# Eliminate Parameter to Find Cartesian Equation of Curve

• Slimsta
In summary, the author attempted to solve for the cartesian equation of a curve but was unsuccessful.
Slimsta

## Homework Statement

eliminate the parameter to find cartesian equation of the curve

## The Attempt at a Solution

@ means delta

x = 4cos@, y=5sin@, -pi/2 <= @ <= pi/2

i have no idea how to do it.. i read the whole chapter and it doesn't make any sense..
so i make x=y in some way.. or what?

do you know any trigonometric identities involving cosδ and sinδ?

rock.freak667 said:
do you know any trigonometric identities involving cosδ and sinδ?

yeah.. obv.. i just didnt know how to put this delta symbol on here

so which trig identity would you use here?

cos2x + sin2x = 1
but then what do i do about the 4 and 5 that are in there?

Slimsta said:
cos2x + sin2x = 1
but then what do i do about the 4 and 5 that are in there?

Right so

cos2δ+sin2δ=1

4 and 5 are constants, so you can just divide by them to get sinδ and cosδ

rock.freak667 said:
Right so

cos2δ+sin2δ=1

4 and 5 are constants, so you can just divide by them to get sinδ and cosδ

so that would mean, cos2δ = x2 = (4cosδ)2?

x2+y2 = (4cosδ)2 + (5sinδ)2 = 1
is this correct?

Slimsta said:
so that would mean, cos2δ = x2 = (4cosδ)2?

x2+y2 = (4cosδ)2 + (5sinδ)2 = 1
is this correct?

no, that doesn't make sense, the identity is:
cos2δ + sin2δ = 1

start with that form and think about what you have to divide x & y by to substitute into it

Slimsta said:
cos2x + sin2x = 1
but then what do i do about the 4 and 5 that are in there?

rock.freak667 said:
Right so

cos2δ+sin2δ=1

4 and 5 are constants, so you can just divide by them to get sinδ and cosδ
$(4sin(\delta))^2+ (5cos(\delta))^2= 16sin^2(\delta)+ 25cos^2(\delta)$ is NOT equal to 1. $sin^2(\delta)+ cos^2(\delta)$ is equal to 1.

If $x= 4 cos(\delta)$ then $cos(\delta)= ?$

HallsofIvy said:
$(4sin(\delta))^2+ (5cos(\delta))^2= 16sin^2(\delta)+ 25cos^2(\delta)$ is NOT equal to 1. $sin^2(\delta)+ cos^2(\delta)$ is equal to 1.

If $x= 4 cos(\delta)$ then $cos(\delta)= ?$

oh so $x/4= cos(\delta)$
and then same thing for $x/5= sin(\delta)$

then plug it in the equation..
$(x/5)^2+ (x/4)^2$ = 1

and solve for x or how do i find cartesian equation of the curve?

Slimsta said:
oh so $x/4= cos(\delta)$
and then same thing for $x/5= sin(\delta)$

then plug it in the equation..
$(x/5)^2+ (x/4)^2$ = 1

and solve for x or how do i find cartesian equation of the curve?

$x/4= cos(\delta)$ is true

$x/5= sin(\delta)$ is definitely not, where did you get that?

lanedance said:
$x/4= cos(\delta)$ is true

$x/5= sin(\delta)$ is definitely not, where did you get that?

that was my mistake... i meant $y/5= sin(\delta)$

so from there $(y/5)^2+ (x/4)^2$ = 1
==> $(y^2/25)+ (x^2/16)$ = 1

what i did next on my paper is, solved for y, and this cartesian equation of the curve..
is that right?

be careful though, there is two solutions for y... the one you should use is based on the orginal domainof $\delta$

how do you know what the graph going to look like though?
i mean, what if you have a question like this:
x = 3cos6t
y = 4sin2t

yeah, that is a bit more complicated, the first one simplfies a lot because it is an ellipse

what do you think will happen? think about cycles

well cos6t just means it going to be horizontally compressed (by 1/6 from normal)
same for sin2t just by 1/2

the only thing i can come up with is: cos2x = 2sinxcosx
so would cos6x = 6sinxcosx ? my guess is no..
then i have no idea what to do :/

## 1. What does it mean to eliminate a parameter in finding the Cartesian equation of a curve?

Eliminating a parameter in finding the Cartesian equation of a curve means to express the curve in terms of x and y instead of a parameter (usually represented by t). This allows for the equation to be graphed and analyzed more easily.

## 2. Why is it important to eliminate a parameter in finding the Cartesian equation of a curve?

Eliminating a parameter allows for a clearer understanding of the curve and its properties. It also allows for easier graphing and analysis of the curve.

## 3. What is the process for eliminating a parameter in finding the Cartesian equation of a curve?

The process involves solving for the parameter in terms of x and y, and then substituting that value into the original equation to eliminate the parameter.

## 4. Are there any limitations to eliminating a parameter in finding the Cartesian equation of a curve?

Eliminating a parameter may not always be possible, especially if the curve is complex. In some cases, the resulting equation may also be more difficult to work with than the original parametric equation.

## 5. Can eliminating a parameter be applied to all types of curves?

Yes, the concept of eliminating a parameter can be applied to any type of curve that is defined by a parametric equation. However, the difficulty of the process may vary depending on the complexity of the curve.

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