Find the Cartesian equation given the parametric equations

chwala
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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 :wink:

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...
 
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You should consider negative x y case.
 
anuttarasammyak said:
You should consider negative x y case.
I seem not to get you. Consider negative cases in what way?
 
I interpreted the equation as well as Wolfram do as below shown.

1674349062272.png
1674349135976.png

As for your
\sqrt[3] x=\cos t ,\sqrt[3] y=\sin t
I am not sure whether \sqrt[3]{-1}=-1 is a conventional expression. Say it is so,
(-1)^{2/6}=1^{1/6}=1 \neq (-1)^{1/3}
That seems inconvenient.
 
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anuttarasammyak said:
I am not sure whether \sqrt[3]{-1}=-1 is a conventional expression.
Sure it is. A negative real number has a real and negative root, plus a couple complex roots.
 
If we allow negative x for ##x^{m/n}##
x^{m/n}=x^{2m/2n}=(x^2)^{m/2n}=(|x|^2)^{m/2n}=|x|^{m/n}
for an example
\sqrt[n]{-1}=\sqrt[n]{1}=1
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,
\sqrt{-1}=i, remains sound, we should just watch as it is and never apply the procedures above.
 
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anuttarasammyak said:
I am not sure whether
\sqrt[3]{-1}=-1 is a conventional expression. Say it is so,
(-1)^{2/6}=1^{1/6}=1 \neq (-1)^{1/3}
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.
 
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