Question

Use the Law of Cosines, b^2 = a^2 + c^2 - 2ac * cos B, to find the value of angle B in a right triangle, where a = 9, b = 40, and c = 41.

a. 84.55 degrees
b. 77.32 degrees
c. 90.01 degrees
d. 78.01 degrees

Answer 1

To find the angle \( B \) in the right triangle using the Law of Cosines, we can begin by plugging in the given values into the formula:

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

Where:
- \( a = 9 \)
- \( b = 40 \)
- \( c = 41 \)

We first calculate \( a^2 \), \( b^2 \), and \( c^2 \):

\[
a^2 = 9^2 = 81
\]
\[
b^2 = 40^2 = 1600
\]
\[
c^2 = 41^2 = 1681
\]

Now, substituting into the Law of Cosines formula:

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

Next, we simplify the equation:

\[
1600 = 81 + 1681 - 738 \cdot \cos B
\]
\[
1600 = 1762 - 738 \cdot \cos B
\]

Now, isolate \( \cos B \):

\[
738 \cdot \cos B = 1762 - 1600
\]
\[
738 \cdot \cos B = 162
\]
\[
\cos B = \frac{162}{738}
\]
\[
\cos B = \frac{81}{369} \approx 0.219
\]

Now, we calculate \( B \):

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

Using a calculator:

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

Thus, the value of angle \( B \) is:

\[
\boxed{77.32} \text{ degrees}
\]

This corresponds to option b.