Question

Use the Law of Cosines, a2=b2+c2−2bc⋅cosA , find the value of ∠A in a right triangle, where a=12 , b=35 , and c=37 .

(1 point)
Responses

89.9°89.9 degrees

42.1°42.1 degrees

18.9°18.9 degrees

71.0°

Answer 1

In a right triangle, if \( a \) is the length of the side opposite angle \( A \), we can use the Law of Cosines to find the angle \( A \). Given that \( a = 12 \), \( b = 35 \), and \( c = 37 \):

Using the Law of Cosines formula:

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

We will plug in the values:

\[
12^2 = 35^2 + 37^2 - 2 \cdot 35 \cdot 37 \cdot \cos A
\]

Calculating the squares:

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

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

\[
144 = 2594 - 2 \cdot 35 \cdot 37 \cdot \cos A
\]

Now, we isolate the cosine term:

\[
2 \cdot 35 \cdot 37 \cdot \cos A = 2594 - 144
\]

This simplifies to:

\[
2 \cdot 35 \cdot 37 \cdot \cos A = 2450
\]

Calculating \( 2 \cdot 35 \cdot 37 \):

\[
2 \cdot 35 = 70
\]
\[
70 \cdot 37 = 2590
\]

Now, substituting back into the equation:

\[
2590 \cdot \cos A = 2450
\]

Dividing both sides by 2590:

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

Calculating the right side:

\[
\cos A \approx 0.9456
\]

Now, we use the inverse cosine function to find angle \( A \):

\[
A = \cos^{-1}(0.9456)
\]

Calculating \( A \) using a calculator or cosine inverse table gives:

\[
A \approx 18.9^\circ
\]

Thus, the value of \( \angle A \) is:

\[
\boxed{18.9^\circ}
\]