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 .

Answer 1

To find the angle \( A \) in the right triangle using the Law of Cosines, we can plug the provided values into the formula:

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

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

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

\[
a^2 = 12^2 = 144
\]

\[
b^2 = 35^2 = 1225
\]

\[
c^2 = 37^2 = 1369
\]

Now, substituting into the Law of Cosines equation:

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

Calculating \( 1225 + 1369 \):

\[
1225 + 1369 = 2594
\]

Now, substitute that into the equation:

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

Next, we simplify to isolate \( \cos A \):

Subtract \( 2594 \) from both sides:

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

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

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

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

Now substitute that back into the equation:

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

Dividing both sides by \(-2590\):

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

Calculating this fraction:

\[
\cos A \approx 0.9465
\]

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

\[
A = \cos^{-1}(0.9465)
\]

Using a calculator to find \( A \):

\[
A \approx 18.19^\circ
\]

Thus, the value of angle \( A \) is approximately \( \boxed{18.19^\circ} \).