In any triangle \(ABC\) with corresponding side lengths \(a\), \(b\), and \(c\), the law of cosines is a fundamental equation used to relate the lengths of the sides of the triangle to the cosine of one of its angles.
The law of cosines states that for any triangle \(ABC\):
[tex]\[a^2 = b^2 + c^2 - 2bc \cos(A)\][/tex]
Let's carefully go through each option to determine which one correctly represents the law of cosines:
Option A:
[tex]\[a^2 = b^2 + c^2 - 2bc \cos(A)\][/tex]
This equation correctly matches the standard form of the law of cosines.
Option B:
[tex]\[a^2 = b^2 + c^2 - 2ac \cos(B)\][/tex]
This is incorrect because it should relate to angle \(A\) using sides \(b\) and \(c\). The correct form should use angles \(B\) as:
[tex]\[b^2 = a^2 + c^2 - 2ac \cos(B)\][/tex]
Option C:
[tex]\[a^2 = b^2 - c^2 - 2bc \cos(A)\][/tex]
This is incorrect because it wrongly subtracts \(c^2\) instead of adding it. Additionally, the term involving \(2bc \cos(A)\) is correct, but due to the incorrect operation involving \(c^2\), this form is invalid.
Option D:
[tex]\[a^2 = b^2 + c^2 - 25c \cos(C)\][/tex]
This form is incorrect because the coefficient \(25\) is incorrect; it should be \(2\) instead. Also, the correct form in relation to \(a\), \(b\), \(c\), and angle \(C\) would be:
[tex]\[c^2 = a^2 + b^2 - 2ab \cos(C)\][/tex]
Among all the given options, option A correctly represents the law of cosines.
Therefore, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
