Question

Use the Law of Cosines,  a2=b2+c2−2bc⋅cosA , to find the value of angle A in a right triangle, where a=12 , b=35 , and c=37 .(1 point)Responses71.0 degrees71.0 degrees42.1 degrees42.1 degrees89.9 degrees89.9 degrees18.9 degrees

Answer 1

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

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

Plugging in the values:

- \( a = 12 \)
- \( b = 35 \)
- \( c = 37 \)

We will rearrange the formula to solve for \( \cos A \):

\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]

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

\[
b^2 = 35^2 = 1225
\]
\[
c^2 = 37^2 = 1369
\]
\[
a^2 = 12^2 = 144
\]

Now, substitute these values into the cosine formula:

\[
\cos A = \frac{1225 + 1369 - 144}{2 \cdot 35 \cdot 37}
\]

Calculate \( 1225 + 1369 - 144 \):

\[
1225 + 1369 = 2594
\]
\[
2594 - 144 = 2450
\]

Now calculate \( 2 \cdot 35 \cdot 37 \):

\[
2 \cdot 35 \cdot 37 = 2 \cdot 1295 = 2590
\]

So, we have:

\[
\cos A = \frac{2450}{2590}
\]

Calculating \( \frac{2450}{2590} \):

\[
\cos A \approx 0.9467
\]

Now, find angle \( A \):

\[
A = \cos^{-1}(0.9467)
\]

Using a calculator to calculate \( A \):

\[
A \approx 18.9 \text{ degrees}
\]

Thus, the value of angle \( A \) is approximately **18.9 degrees**.