Question

Use the Law of Cosines, a2=b2+c2−2bc⋅cosA
, to find the value of angle A
in a right triangle, where a=7
, b=24
, and c=25
.(1 point)
Responses

89.9 degrees
89.9 degrees

25.1 degrees
25.1 degrees

73.6 degrees
73.6 degrees

16.3 degrees

Answer 1

In a right triangle, one of the angles is \(90^\circ\). However, we can use the Law of Cosines to find the angle \(A\) given the sides. The Law of Cosines states:

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

Given the values:
- \(a = 7\)
- \(b = 24\)
- \(c = 25\)

Let's substitute these values into the formula to find \(\cos A\):

1. Substituting the values:
\[
7^2 = 24^2 + 25^2 - 2 \cdot 24 \cdot 25 \cdot \cos A
\]

2. Calculate \(7^2\), \(24^2\), and \(25^2\):
\[
49 = 576 + 625 - 1200 \cdot \cos A
\]

3. Combine \(576\) and \(625\):
\[
49 = 1201 - 1200 \cdot \cos A
\]

4. Isolate the cosine term:
\[
1200 \cdot \cos A = 1201 - 49
\]
\[
1200 \cdot \cos A = 1152
\]

5. Divide both sides by \(1200\):
\[
\cos A = \frac{1152}{1200}
\]
\[
\cos A = 0.96
\]

6. Now we find angle \(A\) by taking the arccos of \(0.96\):
\[
A = \cos^{-1}(0.96)
\]

Calculating this gives approximately:
\[
A \approx 16.3 \text{ degrees}
\]

Thus, the value of angle \(A\) is **16.3 degrees**.