 37 \cdot \cos A \]
\[ -2450 = -2 \cdot 35 \cdot 37 \cdot \cos A \]
Now calculate \( 2 \cdot 35 \cdot 37 \):
\[ 2 \cdot 35 \cdot 37 = 2 \cdot 1295 = 2590 \]
Now substitute that back into the equation:
\[ -2450 = -2590 \cdot \cos A \]
Dividing both sides by \(-2590\):
\[ \cos A = \frac{2450}{2590} \]
Calculating this fraction:
\[ \cos A \approx 0.9465 \]
Now, we find angle \( A \) using the inverse cosine:
\[ A = \cos^{-1}(0.9465) \]
Using a calculator to find \( A \):
\[ A \approx 18.19^\circ \]
Thus, the value of angle \( A \) is approximately \( \boxed{18.19^\circ} \).