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

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In summary, the conversation is discussing how to sketch the formula r^2=a^2*cos2t in the x-y plane using polar coordinates. The formula is already in polar form, so there is no need to convert it to Cartesian coordinates. The suggestion is to plot points for different values of theta and a, and then connect them to create a graph. This method will give a better understanding of the relation between r and theta.
  • #1
ponjavic
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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.
 
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  • #2
I don't see why you need to convert to Cartesian coordinates. You can make a sketch of the relation directly. [itex]\theta[/itex] is the angle relative to the x-axis and r is the distance from the origin.
 
  • #3
Try this: take [itex]\theta= 0, \pi/4, \pi/2, \pi/3,[/itex] etc. and see what you get for r: graph those points in polar coordinates.
 
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  • #4
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.
 
  • #5
Tide said:
I don't see why you need to convert to Cartesian coordinates. You can make a sketch of the relation directly. [itex]\theta[/itex] 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?

[itex]\theta= 0, \pi/4, \pi/2, \pi/3,[/itex]
I'll try to work around with this and see what it gets me, what's a^2 though?
 
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  • #6
pon,

Do as Halls suggested. For example, when [itex]\theta = 0[/itex] you know that the point lies on the x axis. Evaluate the expresstion when [itex]\theta = 0[/itex] 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 [itex]\theta = \pi / 4[/itex] 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 [itex]\theta[/itex].

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.
 

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

What is the purpose of sketching/solving polar equations?

The purpose of sketching/solving polar equations is to visually represent and solve mathematical equations that involve polar coordinates. This can help in understanding and analyzing the behavior and patterns of equations in a polar coordinate system.

How do you graph a polar equation like r^2=a^2*cos2t?

To graph a polar equation like r^2=a^2*cos2t, you can use a polar coordinate graphing calculator or plot points manually by substituting different values of theta (t) into the equation and calculating the corresponding values of r. Then, plot these points on a polar graph and connect them to create a curve.

What is the significance of the constant "a" in the equation r^2=a^2*cos2t?

The constant "a" in the equation r^2=a^2*cos2t represents the length of the radius of the polar equation. This means that as the value of "a" changes, the size of the graph will also change. It can also affect the shape and symmetry of the curve.

How do you solve a polar equation like r^2=a^2*cos2t for specific values of theta (t)?

To solve a polar equation like r^2=a^2*cos2t for specific values of theta (t), you can substitute the given value of theta into the equation and solve for the corresponding value of r. This will give you a set of coordinates that can be plotted on the polar graph.

What are some real-life applications of sketching/solving polar equations?

Sketching/solving polar equations can be used in various real-life applications, such as in physics, engineering, and astronomy. For example, it can be used to represent the trajectory of a projectile, the motion of a pendulum, or the orbits of planets and satellites. It can also be used in designing structures or machinery that follow circular or spiral patterns.

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