8.4 Polar coordinates: graphs (Page 5/16)

Precalculus

Page 5 / 16

Try It

Sketch the graph of $r = 3 - 2 \cos θ .$

Graph of the limaçon r=3-2cos(theta). Extending to the left.

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.

A General Note

Formulas for inner-loop limaçons

The formulas that generate the inner-loop limaçons are given by $r = a \pm b \cos θ$ and $r = a \pm b \sin θ$ 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] .

$θ$	$0$	$\frac{π}{6}$	$\frac{π}{3}$	$\frac{π}{2}$	$\frac{2 π}{3}$	$\frac{5 π}{6}$	$π$	$\frac{7 π}{6}$	$\frac{4 π}{3}$	$\frac{3 π}{2}$	$\frac{5 π}{3}$	$\frac{11 π}{6}$	$2 π$
$r$	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!

Investigating lemniscates

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

A General Note

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 $a \neq 0.$ 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 $θ = \frac{π}{2},$ 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 $θ = \frac{π}{2},$ 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 $u = 2 θ .$

\begin{array}{l} 0 = 4 \cos 2 θ \\ 0 = 4 \cos u \\ 0 = \cos u \\ \cos^{- 1} 0 = \frac{π}{2} \\ u = \frac{π}{2} & Substitute 2 θ back in for u . \\ 2 θ = \frac{π}{2} \\ θ = \frac{π}{4} \end{array}

So, the point $(0, \frac{π}{4})$ 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,

\begin{matrix} r^{2} = 4 \cos (0) \\ r^{2} = 4 (1) = 4 \\ r = \pm \sqrt{4} = 2 \end{matrix}

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

Make a table similar to [link] .

$θ$	0	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
$r$	2	$\sqrt{2}$	0	$\sqrt{2}$	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!

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.

A General Note

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 $a \neq 0.$ If $n$ is even, the curve has $2 n$ petals. If $n$ is odd, the curve has $n$ petals. See [link] .

Questions & Answers

what does mean opportunity cost?

Aster Reply

what is poetive effect of population growth

Solomon Reply

what is inflation

Nasir Reply

what is demand

Eleni

what is economics

IMLAN Reply

economics theory describes individual behavior as the result of a process of optimization under constraints the objective to be reached being determined by

Kalkidan

Economics is a branch of social science that deal with How to wise use of resource ,s

Kassie

need

WARKISA

Economic Needs: In economics, needs are goods or services that are necessary for maintaining a certain standard of living. This includes things like healthcare, education, and transportation.

Kalkidan

What is demand and supply

EMPEROR Reply

deman means?

Alex

what is supply?

Alex

ex play supply?

Alex

Money market is a branch or segment of financial market where short-term debt instruments are traded upon. The instruments in this market includes Treasury bills, Bonds, Commercial Papers, Call money among other.

murana Reply

good

Kayode

what is money market

umar Reply

Examine the distinction between theory of comparative cost Advantage and theory of factor proportion

Fatima Reply

What is inflation

Bright Reply

a general and ongoing rise in the level of prices in an economy

AI-Robot

What are the factors that affect demand for a commodity

Florence Reply

price

Kenu

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 9

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

	9 Investigating the archimedes’ spiral By OpenStax Read Online Course
	7 Investigating lemniscates By OpenStax Read Online Course
	7 AP 07 Axial Skeleton MCQ By OpenStax Start Quiz
	8 AP 08 Appendicular Skeleton Essay By OpenStax Start Flashcards
	1 Human A&P 2- Test 1 By Madison Christian Start Flashcards
	4 BOD Hemolymphatic -Dr. Han By Brooke Delaney Start Exam
	23 AP 23 Digestive System MCQ By OpenStax Start Quiz
	40 Biology 40 The Circulatory System MCQ By OpenStax Start Quiz
	Computer System Engineering By Robert Morris Start Exam
	3 Biology 03 Biological Macromolecules MCQ By OpenStax Start Quiz
	7 Neuroanatomy 07 The Visual System By Stephen Voron Start Quiz
	6 Psychology MCQ 2010 2 Exam By John Gabrieli Start Exam