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)
Responses

18.9 degrees
18.9 degrees

42.1 degrees
42.1 degrees

71.0 degrees
71.0 degrees

89.9 degrees

Answer 1

To find the value of angle \( A \) in the triangle using the Law of Cosines, we can rearrange the formula as follows:

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

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

We can substitute these values into the formula:

1. Calculate \( b^2 \), \( c^2 \), and \( a^2 \):
- \( b^2 = 35^2 = 1225 \)
- \( c^2 = 37^2 = 1369 \)
- \( a^2 = 12^2 = 144 \)

2. Substitute into the cosine formula:
\[
\cos A = \frac{1225 + 1369 - 144}{2 \cdot 35 \cdot 37}
\]

3. Simplify the numerator:
\[
\cos A = \frac{1225 + 1369 - 144}{2 \cdot 35 \cdot 37} = \frac{2450}{2590}
\]

4. Calculate \( 2 \cdot 35 \cdot 37 \):
\[
2 \cdot 35 = 70,
\]
\[
70 \cdot 37 = 2590
\]

5. Next, simplify \( \frac{2450}{2590} \):
\[
\cos A \approx 0.945
\]

6. Finally, find \( A \) by taking the inverse cosine:
\[
A \approx \cos^{-1}(0.945)
\]

Using a calculator, we find:
\[
A \approx 18.9 \text{ degrees}
\]

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