Answer
(a)$$(12, \frac{8\pi }{3})$$
(b)$$(-12, \frac{5\pi }{3})$$
(c)$$(12, -\frac{4\pi }{3})$$
Work Step by Step
To plot the point $(r, \theta )=(12, \frac{2\pi }{3})$, begin with the $\frac{2\pi }{3}$ angle. Because $\frac{2\pi }{3}$ is a positive angle, draw $\theta = \frac{2\pi }{3}$ counterclockwise from the polar axis. Now consider $r=12$. Because $r \gt 0$, plot the point by going out twelve 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)$$(12, \frac{2\pi }{3})= (12, \frac{2\pi }{3}+2(1)\pi)=(12, \frac{8\pi }{3}),$$(b)$$(12, \frac{2\pi }{3})= (-12, \frac{2\pi }{3}+\pi +2(0)\pi )=(-12, \frac{5\pi }{3}),$$(c)$$(12, \frac{2\pi }{3})= (12, \frac{2\pi }{3}+2(-1)\pi)=(12, -\frac{4\pi }{3}).$$
