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

16.3 degrees
16.3 degrees

73.6 degrees
73.6 degrees

89.9 degrees
89.9 degrees

25.1 degrees

Answer 1

In a right triangle, we can use the Law of Cosines to find the cosine of angle \( A \) using the formula:

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

We have:

- \( a = 7 \)
- \( b = 24 \)
- \( c = 25 \)

First, let's plug the values into the Law of Cosines formula:

\[
7^2 = 24^2 + 25^2 - 2 \cdot 24 \cdot 25 \cdot \cos A
\]

Calculating each term:

\[
49 = 576 + 625 - 1200 \cdot \cos A
\]

Adding the squares of \( b \) and \( c \):

\[
49 = 1201 - 1200 \cdot \cos A
\]

Now let's solve for \( \cos A \):

\[
1200 \cdot \cos A = 1201 - 49
\]

\[
1200 \cdot \cos A = 1152
\]

\[
\cos A = \frac{1152}{1200}
\]

\[
\cos A = 0.96
\]

Now we find angle \( A \) using the inverse cosine:

\[
A = \cos^{-1}(0.96)
\]

Calculating that gives:

\[
A \approx 16.3 \text{ degrees}
\]

Therefore, the value of angle \( A \) in the triangle is approximately **16.3 degrees**. Thus the correct response is:

**16.3 degrees**.