<< Chapter < Page Chapter >> Page >

Sketch the graph of r = 3 2 cos θ .

Graph of the limaçon r=3-2cos(theta). Extending to the left.
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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).
Lemniscate
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?
Practice Key Terms 9

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask