Chapter 9.1, Problem 30E

To determine

To find: the four pairs of polar coordinates that represent the point .

Answer to Problem 30E

The four pairs of polar coordinates are $\left(-1,\frac{7\pi }{3}\right),\left(1,\frac{4\pi }{3}\right),\left(-1,-\frac{5\pi }{3}\right),\left(1,-\frac{2\pi }{3}\right)$ .

Explanation of Solution

Given information:

The point is $U\left(-1,\frac{\pi }{3}\right)$ .

Calculation: to find the four pairs that represents the same point, first find the polar coordinates of points,

The other polar coordinates can be obtained as:

  $\left(r,\theta +2\pi k\right)\text{ or }\left(-r,\theta +\left(2k+1\right)\pi \right)$

Now, substitute the value of $r\text{ and }\theta$ ,

Thus, the four pair of polar coordinates can be $\left(-1,\frac{7\pi }{3}\right),\left(1,\frac{4\pi }{3}\right),\left(-1,-\frac{5\pi }{3}\right),\left(1,-\frac{2\pi }{3}\right)$ .