Answer
(a)$$(4, \frac{5\pi }{2})$$
(b)$$(-4, \frac{3\pi }{2})$$
(c)$$(4, -\frac{3\pi }{2})$$
Work Step by Step
To plot the point $(r, \theta )=(4, \frac{\pi }{2})$, begin with the $\frac{\pi }{2}$ angle. Because $\frac{\pi }{2}$ is a positive angle, draw $\theta = \frac{\pi }{2}$ counterclockwise from the polar axis. Now consider $r=4$. Because $r \gt 0$, plot the point by going out four units on the terminal side of $\theta$.
Please note that if $n$ is any integer, the point $(r, \theta )$ can be represented as$$(r, \theta ) = (r, \theta +2n\pi ) \\ \text{or} \\ (r, \theta )= (-r, \theta +\pi + 2n\pi ).$$So we have
(a)$$(4, \frac{\pi }{2})= (4, \frac{\pi }{2}+2(1)\pi)=(4, \frac{5\pi }{2}),$$(b)$$(4, \frac{\pi }{2})= (-4, \frac{\pi }{2}+\pi +2(0)\pi )=(-4, \frac{3\pi }{2}),$$(c)$$(4, \frac{\pi }{2})= (4, \frac{\pi }{2}+2(-1)\pi)=(4, -\frac{3\pi }{2}).$$
