Let's solve the problem step by step using the given values of \(a = 3\), \(b = 4\), and \(c = 5\).
First, recall the law of cosines formula:
[tex]\[ a^2 + b^2 - 2ab \cos C = c^2 \][/tex]
Step 1: Calculate \(c^2\):
[tex]\[ c^2 = 5^2 = 25 \][/tex]
Step 2: Calculate \(a^2 + b^2\):
[tex]\[ a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 \][/tex]
Step 3: Substitute the values we calculated into the law of cosines formula and solve for \(2ab \cos C\):
[tex]\[ a^2 + b^2 - 2ab \cos C = c^2 \][/tex]
[tex]\[ 25 - 2ab \cos C = 25 \][/tex]
Step 4: Rearrange the equation to isolate \(2ab \cos C\):
[tex]\[ 25 - 25 = 2ab \cos C \][/tex]
[tex]\[ 0 = 2ab \cos C \][/tex]
Therefore, we find that:
[tex]\[ 2ab \cos C = 0 \][/tex]
So, the value of \(2ab \cos C\) is \(0\).
Given the choices:
A. -21
B. 24
C. 21
D. -24
The answer is none of the options because [tex]\(2ab \cos C = 0\)[/tex].
