esiLimelawcuu

2023-02-18

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

Julien Zavala

Step $1$: Angle conversion from radians to degrees:
$\frac{3\pi }{2}=\frac{3×\pi ×180}{\pi ×2}\frac{3\pi }{2}=270°$
$\mathrm{sin}\left(\frac{3\pi }{2}\right)=\mathrm{sin}\left(270°\right)$
Step $2$: Find the required value:
As we know, $\mathrm{sin}\left(3x\right)=3\mathrm{sin}\left(x\right)-4{\mathrm{sin}}^{3}\left(x\right)$
$270$ can be written as :
$270=3×\left(90\right)$
$\mathrm{sin}\left(270°\right)=\mathrm{sin}\left(3×90°\right)\mathrm{sin}\left(270°\right)=3\mathrm{sin}\left(90°\right)-4{\mathrm{sin}}^{3}\left(90°\right)\mathrm{sin}\left(270°\right)=3-4\mathrm{sin}\left(270°\right)=-1$$\left(\mathrm{sin}\left(90°\right)=1\right)$
Hence, the value of $\mathrm{sin}\left(\frac{3\pi }{2}\right)$ is $-1$.

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