Using the unit circle, we can see that 3π/2 is equal to 270º
Then we can say that
[tex]\begin{gathered} \cos \frac{3\pi}{2}=0 \\ \\ \sin \frac{3\pi}{2}=-1 \end{gathered}[/tex]
But the csc(x) is
[tex]\csc (x)=\frac{1}{\sin (x)}[/tex]
Therefore
[tex]\csc \frac{3\pi}{2}=\frac{1}{\sin \frac{3\pi}{2}}[/tex]
As we saw, sin(3π/2) = -1, then
[tex]\csc \frac{3\pi}{2}=\frac{1}{-1}=-1[/tex]
Hence, the final answer is:
[tex]\csc \mleft(\frac{3\pi}{2}\mright)=-1[/tex]
