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

84.55 degrees
84.55 degrees

78.01 degrees
78.01 degrees

90.01 degrees
90.01 degrees

77.32 degrees

Answer 1

To find angle \( B \) in the triangle using the Law of Cosines, we use the formula:

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

Given the values \( a = 9 \), \( b = 40 \), and \( c = 41 \), we can substitute these into the formula:

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

Calculating the squares:

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

Now, add \( 81 \) and \( 1681 \):

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

Next, we isolate the \( \cos B \) term:

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

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

Dividing both sides by \(-738\):

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

Calculating \( \frac{162}{738} \):

\[
\cos B \approx 0.219
\]

Now, we need to find \( B \) by taking the arccosine:

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

Using a calculator:

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

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