Answer
(a)$$(5, \frac{13\pi }{6})$$
(b)$$(-5, \frac{7\pi }{6})$$
(c)$$(5, -\frac{11\pi }{6})$$
Work Step by Step
To plot the point $(r, \theta )=(5, \frac{\pi }{6})$, begin with the $\frac{\pi }{6}$ angle. Because $\frac{\pi }{6}$ is a positive angle, draw $\theta = \frac{\pi }{6}$ counterclockwise from the polar axis. Now consider $r=5$. Because $r \gt 0$, plot the point by going out five 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)$$(5, \frac{\pi }{6})= (5, \frac{\pi }{6}+2(1)\pi)=(5, \frac{13\pi }{6}),$$(b)$$(5, \frac{\pi }{6})= (-5, \frac{\pi }{6}+\pi +2(0)\pi )=(-5, \frac{7\pi }{6}),$$(c)$$(5, \frac{\pi }{6})= (5, \frac{\pi }{6}+2(-1)\pi)=(5, -\frac{11\pi }{6}).$$
