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

90.01 degrees
90.01 degrees

78.01 degrees
78.01 degrees

77.32 degrees
77.32 degrees

84.55 degrees

Answer 1

To find angle \( B \) in the right triangle using the Law of Cosines, we are given the sides \( a = 9 \), \( b = 40 \), and \( c = 41 \).

According to the Law of Cosines:

\[
b^2 = a^2 + c^2 - 2ac \cdot \cos B
\]

First, let's plug in the values of \( a \), \( b \), and \( c \):

\[
40^2 = 9^2 + 41^2 - 2 \cdot 9 \cdot 41 \cdot \cos B
\]

Calculating the squares:

\[
1600 = 81 + 1681 - 738 \cdot \cos B
\]

Now, simplify the left side:

\[
1600 = 1762 - 738 \cdot \cos B
\]

Rearranging to isolate \( \cos B \):

\[
738 \cdot \cos B = 1762 - 1600
\]

\[
738 \cdot \cos B = 162
\]

Dividing both sides by 738 to solve for \( \cos B \):

\[
\cos B = \frac{162}{738}
\]

Calculating this value:

\[
\cos B = 0.219
\]

Now using the arccosine function to find angle \( B \):

\[
B = \cos^{-1}(0.219)
\]

Calculating \( B \):

\[
B \approx 77.32 \text{ degrees}
\]

So the value of angle \( B \) is approximately \( 77.32 \) degrees.

The correct response is:

**77.32 degrees**.