Question

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

a. 13.1 degrees
b. 12.7 degrees
c. 17.3 degrees
d. 14.5 degrees

Answer 1

In the given right triangle, we have the following sides:

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

We need to apply the Law of Cosines to find the angle \( A \). According to the Law of Cosines, the formula is:

\[
a^2 = b^2 + c^2 - 2bc \cdot \cos A
\]

For our triangle, we can plug in the known values:

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

Calculating the squares:

\[
81 = 1600 + 1681 - 2 \cdot 40 \cdot 41 \cdot \cos A
\]

Now, combine \( 1600 \) and \( 1681 \):

\[
81 = 3281 - 3280 \cdot \cos A
\]

Rearranging the equation gives:

\[
3280 \cdot \cos A = 3281 - 81
\]

\[
3280 \cdot \cos A = 3200
\]

Now, divide both sides by \( 3280 \):

\[
\cos A = \frac{3200}{3280}
\]

\[
\cos A = \frac{320}{328} = \frac{40}{41}
\]

Now we will find angle \( A \) by taking the inverse cosine:

\[
A = \cos^{-1}\left(\frac{40}{41}\right)
\]

Using a calculator:

\[
A \approx 12.7 \text{ degrees}
\]

Thus, the value of angle \( A \) is approximately \( 12.7 \) degrees.

Therefore, the correct answer is:

**b. 12.7 degrees**