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 answer

To find the value of angle \( A \) in a triangle using the Law of Cosines, the formula is given by:

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

In this case, we have \( a = 7 \), \( b = 24 \), and \( c = 25 \).

Plugging in the values, we can rearrange the Law of Cosines to find \( \cos A \):

  1. Substitute the values into the equation:

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

  1. Calculate \( 7^2 \), \( 24^2 \), and \( 25^2 \):

\[ 49 = 576 + 625 - 1200 \cdot \cos A \] \[ 49 = 1201 - 1200 \cdot \cos A \]

  1. Rearranging to isolate \( \cos A \):

\[ 1200 \cdot \cos A = 1201 - 49 \] \[ 1200 \cdot \cos A = 1152 \] \[ \cos A = \frac{1152}{1200} \]

  1. Simplify the fraction:

\[ \cos A = \frac{96}{100} = 0.96 \]

  1. Now, to find the angle \( A \), take the inverse cosine:

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

Calculating \( A \):

\[ A \approx 16.26^\circ \]

Thus, the value of angle \( A \) is approximately \( 16.26^\circ \).