# Chapter 6 - Section 6.3 - Polar Coordinates - Exercise Set - Page 742: 22

(a)$$(8, \frac{13\pi }{6})$$ (b)$$(-8, \frac{7\pi }{6})$$ (c)$$(8, -\frac{11\pi }{6})$$

#### Work Step by Step

To plot the point $(r, \theta )=(8, \frac{\pi }{6})$, 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 $r=8$. Because $r \gt 0$, plot the point by going out eight 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)$$(8, \frac{\pi }{6})= (8, \frac{\pi }{6}+2(1)\pi)=(8, \frac{13\pi }{6}),$$(b)$$(8, \frac{\pi }{6})= (-8, \frac{\pi }{6}+\pi +2(0)\pi )=(-8, \frac{7\pi }{6}),$$(c)$$(8, \frac{\pi }{6})= (8, \frac{\pi }{6}+2(-1)\pi)=(8, -\frac{11\pi }{6}).$$

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