Further Applications of Trigonometry

Parametric Equations

Learning Objectives

In this section, you will:

• Parameterize a curve.
• Eliminate the parameter.
• Find a rectangular equation for a curve defined parametrically.
• Find parametric equations for curves defined by rectangular equations.

Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in (Figure). 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 ofthe 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, bothand
vary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which bothandare dependent functions. In the example in the section opener, the parameter is time,Theposition of the moon at time,is represented as the functionand theposition of the moon at time,is represented as the functionTogether, and are called parametric equations, and generate an ordered pairParametric equations primarily describe motion and direction.

When we parameterize a curve, we are translating a single equation in two variables, such asandinto an equivalent pair of equations in three variables,andOne 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 ofand ofThere are a number of shapes that cannot be represented in the formmeaning that they are not functions. For example, consider the graph of a circle, given asSolving forgivesor two equations:andIf we graphandtogether, the graph will not pass the vertical line test, as shown in (Figure). 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.

Parametric Equations

Supposeis a number on an interval,The set of ordered pairs,whereandforms a plane curve based on the parameterThe equationsandare the parametric equations.

Parameterizing a Curve

Parameterize the curvelettingGraph both equations.

Ifthen to findwe replace the variablewith the expression given inIn other words, Make a table of values similar to (Figure), and sketch the graph.

See the graphs in (Figure). It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated asincreases.

Analysis

The arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of

Try It

Construct a table of values and plot the parametric equations:

Finding a Pair of Parametric Equations

Find a pair of parametric equations that models the graph ofusing the parameterPlot some points and sketch the graph.

Ifand we substituteforinto theequation, thenOur pair of parametric equations is

To graph the equations, first we construct a table of values like that in (Figure). We can choose values aroundfromtoThe values in thecolumn will be the same as those in thecolumn becauseCalculate values for the column

The graph ofis a parabola facing downward, as shown in (Figure). We have mapped the curve over the interval shown as a solid line with arrows indicating the orientation of the curve according toOrientation refers to the path traced along the curve in terms of increasing values ofAs this parabola is symmetric with respect to the linethe values ofare reflected across the y-axis.

Try It

Parameterize the curve given by

Finding Parametric Equations That Model Given Criteria

An object travels at a steady rate along a straight path toin the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.

The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The x-value of the object starts atmeters and goes to 3 meters. This means the distance x has changed by 8 meters in 4 seconds, which is a rate of orWe can write the x-coordinate as a linear function with respect to time asIn the linear function templateand

Similarly, the y-value of the object starts at 3 and goes towhich is a change in the distance y of −4 meters in 4 seconds, which is a rate of orWe can also write the y-coordinate as the linear functionTogether, these are the parametric equations for the position of the object, where
and
are expressed in meters and
represents time:

Using these equations, we can build a table of values for and (see (Figure)). In this example, we limited values ofto non-negative numbers. In general, any value ofcan be used.

From this table, we can create three graphs, as shown in (Figure).

Analysis

Again, we see that, in (Figure)(c), when the parameter represents time, we can indicate the movement of the object along the path with arrows.

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 asandEliminating 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 parameterfrom 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 forWe substitute the resulting expression for
into the second equation. This gives one equation inand

Eliminating the Parameter in Polynomials

Givenandeliminate the parameter, and write the parametric equations as a Cartesian equation.

We will begin with the equation forbecause the linear equation is easier to solve for

Next, substituteforin

Analysis

This is an equation for a parabola in which, in rectangular terms,is dependent onFrom the curve’s vertex atthe graph sweeps out to the right. See (Figure). In this section, we consider sets of equations given by the functionsandwhereis the independent variable of time. Notice, bothandare functions of time; so in generalis not a function of

Try It

Given the equations below, eliminate the parameter and write as a rectangular equation foras a function

Eliminating the Parameter in Exponential Equations

Eliminate the parameter and write as a Cartesian equation: and

Isolate

Substitute the expression into

Analysis

The graph of the parametric equation is shown in (Figure)(a). The domain is restricted toThe Cartesian equation,is shown in (Figure)(b) and has only one restriction on the domain,

Eliminating the Parameter in Logarithmic Equations

Eliminate the parameter and write as a Cartesian equation:and

Solve the first equation for

Then, substitute the expression for into the equation.

Analysis

To be sure that the parametric equations are equivalent to the Cartesian equation, check the domains. The parametric equations restrict the domain onto we restrict the domain ontoThe domain for the parametric equationis restricted to we limit the domain onto

Try It

Eliminate the parameter and write as a rectangular equation.

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 forandwe have

Then, use the Pythagorean Theorem:

Substituting gives

Eliminating the Parameter from a Pair of Trigonometric Parametric Equations

Eliminate the parameter from the given pair of trigonometric equations whereand sketch the graph.

Solving forand we have

Next, use the Pythagorean identity and make the substitutions.

The graph for the equation is shown in (Figure).[/hidden-answer]

Analysis

Applying the general equations for conic sections (introduced in Analytic Geometry, we can identifyas an ellipse centered atNotice that whenthe coordinates areand whenthe coordinates areThis shows the orientation of the curve with increasing values of

Try It

Eliminate the parameter from the given pair of parametric equations and write as a Cartesian equation:and

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 asIn this case, can be any expression. For example, consider the following pair of equations.

Rewriting this set of parametric equations is a matter of substitutingforThus, the Cartesian equation is

Finding a Cartesian Equation Using Alternate Methods

Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.

Method 1. First, let’s solve theequation forThen we can substitute the result into the equation.

Now substitute the expression forinto theequation.

Method 2. Solve theequation forand substitute this expression in theequation.

Make the substitution and then solve for

Try It

Write the given parametric equations as a Cartesian equation: and

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 representand then substitute it into theequation, 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 foras the domain of the rectangular equation, then the graphs will be different.

Finding a Set of Parametric Equations for Curves Defined by Rectangular Equations

Find a set of equivalent parametric equations for

An obvious choice would be to letThen But let’s try something more interesting. What if we letThen we have

The set of parametric equations is

See (Figure).

Access these online resources for additional instruction and practice with parametric equations.

Key Concepts

• Parameterizing a curve involves translating a rectangular equation in two variables,andinto two equations in three variables, x, y, and t. Often, more information is obtained from a set of parametric equations. See (Figure), (Figure), and (Figure).
• Sometimes equations are simpler to graph when written in rectangular form. By eliminatingan equation inandis the result.
• To eliminatesolve one of the equations forand substitute the expression into the second equation. See (Figure), (Figure), (Figure), and (Figure).
• Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve forin one of the equations, and substitute the expression into the second equation. See (Figure).
• There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.
• Find an expression forsuch that the domain of the set of parametric equations remains the same as the original rectangular equation. See (Figure).

Section Exercises

Verbal

What is a system of parametric equations?

A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example,and

Some examples of a third parameter are time, length, speed, and scale. Explain when time is used as a parameter.

Explain how to eliminate a parameter given a set of parametric equations.

Choose one equation to solve forsubstitute into the other equation and simplify.

What is a benefit of writing a system of parametric equations as a Cartesian equation?

What is a benefit of using parametric equations?

Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.

Why are there many sets of parametric equations to represent on Cartesian function?

Algebraic

For the following exercises, eliminate the parameterto rewrite the parametric equation as a Cartesian equation.

or

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 andIdentify the curve.

Ellipse

Circle

Parameterize the line fromtoso that the line is atatand atat

Parameterize the line fromtoso that the line is atatand atat

Parameterize the line fromtoso that the line is atatand atat

Parameterize the line fromtoso that the line is atatand atat

Technology

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.

yes, at

For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

–1
0
1

1
2
3

1 -3 1
2 0 7
3 5 17

-1
0
1
2

Extensions

Find two different sets of parametric equations for

Find two different sets of parametric equations for

Find two different sets of parametric equations for