In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−2, − π/2)

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 (polar axis).
2. Note that the given point is \((-2, -\frac{\pi}{2})\). The negative radius means we first consider the point at radius \(2\) but in the opposite direction of the angle \(-\frac{\pi}{2}\).
3. Since the angle \(-\frac{\pi}{2}\) corresponds to rotating \(\frac{\pi}{2}\) radians clockwise from the positive x-axis, identify this direction on the polar coordinate system (which points downward along the negative y-axis).
4. Because the radius is negative, move in the opposite direction of the angle \(-\frac{\pi}{2}\), which means moving \(2\) units in the direction of \(-\frac{\pi}{2} + \pi = \frac{\pi}{2}\) (upward along the positive y-axis).
5. Plot the point \(2\) units away from the origin along the positive y-axis, which corresponds to the adjusted angle \(\frac{\pi}{2}\), completing the plotting of the point with coordinates \((-2, -\frac{\pi}{2})\).

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from the origin (radius r) and an angle θ measured from the positive x-axis. Each point is expressed as (r, θ), where r can be positive or negative, and θ is typically in radians or degrees.

Negative Radius in Polar Coordinates

A negative radius means the point is plotted in the direction opposite to the angle θ. For (−r, θ), you move |r| units along the line at angle θ + π (180 degrees), effectively reflecting the point across the origin.

Angle Measurement and Direction

Angles in polar coordinates are measured counterclockwise from the positive x-axis. Negative angles indicate clockwise rotation. For example, an angle of −π/2 corresponds to a 90-degree rotation clockwise from the positive x-axis.