# Further Applications of Trigonometry

## Parametric Equations

Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in . At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon’s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.
In this section, we will consider sets of equations given by 
   and 
   where 
   is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of 
  the orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations: we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.

### Parameterizing a Curve

When an object moves along a curve—or curvilinear path—in a given direction and in a given amount of time, the position of the object in the plane is given by the x-coordinate and the y-coordinate. However, both 
  and 
 
vary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which both 
  and 
  are dependent functions. In the example in the section opener, the parameter is time, 
  The 
  position of the moon at time, 
  is represented as the function 
  and the 
  position of the moon at time, 
  is represented as the function 
  Together, 
   and 
   are called parametric equations, and generate an ordered pair 
  Parametric equations primarily describe motion and direction.
When we parameterize a curve, we are translating a single equation in two variables, such as 
  and 
  into an equivalent pair of equations in three variables, 
  and 
  One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object’s motion over time.
When we graph parametric equations, we can observe the individual behaviors of 
  and of 
  There are a number of shapes that cannot be represented in the form 
  meaning that they are not functions. For example, consider the graph of a circle, given as 
  Solving for 
  gives 
  or two equations: 
  and 
  If we graph 
  and 
  together, the graph will not pass the vertical line test, as shown in . Thus, the equation for the graph of a circle is not a function.
However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.

### Eliminating the Parameter

In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as 
  and 
  Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter 
  from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.

### Eliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations

For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for 
  We substitute the resulting expression for 
 
into the second equation. This gives one equation in 
  and 
 


### Eliminating the Parameter from Trigonometric Equations

Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.
First, we use the identities:
Solving for 
  and 
  we have
Then, use the Pythagorean Theorem:
Substituting gives

### Finding Cartesian Equations from Curves Defined Parametrically

When we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially “eliminating the parameter.” However, there are various methods we can use to rewrite a set of parametric equations as a Cartesian equation. The simplest method is to set one equation equal to the parameter, such as 
  In this case, 
  can be any expression. For example, consider the following pair of equations.
Rewriting this set of parametric equations is a matter of substituting 
  for 
  Thus, the Cartesian equation is 
 


### Finding Parametric Equations for Curves Defined by Rectangular Equations

Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression to represent 
  and then substitute it into the 
  equation, and it produces the same graph over the same domain as the rectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of parametric equations, and the function does not allow the same values for 
  as the domain of the rectangular equation, then the graphs will be different.

### Key Concepts

1. Parameterizing a curve involves translating a rectangular equation in two variables, 
  and 
  into two equations in three variables, x, y, and t. Often, more information is obtained from a set of parametric equations. See , , and .
2. Sometimes equations are simpler to graph when written in rectangular form. By eliminating 
  an equation in 
  and 
  is the result.
3. To eliminate 
  solve one of the equations for 
  and substitute the expression into the second equation. See , , , and .
4. Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for 
  in one of the equations, and substitute the expression into the second equation. See .
5. There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.
6. Find an expression for 
  such that the domain of the set of parametric equations remains the same as the original rectangular equation. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, eliminate the parameter 
  to rewrite the parametric equation as a Cartesian equation.
For the following exercises, rewrite the parametric equation as a Cartesian equation by building an 
   table.
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting 
   or by setting 
 

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using 
   and 
  Identify the curve.

### Technology

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.
For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

### Extensions
