The Law of Cosines is a fundamental rule in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its included angles. The standard form of the Law of Cosines for a triangle [tex]\(ABC\)[/tex] with corresponding sides [tex]\(a, b,\)[/tex] and [tex]\(c\)[/tex] is given by:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Let's verify which of the given options matches this standard form:
Option A: [tex]\( a^2 = b^2 + c^2 - 2ac \cos (B) \)[/tex]
- The cosine term here is [tex]\( \cos(B) \)[/tex], and the factor involving the cosine term is [tex]\( 2ac \)[/tex].
- According to the Law of Cosines, the factor should be [tex]\( 2bc \cos (A) \)[/tex] when dealing with side [tex]\(a\)[/tex].
- This option does not match the standard form.
Option B: [tex]\( a^2 = b^2 + c^2 - 2bc \cos (C) \)[/tex]
- The cosine term here is [tex]\( \cos(C) \)[/tex], but the formula should be associating side [tex]\(a\)[/tex] with angle [tex]\(A\)[/tex].
- Additionally, the factor [tex]\( 2bc \cos(C) \)[/tex] should relate sides and angles differently.
- This option does not match the standard form.
Option C: [tex]\( a^2 = b^2 - c^2 - 2bc \cos (A) \)[/tex]
- Here, the formula incorrectly subtracts [tex]\( c^2 \)[/tex] instead of adding it.
- This option is incorrectly arranged and thus does not match the standard form.
Option D: [tex]\( a^2 = b^2 + c^2 - 2bc \cos (A) \)[/tex]
- This option correctly follows the form stated in the Law of Cosines.
- [tex]\(a^2\)[/tex] is correctly expressed in terms of [tex]\(b^2 + c^2\)[/tex], and the cosine term is [tex]\(2bc \cos(A)\)[/tex], which is accurate.
Therefore, the correct option that matches the law of cosines is:
[tex]\[ D. \, a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
So, the correct choice is:
[tex]\[ \boxed{4} \][/tex]
