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

90.01 degrees
90.01 degrees

84.55 degrees

Answer 1

To find the value of angle \( B \) in a triangle using the Law of Cosines, we can rearrange the formula \( b^2 = a^2 + c^2 - 2ac \cdot \cos B \) to solve for \( \cos B \):

1. Start with the formula:
\[
b^2 = a^2 + c^2 - 2ac \cdot \cos B
\]

2. Rearranging for \( \cos B \):
\[
2ac \cdot \cos B = a^2 + c^2 - b^2
\]
\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac}
\]

3. Plug in the values \( a = 9 \), \( b = 40 \), and \( c = 41 \):
\[
\cos B = \frac{9^2 + 41^2 - 40^2}{2 \cdot 9 \cdot 41}
\]

4. Calculate \( 9^2 \), \( 41^2 \), and \( 40^2 \):
- \( 9^2 = 81 \)
- \( 41^2 = 1681 \)
- \( 40^2 = 1600 \)

5. Substitute these values into the equation:
\[
\cos B = \frac{81 + 1681 - 1600}{2 \cdot 9 \cdot 41}
\]
\[
\cos B = \frac{81 + 81}{738}
\]
\[
\cos B = \frac{162}{738}
\]

6. Simplifying the fraction:
\[
\cos B = \frac{81}{369} \approx 0.2196
\]

7. To find angle \( B \), take the inverse cosine:
\[
B = \cos^{-1}(0.2196)
\]

8. Using a calculator (make sure it's in degrees), we find:
\[
B \approx 77.32 degrees
\]

Therefore, the value of angle \( B \) is **77.32 degrees**.