Question

How to find the value of $\cos \left(\frac{3\mathrm{\pi }}{2}\right)$?

Answer 1

Find the value of $\cos \left(\frac{3\mathrm{\pi }}{2}\right)$

We know that

$\cos \left(\alpha +\beta \right)=\cos \left(\alpha \right)\cos \left(\beta \right)-\sin \left(\alpha \right)\sin \left(\beta \right)$

Now by substituting the value of $\alpha =\pi$ and $\beta =\frac{\pi }{2}$in the trigonometric identity we get

$$\begin{gathered} \cos \left(\pi +\frac{\pi }{2}\right)=\cos \left(\pi \right)\cos \left(\frac{\pi }{2}\right)-\sin \left(\pi \right)\sin \left(\frac{\pi }{2}\right) \\ \Rightarrow \cos \left(\pi +\frac{\pi }{2}\right)=0-0\left[\because \cos \left(\frac{\pi }{2}\right)=0,\sin \left(\frac{\pi }{2}\right)=1,\cos \left(\pi \right)=-1\mathrm{and}\sin \left(\pi \right)=0\right] \\ \Rightarrow \cos \left(\frac{3\pi }{2}\right)=0 \end{gathered}$$

Hence the value of $\cos \left(\frac{3\mathrm{\pi }}{2}\right)$ is $0$.