Parametric -> Implicit Equations

[JC3187]

Hi guys,

I have done what I can with the following:

Given a parametric curve x = xsint, y = sin(2t) where t is in R.

Find an implicit equation of this curve.

MY ANSWER:
y = 2costsint = costx

Therefore sint = x/2, cost = y/x

sin^2(t) + cos^2(t) = x^2 / 4 + y^2 / x^2 = 1

Would this be right?
I understand that x != 0 but why can't I just multiply every value underlined by 4x^2?

Thank you, any input is appreciated!

 

[LCKurtz]

Hi guys,

I have done what I can with the following:

Given a parametric curve x = xsint, y - sin(2t) where t is in R.

Please correct the typo's so we know what the actual equations are.

 

[LCKurtz]

Hi guys,

I have done what I can with the following:

Given a parametric curve x = xsint, y - sin(2t) where t is in R.

I will assume, from your work below, that the equations are $x=2\sin t,~y=\sin(2t)$

Find an implicit equation of this curve.

MY ANSWER:
y = 2costsint = costx

Therefore sint = x/2, cost = y/x

sin^2(t) + cos^2(t) = x^2 / 4 + y^2 / x^2 = 1

Would this be right?
I understand that x != 0 but why can't I just multiply every value underlined by 4x^2?

Thank you, any input is appreciated!

Yes, you can multiply through by $4x^2$. The original equations certainly allow $x=0$ and the only problem is that you divided by $x$ in your solution. You could have derived the same equation as you get when you multiply through by $4x^2$ without ever dividing by $x$. So you should give the answer as $$
4x^2 = 4y^2+x^4$$or some variation of that.

 

[SammyS]

Hi guys,

I have done what I can with the following:

Given a parametric curve x = xsint, y - sin(2t) where t is in R.

Find an implicit equation of this curve.

MY ANSWER:
y = 2costsint = costx

Therefore sint = x/2, cost = y/x

sin^2(t) + cos^2(t) = x^2 / 4 + y^2 / x^2 = 1

Would this be right?
I understand that x != 0 but why can't I just multiply every value underlined by 4x^2?

Thank you, any input is appreciated!

Do you mean

x = sin(t)

and

y = sin(2t)

?​

 

[JC3187]

Yes sorry That is what I meant.

How do you derive it from the original equation?
x=2sint, y=sin(2t) without dividing by x?

 

[LCKurtz]

Yes sorry That is what I meant.

How do you derive it from the original equation?
x=2sint, y=sin(2t) without dividing by x?

$$x^2+y^2=4\sin^2 t + (2\sin t\cos t)^2=4\sin^2t(1+\cos^2 t)
=4\sin^2 t(2-\sin^2 t) = x^2(2-\frac{x^2} 4) = \frac{x^2(8-x^2)}{4}$$