<< 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

The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
Practice Key Terms 9

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask