Converting Polar to Cartesian Coordinates

[tina_081493]

I was given the problem r=2sin(2(θ)). I'm supposed to write the equation in the Cartesian Coordinates. I understand the basics to this but I'm not really sure how I'm supposed to write the equation when I have x=2sin(2(θ))cos(θ) and y=2sin(2θ)sin(θ).

 

[Trifis]

Tip: Square x and y and try to add them. Then try to subtitue θ.

 

[tina_081493]

I ended up getting the equation for a circle in the cartesian coordinates which does not match the graph for the polar equation.

 

[Trifis]

I ended up getting the equation for a circle in the cartesian coordinates which does not match the graph for the polar equation.

Could you write down your final result? I doubt that it is a circle equation...

 

[tina_081493]

Ok so as I was typing out my work, I realized a mistake I made. So now I'm down to:

(cosθ)^2+(sinθ)^2+2cosθsinθ=1

Am I going to need to use some kind of trig substitution to solve this now?

 

[Trifis]

I don't know how did you come up with that :/

If you square x and y, you get respectively: x²=4sin²(2θ)cos²(θ) and y²=4sin²(2θ)sin²(θ).
Now you've got to add the two equations and apply the distributive law to the second part in order to get rid of the cos²(θ)+sin²(θ) terms. It should be straightforward from now on. Your final step is to subtitue θ in the remaining 4sin²(2θ) term.

 

[tina_081493]

I took x²+y²=r²

Because of the r², the 4sin²2θ canceled out, leaving me with what I got.

 

[Trifis]

Your final equation should only contain x's and y's.

x²+y²=4sin²(2θ) right? Subtitue θ and you are done...

 

[tina_081493]

What do I substitute theta with?