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Rewriting a polar equation in cartesian form

Rewrite the polar equation r = 3 1 2 cos θ as a Cartesian equation.

The goal is to eliminate θ and r , and introduce x and y . We clear the fraction, and then use substitution. In order to replace r with x and y , we must use the expression x 2 + y 2 = r 2 .

                    r = 3 1 2 cos θ r ( 1 2 cos θ ) = 3 r ( 1 2 ( x r ) ) = 3 Use  cos θ = x r  to eliminate  θ .           r 2 x = 3                     r = 3 + 2 x Isolate  r .                   r 2 = ( 3 + 2 x ) 2 Square both sides .          x 2 + y 2 = ( 3 + 2 x ) 2 Use  x 2 + y 2 = r 2 .

The Cartesian equation is x 2 + y 2 = ( 3 + 2 x ) 2 . However, to graph it, especially using a graphing calculator or computer program, we want to isolate y .

x 2 + y 2 = ( 3 + 2 x ) 2          y 2 = ( 3 + 2 x ) 2 x 2            y = ± ( 3 + 2 x ) 2 x 2

When our entire equation has been changed from r and θ to x and y , we can stop, unless asked to solve for y or simplify. See [link] .

Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are hyperbolas.

The “hour-glass” shape of the graph is called a hyperbola . Hyperbolas have many interesting geometric features and applications, which we will investigate further in Analytic Geometry .

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Rewrite the polar equation r = 2 sin θ in Cartesian form.

x 2 + y 2 = 2 y or, in the standard form for a circle, x 2 + ( y 1 ) 2 = 1

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Rewriting a polar equation in cartesian form

Rewrite the polar equation r = sin ( 2 θ ) in Cartesian form.

                  r = sin ( 2 θ ) Use the double angle identity for sine .                   r = 2 sin θ cos θ Use  cos θ = x r  and  sin θ = y r .                   r = 2 ( x r ) ( y r ) Simplify .                   r = 2 x y r 2  Multiply both sides by  r 2 .                 r 3 = 2 x y ( x 2 + y 2 ) 3 = 2 x y As x 2 + y 2 = r 2 , r = x 2 + y 2 .

This equation can also be written as

( x 2 + y 2 ) 3 2 = 2 x y or x 2 + y 2 = ( 2 x y ) 2 3
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Access these online resources for additional instruction and practice with polar coordinates.

Key equations

Conversion formulas cos θ = x r x = r cos θ sin θ = y r y = r sin θ r 2 = x 2 + y 2 tan θ = y x

Key concepts

  • The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
  • To plot a point in the form ( r , θ ) , θ > 0 , move in a counterclockwise direction from the polar axis by an angle of θ , and then extend a directed line segment from the pole the length of r in the direction of θ . If θ is negative, move in a clockwise direction, and extend a directed line segment the length of r in the direction of θ . See [link] .
  • If r is negative, extend the directed line segment in the opposite direction of θ . See [link] .
  • To convert from polar coordinates to rectangular coordinates, use the formulas x = r cos θ and y = r sin θ . See [link] and [link] .
  • To convert from rectangular coordinates to polar coordinates, use one or more of the formulas: cos θ = x r , sin θ = y r , tan θ = y x , and r = x 2 + y 2 . See [link] .
  • Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See [link] , [link] , and [link] .
  • Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See [link] , [link] , and [link] .

Section exercises

Verbal

How are polar coordinates different from rectangular coordinates?

For polar coordinates, the point in the plane depends on the angle from the positive x- axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.

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Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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