Solve one parametric equation in terms of the parameter. Substitute the resulting expression for the parameter into the other parametric equation.
Step 2: Substitute the resulting expression for the parameter into the other parametric equation and simplify. The domain for t in both parametric equations is all real numbers and the domain for x in the rectangular equation is all real numbers. Thus no adjustments are need to the domain of the rectangular equation. To link to this Parametric Equations: Eliminating Parameters page, copy the following code to your site:.
Toggle navigation. Example 1: Eliminate the parameter and obtain the rectangular equation, state the domain and sketch the curve for the pair of parametric equations. Example 2: Eliminate the parameter and obtain the rectangular equation, state the domain and sketch the curve for the pair of parametric equations.
Forums Mathematics Calculus. Eliminate Parameter with Sin and Cos. Thread starter Mikeylikesit Start date May 1, Is there a way to get it into the argument of sin? I need this to then figure out intersection points and tangents later. Thanks in advance for any insight!
Related Calculus News on Phys. Homework Helper. Gold Member. Mikeylikesit said:. There is an interesting geometric interpretation of the equation that we can discuss after you try to find it. Thank you for the help! I can't figure what that equation would be I can see one involving arccos and substitution but I think that would be a dead end. Using the derivatives might be of help later, but for now do something simpler. Can you interpret the resulting quadratic equation?
So it seems at this point to find where the graph would intersect itself, we would ask what value for t would give us 2 sets of identical x,y coordinates? Svein Science Advisor. Insights Author. Log in or register to reply now!Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in [link].
Eliminating the parameter when sin and cos are involved.
At any moment, the moon is located at a particular spot relative to the planet. We can solve only for one variable at a 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.
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.
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. 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.
See the graphs in [link]. The arrows indicate the direction in which the curve is generated. To graph the equations, first we construct a table of values like that in [link]. 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. Again, we see that, in [link] cwhen the parameter represents time, we can indicate the movement of the object along the path with arrows. 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.
Here we will review the methods for the most common types of equations. See [link]. The graph of the parametric equation is shown in [link] a. Then, substitute the expression for t. To be sure that the parametric equations are equivalent to the Cartesian equation, check the domains. Eliminate the parameter and write as a rectangular equation. Eliminating the parameter from trigonometric equations is a straightforward substitution.
We can use a few of the familiar trigonometric identities and the Pythagorean Theorem. Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.
Method 1. Then we can substitute the result into the y. Method 2. 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.
Often, more information is obtained from a set of parametric equations. See [link][link]and [link]. See [link][link][link]and [link]. A pair of functions that is dependent on an external factor.Take, for example, a circle.
However, we will never be able to write the equation of a circle down as a single equation in either of the forms above. Each formula gives a portion of the circle. Even if we can narrow things down to only one of these portions the function is still often fairly unpleasant to work with.
So, to deal with some of these problems we introduce parametric equations. This will often be dependent on the problem and just what we are attempting to do. To help visualize just what a parametric curve is pretend that we have a big tank of water that is in constant motion and we drop a ping pong ball into the tank.
Note that this is not always a correct analogy but it is useful initially to help visualize just what a parametric curve is.
Sketching a parametric curve is not always an easy thing to do. This example will also illustrate why this method is usually not the best. Unfortunately, there is no real answer to this question at this point. Sometimes we have no choice, but if we do have a choice we should avoid it. We have one more idea to discuss before we actually sketch the curve. Parametric curves have a direction of motion. So, when plotting parametric curves, we also include arrows that show the direction of motion.
Before we end this example there is a somewhat important and subtle point that we need to discuss first. Had we simply stopped the sketch at those points we are indicating that there was no portion of the curve to the right of those points and there clearly will be. Without limits on the parameter the graph will continue in both directions as shown in the sketch above.
We will often have limits on the parameter however and this will affect the sketch of the parametric equations. Therefore, the parametric curve will only be a portion of the curve above. Here is the parametric curve for this example.
It is now time to take a look at an easier method of sketching this parametric curve. We will sometimes call this the algebraic equation to differentiate it from the original parametric equations.
