Polar curves

 Goal: graph a curve in the polar coordinate system

Polar curve

The same way we graph a function y = f(x) in the cartesian plane, we can generate a curve for the function r = f(θ) in the polar coordinate system. We start by creating a list of values for the independent values θ. Then we create another list for the dependent variable r. This process generates a list of ordered pairs that can be plotted in the polar coordinate system.

Problem-solving strategy

1. Create a table with 2 columns: one for θ and the other for r.

2. Create a list of values for θ

3. Calculate the corresponding values r for each value of θ.

4. Plot each pair (r,θ) in the coordinate plane

5. Connect the points and look for a pattern.

Example

Graph the curve defined by the function r = 4 sinθ. Identify the curve and rewrite the equation in rectangular coordinates.

Solution

The function is a multiple of the sine function. Since sine is periodic the given function is periodic. The period of sine is 2ℼ. The period of the function is 2ℼ.. We choose the values of θ between 0 and 2ℼ. Here is the table of values:

Here is the graph of the function:                                                                                                            

 Rewrite the given equation in rectangular coordinates means writing in function of x and y. To do so, let's use the equation x² + y² = r². Let's multiply the equation r = 4sinθ by r, This gives r² = 4rsinθ. We know that y = 4sinθ. Therefore we have r² = 4y. . Let's substitute r² in the initial equation. We have 

x² + y² = 4y. Let's put it in standard form.   

                                                                                      

This is the equation of a circle of radius 2 and center (0,2) in the rectangular coordinate system'

Practice