Find the Cartesian equation given the parametric equations

[chwala]

Homework Statement

Find the cartesian equation given the parametric equations;

##x=\cos ^3t, y=\sin^3 t ##

Relevant Equations

parametric equations

hmmmmm nice one...boggled me a bit; was trying to figure out which trig identity and then alas it clicked

My take;

##x=(\cos t)^3 ## and ##y=(\sin t)^3##

##\sqrt[3] x=\cos t## and ##\sqrt[3] y=\sin t##

we know that

##\cos^2 t + \sin^2t=1##

therefore we shall have,

##x^{\frac{2}{3}} + y^{\frac{2}{3}}=1##

Any other way is highly appreciated...

 

[anuttarasammyak]

You should consider negative x y case.

 

[chwala]

You should consider negative x y case.

I seem not to get you. Consider negative cases in what way?

 

[anuttarasammyak]

I interpreted the equation as well as Wolfram do as below shown.

As for your
[tex]\sqrt[3] x=\cos t ,\sqrt[3] y=\sin t[/tex]
I am not sure whether [tex]\sqrt[3]{-1}=-1[/tex] is a conventional expression. Say it is so,
[tex](-1)^{2/6}=1^{1/6}=1 \neq (-1)^{1/3}[/tex]
That seems inconvenient.

 

[Mark44]

I am not sure whether [tex]\sqrt[3]{-1}=-1[/tex] is a conventional expression.

Sure it is. A negative real number has a real and negative root, plus a couple complex roots.

 

[anuttarasammyak]

If we allow negative x for ##x^{m/n}##
[tex]x^{m/n}=x^{2m/2n}=(x^2)^{m/2n}=(|x|^2)^{m/2n}=|x|^{m/n}[/tex]
for an example
[tex]\sqrt[n]{-1}=\sqrt[n]{1}=1[/tex]
which is a false statement. So I think we had better use ##x^{m/n}## for only positive x.

[EDIT] In order the defnition of imaginary number unit,
[tex]\sqrt{-1}=i[/tex], remains sound, we should just watch as it is and never apply the procedures above.

 

[Mark44]

I am not sure whether
[tex]\sqrt[3]{-1}=-1[/tex] is a conventional expression. Say it is so,
[tex](-1)^{2/6}=1^{1/6}=1 \neq (-1)^{1/3}[/tex]
That seems inconvenient.

I don't believe there is any problem with ##\sqrt[3]{-1} = (-1)^{1/3}##, both of which are equal to -1.
However, when you start manipulating the exponent, using the properties of fractional exponents, then I agree that you run into trouble when the base is negative.

Having thought about things a bit longer, @anuttarasammyak, I agree with what you are saying. The parametric equations ##x = \cos^3(t)## and ##y = \sin^3(t)## agree with the resulting equation that @chwala got: ##x^{2/3} + y^{2/3} = 1## only when both x and y are positive. For negative values of x or y, which map to values of t in the parametric equations, you run into problems of inconsistency in the non-parametric equation.