In a right triangle, we can use the Law of Cosines to find angle \( B \) with the given sides. The Law of Cosines states:
\[ b^2 = a^2 + c^2 - 2ac \cdot \cos B \]
Given:
- \( a = 9 \)
- \( b = 40 \)
- \( c = 41 \)
First, we can plug in the values into the formula:
- Calculate \( a^2 \), \( b^2 \), and \( c^2 \):
\[ a^2 = 9^2 = 81 \] \[ b^2 = 40^2 = 1600 \] \[ c^2 = 41^2 = 1681 \]
- Substitute into the Law of Cosines formula:
\[ 1600 = 81 + 1681 - 2 \cdot 9 \cdot 41 \cdot \cos B \]
- Combine the terms on the right:
\[ 1600 = 81 + 1681 = 1762 - 738 \cdot \cos B \]
- Rearranging gives:
\[ 738 \cdot \cos B = 1762 - 1600 \] \[ 738 \cdot \cos B = 162 \]
- Solve for \( \cos B \):
\[ \cos B = \frac{162}{738} \] \[ \cos B \approx 0.219 \]
- Finally, take the arccos to find angle \( B \):
\[ B \approx \cos^{-1}(0.219) \]
Now using a calculator:
\[ B \approx 77.32 \text{ degrees} \]
Therefore, the value of angle \( B \) is approximately \( 77.32 \) degrees. The correct answer is:
77.32 degrees