Solve the Law of Cosines: [tex]c^2 = a^2 + b^2 - 2ab \cos C[/tex] for [tex]\cos C[/tex].

A. [tex]\cos C = \frac{a^2 + b^2 - c^2}{2ab}[/tex]
B. [tex]\cos C = \frac{a^2 + b^2 + c^2}{-2ab}[/tex]
C. [tex]\cos C = \frac{c^2 - a^2 - b^2}{2ab}[/tex]
D. [tex]\cos C = \frac{-c^2 + a^2 + b^2}{-2ab}[/tex]



Answer :

To solve for \(\cos C\) using the Law of Cosines, we start with the given equation:

[tex]\[ c^2 = a^2 + b^2 - 2ab \cos C \][/tex]

We need to isolate \(\cos C\). Here are the steps to do so:

1. Start with the equation:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos C \][/tex]

2. Rearrange the equation to isolate the term involving \(\cos C\). Subtract \(a^2 + b^2\) from both sides:
[tex]\[ c^2 - a^2 - b^2 = -2ab \cos C \][/tex]

3. Divide both sides of the equation by \(-2ab\) to solve for \(\cos C\):
[tex]\[ \cos C = \frac{c^2 - a^2 - b^2}{-2ab} \][/tex]

4. Simplify the equation by factoring out a negative sign in the numerator:
[tex]\[ \cos C = \frac{-(a^2 + b^2 - c^2)}{2ab} \][/tex]

5. Rearranging the terms in the numerator gives:
[tex]\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]

Therefore, the correct expression for \(\cos C\) is:
[tex]\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]

Thus, the correct choice from the given options is:
[tex]\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]