Help with solving parametric equation

[53Mark53]

Homework Statement

Consider the following parametric curve:

x=5cos^7(t)

y=5sin^7(t)

Write it in cartesian form, giving your answer as an equation of the form F(x,y)=c for some function F and some constant c.

The Attempt at a Solution

[/B]
I know that sin^2(t)+cos^2(t) = 1 but I don't think this will be much help to figure this out and I am also unsure how I would find the constant

Any help would be much appreciated

 

[Fightfish]

I know that sin^2(t)+cos^2(t) = 1 but I don't think this will be much help to figure this out

Why not? How are $x$ and $y$ related to $\sin^2(t)$ and $\cos^2(t)$?

 

[53Mark53]

Why not? How are $x$ and $y$ related to $\sin^2(t)$ and $\cos^2(t)$?

would that mean x^2+y^2 = 1

which would mean

(5cos^7(t))^2 +(5sin^7(t))^2 = 1


25cos^9(t) +25sin^9(t) = 1

But what would I do now?

 

[Fightfish]

would that mean x^2+y^2 = 1

Nope, that is not correct. Try to express $\sin^2(t)$ in terms of $y$ and $\cos^2(t)$ in terms of $x$.

 

[53Mark53]

Nope, that is not correct. Try to express $\sin^2(t)$ in terms of $y$ and $\cos^2(t)$ in terms of $x$.

would that mean

y=1-cos^2(t)

x=1-sin^2(t)

 

[Fightfish]

would that mean

y=1-cos^2(t)

x=1-sin^2(t)

No, why would you think that?
You already have $x$ and $y$ defined in your original post: $x = 5 \cos^7(t)$ and $y = 5 \sin^7(t)$.
Can you manipulate $x = 5 \cos^7(t)$ to get $\cos^2(t)$ in terms of $x$? And likewise for $\sin^2(t)$ in terms of $y$.

 

[SammyS]

Homework Statement

Consider the following parametric curve:

x=5cos^7(t)

y=5sin^7(t)

Write it in cartesian form, giving your answer as an equation of the form F(x,y)=c for some function F and some constant c.

The Attempt at a Solution

[/B]
I know that sin^2(t)+cos^2(t) = 1 but I don't think this will be much help to figure this out and I am also unsure how I would find the constant

Any help would be much appreciated

To start: Solve

$x=5\cos^7(t)$​

for $\ \cos(t)\ .$

 

[53Mark53]

To start: Solve

$x=5\cos^7(t)$​

for $\ \cos(t)\ .$

cos(t)=(x/5)^(7/2)

Is this correct?

 

[SammyS]

cos(t)=(x/5)^(7/2)

Is this correct?

No. The exponent is wrong.

 

[53Mark53]

No. The exponent is wrong.

cos(t)=(x/5)^(1/7)

What would i do now?

 

[SammyS]

cos(t)=(x/5)^(1/7)

What would i do now?

That's better.

Do similar for y.

Square each & add.

 

[53Mark53]

That's better.

Do similar for y.

Square each & add.

Thanks I got the right answer now!