Question

Use the Law of Cosines, b2=a2+c2−2ac⋅cosB , to find the value of angle B in a right triangle, where a=9 , b=40 , and c=41 .(1 point)
Responses

78.01 degrees
78.01 degrees

77.32 degrees
77.32 degrees

84.55 degrees
84.55 degrees

90.01 degrees
90.01 degrees

Answer 1

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 \( c \):

1. Calculate \( a^2 \), \( b^2 \), and \( c^2 \):
- \( a^2 = 9^2 = 81 \)
- \( b^2 = 40^2 = 1600 \)
- \( c^2 = 41^2 = 1681 \)

2. Substitute these values into the equation:

\[
\cos B = \frac{81 + 1681 - 1600}{2 \cdot 9 \cdot 41}
\]

3. Simplify the numerator:

\[
\cos B = \frac{81 + 1681 - 1600}{738} = \frac{162}{738}
\]

4. Now calculate \( \frac{162}{738} \):

\[
\frac{162}{738} = \frac{27}{123} \approx 0.2195
\]

5. 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**.