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 answer

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 \( \boxed{18.19^\circ} \).