Sketching/solving polar: r^2=a^2*cos2t

[ponjavic]

Sketch (in the x-y plane): r^2=a^2*cos2t where r and t are polar coordinates.

I simply am not able to convert this formula to x and y.
I have gotten as far as:
(x^2+y^2)^2=a^2(x^2-y^2)
or r^4=a^2(r^2*cost^2-r^2*sint^2)
using r = x^2+y^2, cos2t=1-sint^2, x = rcost and y = rsint but I simply can not get any further.

Please give any hints you think might help me solve this.

 

[Tide]

I don't see why you need to convert to Cartesian coordinates. You can make a sketch of the relation directly. $\theta$ is the angle relative to the x-axis and r is the distance from the origin.

 

[HallsofIvy]

Try this: take $\theta= 0, \pi/4, \pi/2, \pi/3,$ etc. and see what you get for r: graph those points in polar coordinates.

 

[fargoth]

well, there is now way to isolate y from x using cartesian coordianres, its really easier to see what's going on in polar coordinates.
the best way is to do as suggested, and plot what you get on x-y plane.

 

[ponjavic]

I don't see why you need to convert to Cartesian coordinates. You can make a sketch of the relation directly. $\theta$ is the angle relative to the x-axis and r is the distance from the origin.

I don't know how to use polar coordinates to sketch, If I have it in cartesians I could do y=0, y'=0, x=0, find asymptotes and such.
How can I find this to help me sketch in polar coordinates?

$\theta= 0, \pi/4, \pi/2, \pi/3,$
I'll try to work around with this and see what it gets me, what's a^2 though?

 

[Tide]

pon,

Do as Halls suggested. For example, when $\theta = 0$ you know that the point lies on the x axis. Evaluate the expresstion when $\theta = 0$ to find out how far from the origin the corresponding point is. Then place a point r units from the origin and on the x-axis corresponding to that point.

Next try $\theta = \pi / 4$ which you know lies along a line at 45 degrees above the x axis. Find the distance to that point using your formula and place a point that far from the origin and along the line y = x on your graph. Do this for several values of $\theta$.

Also, your graph will depend on the parameter a. Do all of the above for different values of a like a = 1, a = 1/2, a = 2 etc.