Question

The law of cosines is a^2+b^2-2abcos(C). Find the value of 2abcos(C).A. 37B. -40C. 40D. 20

Answer 1

The cosine rule is shown below:

$c^2=a^2+b^2-2ab\cos C$

The small letters are the side lengths and capital letters are the angles.

From the triangle shown, we can write:

$\begin{gathered} c^2=a^2+b^2-2ab\cos C \\ 2^2=4^2+5^2-2(4)(5)\cos C \end{gathered}$

We can simplify and solve for the angle C. The steps are shown below:

$\begin{gathered} 2^2=4^2+5^2-2(4)(5)\cos C \\ 4=16+25-40\cos C \\ 4=41-40\cos C \\ 40\cos C=41-4 \\ 40\cos C=37 \\ \cos C=\frac{37}{40} \\ C=\cos ^{-1}(\frac{37}{40}) \\ C=22.33 \end{gathered}$

Now, we can find the value of "2ab cos(C)". Shown below:

$\begin{gathered} 2ab\cos C \\ =2(4)(5)\cos (22.33) \\ =40\cos (22.33) \\ =40\times0.925 \\ =37 \end{gathered}$

Thus, the answer is 37.

Correct Answer

A