To find the angle \( A \) in the triangle using the Law of Cosines, we can rearrange the formula to solve for \( \cos A \).
The Law of Cosines states:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos A \]
We can isolate \( \cos A \):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Now, plug in the values \( a = 9 \), \( b = 40 \), and \( c = 41 \):
- Calculate \( b^2 \), \( c^2 \), and \( a^2 \):
\[ b^2 = 40^2 = 1600 \] \[ c^2 = 41^2 = 1681 \] \[ a^2 = 9^2 = 81 \]
- Substitute into the formula:
\[ \cos A = \frac{1600 + 1681 - 81}{2 \cdot 40 \cdot 41} \]
Now calculate the numerator:
\[ 1600 + 1681 - 81 = 3200 \]
Calculate the denominator:
\[ 2 \cdot 40 \cdot 41 = 3280 \]
So,
\[ \cos A = \frac{3200}{3280} = \frac{320}{328} = \frac{40}{41} \]
- Use the inverse cosine to find angle \( A \):
\[ A = \cos^{-1}\left(\frac{40}{41}\right) \]
Now we can use a calculator to find \( A \):
\[ A \approx \cos^{-1}(0.975609756) \approx 12.7 \text{ degrees} \]
Thus, the value of angle \( A \) is approximately 12.7 degrees.
The response is 12.7 degrees.