# Further Applications of Trigonometry

## Polar Coordinates: Graphs

The planets move through space in elliptical, periodic orbits about the sun, as shown in . They are in constant motion, so fixing an exact position of any planet is valid only for a moment. In other words, we can fix only a planet’s instantaneous position. This is one application of polar coordinates, represented as 
 
 We interpret 
  as the distance from the center of the sun and 
  as the planet’s angular bearing, or its direction from the center of the sun. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates.

### Testing Polar Equations for Symmetry

Just as a rectangular equation such as 
  describes the relationship between 
  and 
  on a Cartesian grid, a polar equation describes a relationship between 
  and 
  on a polar grid. Recall that the coordinate pair 
  indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of 
  and extend a ray from the pole (origin) 
  units in the direction of 
  All points that satisfy the polar equation are on the graph.
Symmetry is a property that helps us recognize and plot the graph of any equation. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of 
 
to determine the graph of a polar equation.
In the first test, we consider symmetry with respect to the line 
  (y-axis). We replace 
  with 
  to determine if the new equation is equivalent to the original equation. For example, suppose we are given the equation 
 

This equation exhibits symmetry with respect to the line 
 

In the second test, we consider symmetry with respect to the polar axis ( 
 -axis). We replace 
  with 
  or 
  to determine equivalency between the tested equation and the original. For example, suppose we are given the equation 
  
The graph of this equation exhibits symmetry with respect to the polar axis.
In the third test, we consider symmetry with respect to the pole (origin). We replace 
  with 
  to determine if the tested equation is equivalent to the original equation. For example, suppose we are given the equation 
  
The equation has failed the symmetry test, but that does not mean that it is not symmetric with respect to the pole. Passing one or more of the symmetry tests verifies that symmetry will be exhibited in a graph. However, failing the symmetry tests does not necessarily indicate that a graph will not be symmetric about the line 
  the polar axis, or the pole. In these instances, we can confirm that symmetry exists by plotting reflecting points across the apparent axis of symmetry or the pole. Testing for symmetry is a technique that simplifies the graphing of polar equations, but its application is not perfect.

### Graphing Polar Equations by Plotting Points

To graph in the rectangular coordinate system we construct a table of 
  and 
  values. To graph in the polar coordinate system we construct a table of 
  and 
  values. We enter values of 
 
into a polar equation and calculate 
  However, using the properties of symmetry and finding key values of 
  and 
  means fewer calculations will be needed.

### Finding Zeros and Maxima

To find the zeros of a polar equation, we solve for the values of 
  that result in 
   Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for 
  We use the same process for polar equations. Set 
  and solve for 
 

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of 
  into the equation that result in the maximum value of the trigonometric functions. Consider 
  the maximum distance between the curve and the pole is 5 units.  The maximum value of the cosine function is 1 when 
  so our polar equation is 
  and the value 
   will yield the maximum 
 

Similarly, the maximum value of the sine function is 1 when 
  and if our polar equation is 
  the value 
  will yield the maximum 
  We may find additional information by calculating values of 
  when 
  These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

### Investigating Circles

Now we have seen the equation of a circle in the polar coordinate system. In the last two examples, the same equation was used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted points that produced the graphs. However, the circle is only one of many shapes in the set of polar curves.
There are five classic polar curves: cardioids, limaҫons, lemniscates, rose curves, and Archimedes’ spirals. We will briefly touch on the polar formulas for the circle before moving on to the classic curves and their variations.

### Investigating Cardioids

While translating from polar coordinates to Cartesian coordinates may seem simpler in some instances, graphing the classic curves is actually less complicated in the polar system. The next curve is called a cardioid, as it resembles a heart. This shape is often included with the family of curves called limaçons, but here we will discuss the cardioid on its own.

### Investigating Limaçons

The word limaçon is Old French for “snail,” a name that describes the shape of the graph. As mentioned earlier, the cardioid is a member of the limaçon family, and we can see the similarities in the graphs. The other images in this category include the one-loop limaçon and the two-loop (or inner-loop) limaçon. One-loop limaçons are sometimes referred to as dimpled limaçons when 
  and convex limaçons when 
  
Another type of limaçon, the inner-loop limaçon, is named for the loop formed inside the general limaçon shape. It was discovered by the German artist Albrecht Dürer(1471-1528), who revealed a method for drawing the inner-loop limaçon in his 1525 book Underweysung der Messing. A century later, the father of mathematician Blaise Pascal, Étienne Pascal(1588-1651), rediscovered it.

### Investigating Lemniscates

The lemniscate is a polar curve resembling the infinity symbol 
  or a figure 8. Centered at the pole, a lemniscate is symmetrical by definition.

###  Investigating Rose Curves 

The next type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern.

### Investigating the Archimedes’ Spiral

The final polar equation we will discuss is the Archimedes’ spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE-c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.

### Summary of Curves

We have explored a number of seemingly complex polar curves in this section.  and  summarize the graphs and equations for each of these curves.

### Key Concepts

1. It is easier to graph polar equations if we can test the equations for symmetry with respect to the line 
  the polar axis, or the pole.
2. There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an equation fails a symmetry test, the graph may or may not exhibit symmetry. See .
3. Polar equations may be graphed by making a table of values for 
  and 
 

4. The maximum value of a polar equation is found by substituting the value 
  that leads to the maximum value of the trigonometric expression.
5. The zeros of a polar equation are found by setting 
  and solving for 
  See .
6. Some formulas that produce the graph of a circle in polar coordinates are given by 
  and 
  See .
7. The formulas that produce the graphs of a cardioid are given by 
  and 
  for 
   and 
  See .
8. The formulas that produce the graphs of a one-loop limaçon are given by 
  and 
  for 
  See .
9. The formulas that produce the graphs of an inner-loop limaçon are given by 
  and 
  for 
  
and 
 
See .
10. The formulas that produce the graphs of a lemniscates are given by 
  and 
  where 
  See .
11. The formulas that produce the graphs of rose curves are given by 
  and 
  where 
  if 
  is even, there are 
  petals, and if 
  is odd, there are 
  petals. See  and .
12. The formula that produces the graph of an Archimedes’ spiral is given by 
   See .

### Section Exercises


### Verbal


### Graphical

For the following exercises, test the equation for symmetry.
For the following exercises, graph the polar equation. Identify the name of the shape.

### Technology

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.
For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of 
   and then explain the differences shown in the graphs.

### Extensions

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.