How do you express cos( (3 pi)/ 2 ) * cos (( 5 pi) /4 )  without using products of trigonometric functions?

Feb 26, 2016

0

Explanation:

From knowledge of the graph of cosx , we know that

$\cos \left(\frac{3 \pi}{2}\right) = 0$

and $\cos \left(\frac{5 \pi}{4}\right) = - \cos \left(\frac{\pi}{4}\right) = - \frac{1}{\sqrt{2}}$

$\Rightarrow \cos \left(\frac{3 \pi}{2}\right) . \cos \left(\frac{5 \pi}{4}\right) = 0. \left(- \frac{1}{\sqrt{2}}\right) = 0$

Feb 26, 2016

It is equivalent to $0$.

Explanation:

To express $\cos \left(3 \frac{\pi}{2}\right) \cdot \cos \left(5 \frac{\pi}{4}\right)$, without using trigonometric functions, we should first find value of $\cos \left(3 \frac{\pi}{2}\right)$ and $\cos \left(5 \frac{\pi}{4}\right)$ separately.

$\cos \left(3 \frac{\pi}{2}\right)$ is equal $\cos \left(\frac{3 \pi}{2} - 2 \pi\right)$ or $\cos \left(- \frac{\pi}{2}\right)$, which is equal to $\cos \left(\frac{\pi}{2}\right)$. But as latter is equal to zero,

$\cos \left(\frac{3 \pi}{2}\right) = 0$

Although, $\cos \left(\frac{5 \pi}{4}\right) = \cos \left(2 \pi - \frac{5 \pi}{4}\right) = \cos \left(\frac{3 \pi}{4}\right) = - \cos \left(\frac{\pi}{4}\right) = \left(- \frac{1}{\sqrt{2}}\right)$

$\cos \left(3 \frac{\pi}{2}\right) \cdot \cos \left(5 \frac{\pi}{4}\right)$ will still be $0$, as $\cos \left(\frac{3 \pi}{2}\right) = 0$