9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-1, -π/3)

Verified step by step guidance

1. Recall that a point in polar coordinates is given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
2. Given the point \((-1, -\frac{\pi}{3})\), note that the radius \(r\) is negative. A negative radius means the point is located in the direction opposite to the angle \(\theta\).
3. To find an alternative representation, convert the point by changing the radius to positive and adjusting the angle by adding \(\pi\) (180 degrees) to the original angle: \((r, \theta) = (-1, -\frac{\pi}{3})\) becomes \((1, -\frac{\pi}{3} + \pi)\).
4. Another alternative is to add \$2\pi\( to the angle to keep the radius negative but express the angle in a positive coterminal form: \)(-1, -\frac{\pi}{3} + 2\pi)$.
5. Finally, you can also express the point with a positive radius and an angle coterminal to the one found in step 3 by adding or subtracting multiples of \$2\pi\( to the angle, for example \)(1, -\frac{\pi}{3} + \pi + 2\pi)$.

Key Concepts

Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle, denoted as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how to plot points using these two values is fundamental for graphing in polar coordinates.

Negative Radius in Polar Coordinates

A negative radius means the point is plotted in the direction opposite to the angle θ. For example, (−r, θ) is equivalent to (r, θ + π), reflecting the point across the origin. Recognizing this helps in converting and graphing points with negative radii correctly.

Alternative Representations of Polar Coordinates

Polar points can have multiple equivalent representations by adding or subtracting 2π to the angle or changing the sign of the radius while adjusting the angle by π. This flexibility allows expressing the same point in different forms, which is useful for understanding and comparing polar coordinates.