College Trigonometry, 2011a

938 Applications of Trigonometry
11.5 Graphs of Polar Equations
In this section, we discuss how to graph equations in polar coordinates on the rectangular coordinate
plane. Since any given point in the plane has infinitely many different representations in polar
coordinates, our ‘Fundamental Graphing Principle’ in this section is not as clean as it was for
graphs of rectangular equations on page 23. We state it below for completeness.
The Fundamental Graphing Principle for Polar Equations
The graph of an equation in polar coordinates is the set of points which satisfy the equation.
That is, a point P (r, θ) is on the graph of an equation if and only if there is a representation of
P ,say(r ′ ,θ ′ ), such that r ′ and θ ′ satisfy the equation.
Our first example focuses on some of the more structurally simple polar equations.
Example 11.5.1. Graph the following polar equations.
1. r =4 2. r = −3 √ 2 3. θ = 5π 4
4. θ = − 3π 2
Solution. In each of these equations, only one of the variables r and θ is present making the other
variable free. 1 This makes these graphs easier to visualize than others.
1. In the equation r =4,θ is free. The graph of this equation is, therefore, all points which
have a polar coordinate representation (4,θ), for any choice of θ. Graphically this translates
into tracing out all of the points 4 units away from the origin. This is exactly the definition
of circle, centered at the origin, with a radius of 4.
y
y
4
θ>0
θ