Question

Use the Law of Cosines,  a2=b2+c2−2bc⋅cosA , to find the value of angle A in a right triangle, where a=9 ,  b=40 , and c=41 .(1 point) Responses 13.1 degrees 13.1 degrees 14.5 degrees 14.5 degrees 12.7 degrees

Answer 1

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

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

In this case, we have:

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

Plugging in these values:

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

Calculating the squares:

\[
81 = 1600 + 1681 - 3280 \cdot \cos A
\]

Now, summing up the squares of \( b \) and \( c \):

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

Rearranging the equation to solve for \( \cos A \):

\[
3280 \cdot \cos A = 3281 - 81
\]
\[
3280 \cdot \cos A = 3200
\]

Now, divide both sides by 3280:

\[
\cos A = \frac{3200}{3280} = \frac{320}{328} = \frac{80}{82} = \frac{40}{41}
\]

Now we find angle \( A \):

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

Using a calculator:

\[
A \approx \cos^{-1}(0.9756) \approx 12.7 \text{ degrees}
\]

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