In a right triangle, we can use the Law of Cosines to find the angle \( B \). Given the sides \( a = 9 \), \( b = 40 \), and \( c = 41 \), we want to calculate the angle \( B \) using the formula:
\[ b^2 = a^2 + c^2 - 2ac \cdot \cos B \]
We can rearrange this to solve for \( \cos B \):
\[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]
Now, we will plug in the values of \( a \), \( b \), and \( c \):
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Calculate \( a^2 \), \( b^2 \), and \( c^2 \):
- \( a^2 = 9^2 = 81 \)
- \( b^2 = 40^2 = 1600 \)
- \( c^2 = 41^2 = 1681 \)
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Substitute these values into the equation:
\[ \cos B = \frac{81 + 1681 - 1600}{2 \cdot 9 \cdot 41} \]
- Simplify the numerator:
\[ \cos B = \frac{81 + 1681 - 1600}{738} = \frac{162}{738} \]
- Now calculate \( \frac{162}{738} \):
\[ \frac{162}{738} = \frac{27}{123} \approx 0.2195 \]
- Use the inverse cosine to find angle \( B \):
\[ B = \cos^{-1}(0.2195) \]
Calculating this gives:
\[ B \approx 77.32 \text{ degrees} \]
Therefore, the value of angle \( B \) is approximately 77.32 degrees.