Certainly! To solve for the value of [tex]\(2ab \cos C\)[/tex], let's start with the given law of cosines equation:
[tex]\[ a^2 + b^2 - 2ab \cos C = c^2 \][/tex]
We need to isolate the term [tex]\(2ab \cos C\)[/tex] on one side of the equation. Here are the steps:
1. Start with the original equation:
[tex]\[ a^2 + b^2 - 2ab \cos C = c^2 \][/tex]
2. Subtract [tex]\(a^2 + b^2\)[/tex] from both sides to isolate the term involving [tex]\(\cos C\)[/tex]:
[tex]\[ -2ab \cos C = c^2 - a^2 - b^2 \][/tex]
3. Multiply both sides of the equation by [tex]\(-1\)[/tex] to make [tex]\(2ab \cos C\)[/tex] positive:
[tex]\[ 2ab \cos C = a^2 + b^2 - c^2 \][/tex]
Therefore, the expression [tex]\(2ab \cos C\)[/tex] simplifies to [tex]\(a^2 + b^2 - c^2\)[/tex].
Now, comparing this with the answer choices given:
The correct value for [tex]\(2ab \cos C\)[/tex] is:
[tex]\[ \boxed{a^2 + b^2 - c^2} \][/tex]
However, note that none of the answer choices provided (A, B, C, D) match this algebraic expression directly. The choices given were specific numerical values, but our derived result is in terms of variables [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. If you have specific numerical values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], please provide those so we can calculate the exact numerical result.
