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Polar Graphs POLAR EQUATIONS AND THEIR GRAPHS It is often necessary to transform from rectangular to polar form or vice versa. The following polar-rectangular relationships are useful in this regard. Strategy for Changing Equations in Rectangular Form to Polar Form 2 2 2 • Use the conversions r= x + y , x= r cos θ , and x= r sin θ to find a polar equation. • Write the polar equation as r in terms of θ . If the equation contains an r 2 you MUST show factoring in the solution path! DO NOT EVER divide by r to arrive at the final answer! Strategy for Changing Equations in Polar Form to Rectangular Form • If the equation contains r and sines or cosines, multiply both sides of the equation by r unless the sine or cosine is in the denominator. If the equation contains a trigonometric ratio other than sine or cosine, use the Reciprocal or Quotient Identities to change to sines and cosines first. • If the equation contains r and a constant, square both sides of the equation. • If the equation contains an angle θ and a constant, turn both sides into a tangent ratio remembering that y tan θ = . x Polar equations can also be graphed by hand in the Polar Coordinate System , however, this task often becomes extremely difficult. That is why we need to know the shape of some of the graphs in polar coordinates. 1 For the following "special" polar equations you need to be able to associate their name with their characteristics and graphs! Equations of Circles Circles with center along one of the coordinate axes and radius a Circle with center at the origin and radius a. NOTE: a is NOT equal to 0 The graphs of Circles are generated as the angle increases from 0 to 2π . a. (equation of a Circle with center at (1, 0) and radius 1) b. (equation of a Circle with center at (0, 0) and radius 2) 2 Equations of Cardioids a , where =1 b The graphs of Cardioids are generated as the angle increases from 0 to 2π . (Equation of a Cardioid) 2 Note that a = 2 and b = 2 and = 1 2 Equations of Limacons , where a and b are NOT equal to 0 a • If < 1, the graph of the Limacon has an inner loop! b a • If 1< < 2 , the graph of the Limacon is "dimpled." b a • If ≥ 2 , the graph of the Limacon is considered "convex." b 3 The graphs of Limacons are generated as the angle increases from 0 to 2π . a. Limacon with inner loop: 2 Note that a = 2 and b = 3 and <1 3 b. Dimpled Limacon: 3 Note that a = 3 and b = 2 and < 2 2 c. Convex Limacon: . Note that the graph is not quite circular! 8 Note that a = 8 and b = 2 and ≥ 2 2 4 Equations of Rose Curves , where a and n are NOT equal to 0 • When n is odd, the entire curve is generated as increases from 0 to . The curve has n petals. • When n is even, the entire curve is generated as increases from 0 to 2 . The curve has 2n petals. a. (equation of a Rose Curve with 3 petals) Note that n = 3 is odd, therefore the rose curve has 3 petals. b. (Equation of a Rose Curve with 8 petals) Note that n = 4 is even, therefore the rose curve has 2(4) = 8 petals. 5 Equations of Lemniscates , where a is NOT equal to 0 The graphs of Lemniscates are generated as the angle increases from 0 to 2 . (Equation of a Lemniscate) In order to graph a lemniscate with a calculator or a computer program we must solve its equation as r in terms of θ . Therefore, and . We will get the same graph no matter which one of the two equations we graph! 6 .
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