Question

How do you evaluate $\sin \left(\frac{3\mathrm{\pi }}{2}\right)$?

Answer 1

Step $1$: Convert the angle from radians to degrees:

$$\begin{gathered} \frac{3\mathrm{\pi }}{2}=\frac{3\times \mathrm{\pi }\times 180}{\mathrm{\pi }\times 2} \\ \frac{3\mathrm{\pi }}{2}=270° \end{gathered}$$

$\sin \left(\frac{3\mathrm{\pi }}{2}\right)=\sin (270°)$

Step $2$: Find the required value:

As we know, $\sin \left(3x\right)=3\sin \left(x\right)-4{\sin }^3\left(x\right)$

$270$ can be written as :

$270=3\times \left(90\right)$

$$\begin{gathered} \sin (270°)=\sin (3\times 90°) \\ \sin (270°)=3\sin \left(90°\right)-4{\sin }^3\left(90°\right) \\ \sin (270°)=3-4 \\ \sin (270°)=-1 \end{gathered}$$ $\left[\sin \left(90°\right)=1\right]$

Hence, the value of $\sin \left(\frac{3\mathrm{\pi }}{2}\right)$ is $-1$.