Wednesday, April 18, 2007

Section 13.2: polar equations of conics and other curves

POLAR EQUATIONS OF CONICS AND OTHER CURVES

This section gives you the basic information about polar equations and how to relate polar coordinates (r,θ) with cartesian coordinates (x,y).

Polar coordinates are a different way of identifying a point in space. With polar coordinates we use a radius (r) and an angle (θ) to specify a point. The coordinate plane looks like this:











The pole is the origin. The polar axis is the same as the positive x-axis on a cartesian graph.










To relate r, θ, x, and y, we can use the following trigonometric equations.

x^2 + y^2 = r^2
sin θ = y/r








From these equations we can derive values for x,y, and θ.

y= r*sin θ

x=r*cos θ

θ=tan-1(y/x)






Now, we need to know how to plot polar graphs on our calculators. First, we need to change the mode to polar. When you go to the y= menu it will now say r=. The standard equation for a polar graph is r=a+bsinθ or r=a+bcosθ. These equations will result in a limaςon or cardioid.









Limaςon




Cardioid


The last bit of information in this section is the general polar equations for conic sections.




These equations are r= k / (a+bsinθ) or r = k / (a+bcosθ). With these equations you can make a parabola, ellipse, or hyperbola.

Depending on the relationship of a and b you will get a different conic shape.


abs(a) = abs(b) -> parabola


abs(a) is greather than abs(b) -> ellipse


abs(a) is less than abs(b) -> hyperbola




Note: the focii of a parabola will be at the origin as well as one of the focii of a ellipse or hyperbola.



Thats it for now!



Next up is EDWARD




IF YOU DONT KNOW POLAR EQUATIONS THIS WILL HAPPEN TO YOU!



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