To find the value of angle \( A \) in a triangle using the Law of Cosines, we can rearrange the formula \( a^2 = b^2 + c^2 - 2bc \cdot \cos A \) to solve for \( \cos A \):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Given the values:
- \( a = 9 \)
- \( b = 40 \)
- \( c = 41 \)
We can first compute \( b^2 \), \( c^2 \), and \( a^2 \):
\[ b^2 = 40^2 = 1600 \] \[ c^2 = 41^2 = 1681 \] \[ a^2 = 9^2 = 81 \]
Now, we substitute these values into the cosine formula:
\[ \cos A = \frac{1600 + 1681 - 81}{2 \cdot 40 \cdot 41} \] \[ \cos A = \frac{1600 + 1681 - 81}{3280} \] \[ \cos A = \frac{3200}{3280} \] \[ \cos A = \frac{320}{328} \approx 0.9756 \]
Next, we find \( A \) by taking the inverse cosine:
\[ A = \cos^{-1}(0.9756) \]
Using a calculator:
\[ A \approx 12.7^\circ \]
Therefore, the value of angle \( A \) is 12.7 degrees.