In this exercise, we explore the concept of polar coordinates, specifically focusing on the point represented by the coordinates \((-2, \frac{5\pi}{3})\). In polar coordinates, a point is defined by a radius \(r\) and an angle \(\theta\). Here, the radius is negative, indicating that we will plot the point in the opposite direction of the angle.

To begin, we identify the angle \(\theta = \frac{5\pi}{3}\). This angle corresponds to a position on the polar graph. However, since the radius \(r = -2\) is negative, we move in the direction opposite to \(\frac{5\pi}{3}\). By counting 2 units in that direction, we find the point \((-2, \frac{5\pi}{3})\) on the graph.

Next, we can generate additional sets of coordinates for the same point by manipulating the angle while keeping the radius consistent. One method is to add multiples of \(2\pi\) to the angle. For instance, adding \(2\pi\) to \(\frac{5\pi}{3}\) gives us:

\[\frac{5\pi}{3} + 2\pi = \frac{5\pi}{3} + \frac{6\pi}{3} = \frac{11\pi}{3}\]

This results in the coordinates \((-2, \frac{11\pi}{3})\), which represents the same point on the graph. Continuing this process, we can add another \(2\pi\) to \(\frac{11\pi}{3}\):

\[\frac{11\pi}{3} + 2\pi = \frac{11\pi}{3} + \frac{6\pi}{3} = \frac{17\pi}{3}\]

Thus, we obtain another set of coordinates: \((-2, \frac{17\pi}{3})\).

To explore further, we can change the radius \(r\) from negative to positive. By negating the radius, we have \(r = 2\). To maintain the same point, we must adjust the angle by adding \(\pi\) to the original angle:

\[\frac{5\pi}{3} + \pi = \frac{5\pi}{3} + \frac{3\pi}{3} = \frac{8\pi}{3}\]

This gives us the coordinates \((2, \frac{8\pi}{3})\), which also represents the same point on the graph.

In summary, we have derived three sets of coordinates for the same point: \((-2, \frac{5\pi}{3})\), \((-2, \frac{11\pi}{3})\), and \((-2, \frac{17\pi}{3})\), along with \((2, \frac{8\pi}{3})\). This illustrates the concept that a single point in polar coordinates can be represented by infinitely many coordinate pairs, depending on the manipulation of the angle and radius.