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Sketch the graph of r = 3 2 cos θ .

Graph of the limaçon r=3-2cos(theta). Extending to the left.
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Another type of limaçon, the inner-loop limaçon , is named for the loop formed inside the general limaçon shape. It was discovered by the German artist Albrecht Dürer (1471-1528), who revealed a method for drawing the inner-loop limaçon in his 1525 book Underweysung der Messing . A century later, the father of mathematician Blaise Pascal , Étienne Pascal(1588-1651), rediscovered it.

Formulas for inner-loop limaçons

The formulas that generate the inner-loop limaçons are given by r = a ± b cos θ and r = a ± b sin θ where a > 0 , b > 0 , and a < b . The graph of the inner-loop limaçon passes through the pole twice: once for the outer loop, and once for the inner loop. See [link] for the graphs.

Graph of four inner loop limaçons side by side. (A) is r=a+bcos(theta),a<b. Extended to the right. (B) is a-bcos(theta), a<b. Extends to the left. (C) is r=a+bsin(theta), a<b. Extends up. (D) is r=a-bsin(theta), a<b. Extends down.

Sketching the graph of an inner-loop limaçon

Sketch the graph of r = 2 + 5 cos θ .

Testing for symmetry, we find that the graph of the equation is symmetric about the polar axis. Next, finding the zeros reveals that when r = 0 , θ = 1.98. The maximum | r | is found when cos θ = 1 or when θ = 0. Thus, the maximum is found at the point (7, 0).

Even though we have found symmetry, the zero, and the maximum, plotting more points will help to define the shape, and then a pattern will emerge.

See [link] .

θ 0 π 6 π 3 π 2 2 π 3 5 π 6 π 7 π 6 4 π 3 3 π 2 5 π 3 11 π 6 2 π
r 7 6.3 4.5 2 −0.5 −2.3 −3 −2.3 −0.5 2 4.5 6.3 7

As expected, the values begin to repeat after θ = π . The graph is shown in [link] .

Graph of inner loop limaçon r=2+5cos(theta). Extends to the right. Points on edge plotted are (7,0), (4.5, pi/3), (2, pi/2), and (-3, pi).
Inner-loop limaçon
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Investigating lemniscates

The lemniscate is a polar curve resembling the infinity symbol or a figure 8. Centered at the pole, a lemniscate is symmetrical by definition.

Formulas for lemniscates

The formulas that generate the graph of a lemniscate    are given by r 2 = a 2 cos 2 θ and r 2 = a 2 sin 2 θ where a 0. The formula r 2 = a 2 sin 2 θ is symmetric with respect to the pole. The formula r 2 = a 2 cos 2 θ is symmetric with respect to the pole, the line θ = π 2 , and the polar axis. See [link] for the graphs.

Four graphs of lemniscates side by side. (A) is r^2 = a^2 * cos(2theta). Horizonatal figure eight, on x-axis. (B) is r^2 = - a^2 * cos(2theta). Vertical figure eight, on y axis. (C) is r^2 = a^2 * sin(2theta). Diagonal figure eight on line y=x. (D) is r^2 = -a^2 *sin(2theta). Diagonal figure eight on line y=-x.

Sketching the graph of a lemniscate

Sketch the graph of r 2 = 4 cos 2 θ .

The equation exhibits symmetry with respect to the line θ = π 2 , the polar axis, and the pole.

Let’s find the zeros. It should be routine by now, but we will approach this equation a little differently by making the substitution u = 2 θ .

0 = 4 cos 2 θ 0 = 4 cos u 0 = cos u cos 1 0 = π 2 u = π 2 Substitute  2 θ  back in for  u . 2 θ = π 2 θ = π 4

So, the point ( 0 , π 4 ) is a zero of the equation.

Now let’s find the maximum value. Since the maximum of cos u = 1 when u = 0 , the maximum cos 2 θ = 1 when 2 θ = 0. Thus,

r 2 = 4 cos ( 0 ) r 2 = 4 ( 1 ) = 4 r = ± 4 = 2

We have a maximum at (2, 0). Since this graph is symmetric with respect to the pole, the line θ = π 2 , and the polar axis, we only need to plot points in the first quadrant.

Make a table similar to [link] .

θ 0 π 6 π 4 π 3 π 2
r 2 2 0 2 0

Plot the points on the graph, such as the one shown in [link] .

Graph of r^2 = 4cos(2theta). Horizontal lemniscate, along x-axis. Points on edge plotted are (2,0), (rad2, pi/6), (rad2 7pi/6).
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Investigating rose curves

The next type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern.

Rose curves

The formulas that generate the graph of a rose curve    are given by r = a cos n θ and r = a sin n θ where a 0. If n is even, the curve has 2 n petals. If n is odd, the curve has n petals. See [link] .

Graph of two rose curves side by side. (A) is r=acos(ntheta), where n is even. Eight petals extending from origin, equally spaced. (B) is r=asin(ntheta) where n is odd. Three petals extending from the origin, equally spaced.

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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