15–22. Sets in polar coordinates Sketch the following sets of points.


r = 3

Verified step by step guidance

1. Understand that the equation \(r = 3\) in polar coordinates represents all points that are at a distance of 3 units from the origin (pole).
2. Recall that in polar coordinates, \(r\) is the radius or distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.
3. Since \(r\) is constant and equal to 3, this means the set of points forms a circle centered at the origin with radius 3.
4. To sketch this, draw a circle with center at the origin (0,0) and radius 3 units on the polar coordinate plane.
5. Label the circle and note that it includes all points where \(r = 3\) for any angle \(\theta\) between \(0\) and \(2\pi\).

Key Concepts

Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. This system is especially useful for describing curves and regions with circular symmetry.

Graphing r = Constant in Polar Coordinates

The equation r = 3 describes all points that are exactly 3 units away from the origin, regardless of the angle θ. Graphically, this set forms a circle centered at the origin with radius 3. Understanding this helps in sketching curves defined by constant radius values.

Conversion Between Polar and Cartesian Coordinates

To better visualize or analyze polar equations, converting between polar (r, θ) and Cartesian (x, y) coordinates is useful. The formulas x = r cos θ and y = r sin θ allow translation of points, enabling the use of familiar Cartesian graphing techniques for polar curves.