And you should recognize that it is a circle again

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[And you should recognize that it is a circle!] Again, using the definitions: 2 2 cos sin tan x r y r y x r x y θ θ θ = = = = You can proceed: 2 2 2 2 2 2 2 2 ( 1) 1 ( cos 1) ( sin ) 1 cos 2 cos 1 sin 1 r x y r r r r θ θ θ θ θ + =⇒ + = + + = 2 2 co ( 2cos ) s 0 0 r r r r θ θ = = The above equation yields 2 solutions : 0 r = and 2cos r θ = 0 r = gives you a point (which is called a “pole”) 2cos r θ = is the solution that you are looking for. Just like before: 2 2 ( 1) 1 2cos x y r θ + = = You took a rather complicated equation for the circle and convert to a much simpler one in polar system. And that is one of the reasons why you should use polar coordinate system. We will discuss graphing a polar curve next.
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Phong Do Graph paper in polar system looks different than the regular rectangular grid. You still have the origin but from it, you have radial rays branching out at regular angular increments . The below figure starts out with 0 o and go in 15 o increment. Look closely and you will see that there are also 5 o ticks to give you finer grid. How fine should you divide the graph is up to you. For complex graph, you may have to go as fine as 1 o increment. The radial rays are cut by series of concentric circles to construct the circular grid . Note where the 0 o line is located and by convention, counterclockwise is positive . The origin of the axis is called the pole. The below graph provides you 5 points: A(6, 0 o ) B(2, 60 o ) C(7, 150 o ) D(4, 255 o ) E(7, 315 o ) Can you identify them? By doing so, you just learned how to move around on a polar graph. To avoid confusion, negative sign is not often used. But do you see that D(4, -105 o ) or E(7, -45 o ) defines exactly the same points D or E . In the same spirit, do you see that (-7, 330 o ) defines C(7, 150 o ). At this point, there is one thing about polar coordinate system that may be frustrating to you. Take E(7, -45 o ) for example, you can also get to the sample point by E(7, 315 o ) . And if you think about it, there is an INFINITE number of ways that you can define “E”. And this fact will cause a bit of a trouble later on . Once you know how to locate points on a polar graph, you can begin to draw graphs. Graphing an equation in polar system is no more different than in rectangular system: You first generated a bunch of points and then smoothly connect them together.
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