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 \( \cos A \):
- Substitute the values into the equation:
\[ 7^2 = 24^2 + 25^2 - 2 \cdot 24 \cdot 25 \cdot \cos A \]
- Calculate \( 7^2 \), \( 24^2 \), and \( 25^2 \):
\[ 49 = 576 + 625 - 1200 \cdot \cos A \] \[ 49 = 1201 - 1200 \cdot \cos A \]
- Rearranging to isolate \( \cos A \):
\[ 1200 \cdot \cos A = 1201 - 49 \] \[ 1200 \cdot \cos A = 1152 \] \[ \cos A = \frac{1152}{1200} \]
- Simplify the fraction:
\[ \cos A = \frac{96}{100} = 0.96 \]
- 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 \( 16.26^\circ \).