There will be two small problems with this method, but it will be easy to address those problems. The reality is that when writing this material up we actually did this problem first then went back and did the first problem. Plotting points is generally the way most people first learn how to construct graphs and it does illustrate some important concepts, such as direction, so it made sense to do that first in the notes.
In practice however, this example is often done first. Doing this gives. Often we would have gotten two distinct roots from that equation. It is fairly simple however as this example has shown. All we need to do is graph the equation that we found by eliminating the parameter. As noted already however, there are two small problems with this method. The first is direction of motion.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. How can we tell? But let's say you don't want to try things to see how the parameter can be eliminated.
The problem is that this equation is ugly; arcsine and arccosine are annoying functions. Better to try to get rid of them. So this is a good stopping point. Checking an elementary Mathematics book you'll find there is an trigonometric equation See Marvis's answer. Isn't this enough? Sign up to join this community.
The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Finding Cartesian equations from parametric equations Ask Question. Asked 8 years ago.
Active 1 year, 10 months ago. Viewed 21k times. Rodrigo de Azevedo Active Oldest Votes. Arturo Magidin Arturo Magidin k 39 39 gold badges silver badges bronze badges. Please stop commenting on most of my answers with comments that have nothing but expressions of praise.
Not only do some of them seem rather exaggerated, but, frankly, I'm finding it creepy. I'm feeling stalked, and I don't like it. I guess it can get wierd.Parametric Equations Introduction, Eliminating The Paremeter t, Graphing Plane Curves, Precalculus
This basic trigonometric formula is what Marviz noted you. I already know every single thing stated in this thread except the one thing everyone is unwilling to tell me.
Why is the y and x being added and why it is equal to one?A curve in the plane is said to be parameterized if the coordinates of the points on the curve, xyare represented as functions of a variable t. The variable t is called a parameter and the relations between xy and t are called parametric equations.
The set D is called the domain of f and g and it is the set of values t takes. We arrived at this pair of parametric equations as described above. The picture below shows the graph of the parametric equations. We have labeled three points on the graph that are obtained from different values of the parameter. If the parameter t represented time measured in seconds, and the x and y -coordinates are the position of a moving object, then where would the object be after 3 seconds?
The object would be at 3, 9. The diagram shows the graph of the parametric equations. We have marked three points on the curve corresponding to three values of the parameter.
We have determined the corresponding values of x and y and plotted these points. The diagram shows the result of plotting these points. Through these plotted points we have drawn a smooth curve and the result is shown in the diagram to the right. The orientation of a parameterized curve is the direction determined by increasing values of the parameter. Sometimes arrows are drawn on the curve to denote the orientation.
The diagram shows the same parametric curve we have just studied where we have included some arrows to illustrate the orientation. In this case, the direction of t increasing is from left to right. In this set of exercises you are given two parametric equations. You are to eliminate the parameter to find an expression between y and x.
Click "New" for a new problem. Type your function in the typing area. Once you have entered the expression, press "Check" to see if your answers are correct.
The "Help" button will provide a hint. The "Solve" button will reveal the solution if your checked answer is incorrect. Eliminating the Parameter.
To see what curve these equations define, let's square both x and yand add both terms. This is the equation of the unit circle and so the two parametric equations are a parameterization of the unit circle. We apply the same procedure to eliminate the parameter, namely square x and yand add the terms.
This parametric curve is also the unit circle and we have found two different parameterizations of the unit circle.Enter a problem Calculus Examples Popular Problems. Set up the parametric equation for to solve the equation for. Rewrite the equation as.
eliminate the parameter from x=cos(t), y=cos(t)+sin^2(t)?
Divide each term by and simplify. Divide each term in by. Cancel the common factor of. Cancel the common factor. Divide by. Take the inverse cosine of both sides of the equation to extract from inside the cosine. Replace in the equation for to get the equation in terms of. Draw a triangle in the plane with vertices, and the origin.
Then is the angle between the positive x-axis and the ray beginning at the origin and passing through. Therefore, is. Simplify the numerator. Since both terms are perfect squaresfactor using the difference of squares formulawhere and.
To write as a fraction with a common denominatormultiply by. Write each expression with a common denominator ofby multiplying each by an appropriate factor of. Multiply by.