Week 12, Day 2

Today we learned about graphs of polar equations. As a quick review, the polar coordinate system is very similar to that of the rectangular coordinate system. In a polar coordinate grid, as shown below, there will be a series of circles extending out from the pole (or origin) and five different lines passing through the pole to represent the angles at which the extact values are known for the trigonometric functions. Graphing a polar equation is accomplished in pretty much the same manner as rectangular equations are graphed. They can be graphed by point-plotting, using the trigonometric functions period, and using the equation's symmetry (if any). When graphimg rectangular equations by point-plotting you should pick values for x and then evaluate the equation to determine its corresponding y value. For a polar equation, you would pick angle measurements for theta and then evaluate the equation to determine its corresponding r value. 

Symmetry Tests for Polar Coordinates

(1) Replace theta with -theta. If an equivalent equation results, the graph is symmetric with respect to the polar axis (x-axis). (2) Replace theta with -theta and r with -r. If an equivalent equation results, the graph is symmetric with respect to theta = pi/2 (y-axis). (3) Replace r with -r. If an equivalent equation results, the graph is symmetric with respect to the pole (origin). It is possible for a polar equation to fail  test and still exhibit that type of symmetry when you finish graphing the function over a full period.

Polar equations have some general types of equations. 

Learning to recognize the formulas of these equations will help in sketching the graphs. 

Circles in Polar Form

Limacons

1. r = a +/- b sin theta, where a > 0 and b < 0

2. r = a +/- b cos theta, where a > 0 and b < 0

The limacons containing sine will be above the horizontal axis if the sign between a and b is plus or below the horizontal axis if the sign between a and b is minus. If the limacon contains the function cosine then the graph will be either to the right of the vertical axis if the sign is plus or to the left of the vertical axis if the sign is minus. The ratio of a/b will determine the exact shape of the limacon. 

Rose Curves

A rose curve is a graph that is produced from a polar equation in the form of:

r = a sin n theta or r = a cos n theta, where a does not equal 0 and n is an interger > 1

They are called rose curves because the loops that are formed resemble petals. The number of petals that are present will depend on the value of n. The value of a will determine the length of the petals.

Lemniscates

The last type of polar equation that we will cover in this section is lemniscates, which has the shape of a figure-8 or propeller. Lemniscates have the general polar equation of:

r^2 = a^2 sin 2 theta or r^2 = a^2 cos 2 theta, where a does not equal 0

A lemniscate containing the sine function will be symmetric to the pole while the lemniscate containing the cosine function will be symmetric to the polar axis, to theta = pi/2, and the pole. 

Example 1: 

Graph the polar equation r = 1 - 2 cos theta

Identify the type of polar equation. 

The polar equation is in the form of a limacon, r = a - b cos theta.

Find the ratio of a/b to determine the equation's general shape: a/b = 1/2

Since the ratio is less than 1, it will have both an interger and outer loop. The loops will be along the polar axis since the function is cosine and will loop to the left since the sign btwn a and b is minus.

Test for symmetry

Evaluate r at different values of theta. 

Since the equation passes the test for symmetry to the polar axis, we only need to evaluate the equation over the interval [0, theta] and then reflect the graph about the polar axis.

Plot the points

Use the symmetry to complete the graph