To find the angle \( B \) in a right triangle using the Law of Cosines, we can start with the formula:
\[ b^2 = a^2 + c^2 - 2ac \cdot \cos B \]
In this case, we have the following values: \( a = 9 \), \( b = 40 \), \( c = 41 \).
First, we can calculate \( a^2 \), \( b^2 \), and \( c^2 \):
\[ a^2 = 9^2 = 81 \] \[ b^2 = 40^2 = 1600 \] \[ c^2 = 41^2 = 1681 \]
Now, substitute these values into the Law of Cosines formula:
\[ 1600 = 81 + 1681 - 2 \cdot 9 \cdot 41 \cdot \cos B \]
This simplifies to:
\[ 1600 = 81 + 1681 - 738 \cos B \] \[ 1600 = 1762 - 738 \cos B \]
Now, isolate \( \cos B \):
\[ 1600 - 1762 = -738 \cos B \] \[ -162 = -738 \cos B \]
Dividing both sides by -738 gives:
\[ \cos B = \frac{162}{738} \]
Now simplify \( \frac{162}{738} \):
\[ \cos B = \frac{162 \div 6}{738 \div 6} = \frac{27}{123} \]
Next, simplify \( \frac{27}{123} \):
\[ \cos B = \frac{9}{41} \]
Now we can find angle \( B \) by taking the inverse cosine:
\[ B = \cos^{-1}\left(\frac{9}{41}\right) \]
Using a calculator or appropriate software to find the angle, we calculate:
\[ B \approx 77.51^\circ \]
Thus, the measure of angle \( B \) is approximately \( 77.51^\circ \).
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