Suppose a point has polar coordinates (4,-(5pi )/(3)) with the angle measured in radians. Find two additional polar representations of the point. Write each coordinate In simplest form with the angle in [-2pi ,2pi ] square

To find two additional polar representations of the point with polar coordinates $$(4,-\frac {5\pi }{3})$$ , we can add or subtract multiples of $$2\pi$$ to the angle until it falls within the desired range of $$[-2\pi,2\pi ]$$ .

First, let's add $$2\pi$$ to the angle:
$$-\frac {5\pi }{3} + 2\pi = -\frac {5\pi }{3} + \frac {6\pi }{3} = \frac {\pi }{3}$$

So, one additional polar representation of the point is $$(4,\frac {\pi }{3})$$ .

Next, let's subtract $$2\pi$$ from the angle:
$$-\frac {5\pi }{3} - 2\pi = -\frac {5\pi }{3} - \frac {6\pi }{3} = -\frac {11\pi }{3}$$

Since the angle is negative, we can add $$2\pi$$ to it until it falls within the desired range:
$$-\frac {11\pi }{3} + 2\pi = -\frac {11\pi }{3} + \frac {6\pi }{3} = -\frac {5\pi }{3}$$

So, another additional polar representation of the point is $$(4,-\frac {5\pi }{3})$$ .

Therefore, the two additional polar representations of the point with polar coordinates $$(4,-\frac {5\pi }{3})$$ are $$(4,\frac {\pi }{3})$$ and $$(4,-\frac {5\pi }{3})$$ .