Question

Which of the following is the exact value of cosine of pi times cosine of pi over 3 minus sine of pi times sin of pi over 3

Answer 1

We know that sin(π) = 0, so this expression is equivalent to:

$\cos \pi\cdot\cos \frac{\pi}{3}-\sin \pi\cdot\sin \frac{\pi}{3}=\cos \pi\cdot\cos \frac{\pi}{3}$

Then, using the following known results:

$\begin{gathered} \cos \pi=-1 \\ \cos \frac{\pi}{3}=\frac{1}{2} \end{gathered}$

We conclude that:

$\cos \pi\cdot\cos \frac{\pi}{3}-\sin \pi\cdot\sin \frac{\pi}{3}=-\frac{1}{2}$