To find the angle \( A \) in a right triangle using the Law of Cosines, we have the formula:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos A \]
Where:
- \( a = 9 \)
- \( b = 40 \)
- \( c = 41 \)
First, let's plug in the values into the equation:
\[ 9^2 = 40^2 + 41^2 - 2 \cdot 40 \cdot 41 \cdot \cos A \]
Calculating the squares:
\[ 81 = 1600 + 1681 - 3280 \cdot \cos A \]
Now, combine the values on the right:
\[ 81 = 3281 - 3280 \cdot \cos A \]
Next, isolate \( \cos A \):
\[ 3280 \cdot \cos A = 3281 - 81 \] \[ 3280 \cdot \cos A = 3200 \] \[ \cos A = \frac{3200}{3280} \]
Calculating the fraction:
\[ \cos A = \frac{3200}{3280} \approx 0.9756 \]
Now, find angle \( A \) using the inverse cosine function:
\[ A = \cos^{-1}(0.9756) \]
Using a calculator to find the angle:
\[ A \approx 12.7 \text{ degrees} \]
Thus, the value of angle \( A \) is approximately:
\[ \boxed{12.7 \text{ degrees}} \]