# Parametric and Cartesian Equations

• lemurs
In summary, the conversation is about parametric equations and how to solve them. The participants discuss the process of solving parametric equations, specifically using trigonometric functions and Pythagorean theorem. They also mention the process of converting a Cartesian equation into a parametric one.
lemurs
ok I am give a parametric equations of

x= 4 cos t and y=5 sin t

I know that i have to solve the x equation for t then stick it in the y equation but i getting stuck or not rembering some simple stuff i should be.

I believe i get t= cos(inv) (x/4) and substiute it into t in y.

if so how so i simplify

5 sin(cos(inv)(x/4))

$$\frac{x}{4} = \cos t$$$$\frac{y}{5} = \sin t$$$$\sin^{2} t + \cos^{2}t = 1$$

so then i would justs substiute in those in so it would be
x^2/16 +y^2/25

lemurs said:
so then i would justs substiute in those in so it would be
x^2/16 +y^2/25
Don't forget they equal one.

in general, sin(arcos(x)) can be solved by drawing a right triangle. arcos(x) is an angle theta whose cosine is x. So pick one of the angles that's not 90 degrees. Label that theta. Since theta is arcos(x), the adjacent side is x, and the hypotenuse is 1. So sin(theta) is opposite over hypotenuse. You can get the opposite side by pythagoras, and you're done.

ok i understnad that now but how would you do the reverse.. go form given a cartesian equation to a cause given X^2-y^2=1 how would you solve that

You're trying to make that into a parametric form? We know cosh2t - sinh2t = 1. so if you let x=cosht, and y=sinht, it works

In general, if you're completely at a loss, you can try to solve for y=f(x), let x=t, and y=f(t)

## 1. What is the difference between parametric and Cartesian equations?

Parametric equations describe the relationship between two variables, usually denoted as x and y, in terms of a third independent variable, usually denoted as t. Cartesian equations, on the other hand, describe the relationship between x and y directly, without the use of an independent variable.

## 2. When are parametric equations most commonly used?

Parametric equations are most commonly used in situations where the motion or behavior of a system is best described by a changing value of t, such as in physics, engineering, and computer graphics.

## 3. How do you convert between parametric and Cartesian equations?

To convert from parametric equations to Cartesian equations, simply eliminate the parameter t by solving for it in one equation and substituting it into the other equation. To convert from Cartesian equations to parametric equations, solve for both x and y in terms of t and set them equal to the original equations.

## 4. What are the benefits of using parametric equations?

Parametric equations allow for a more flexible and intuitive way of representing curves and shapes, as well as describing the motion and behavior of systems. They also allow for easier calculations and graphing of complex curves.

## 5. Can parametric equations be used to describe three-dimensional objects?

Yes, parametric equations can be extended to three dimensions, where an additional parameter, usually denoted as u, is used to describe the z-coordinate. This allows for the representation of complex three-dimensional curves and shapes.

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