10.4 Polar coordinates: graphs (Page 5/16)

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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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, 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

	8 Investigating rose curves By OpenStax Read Online Course
	6 Investigating limaçons By OpenStax Read Online Course
	Anthropology Marriage Family Household By Richley Crapo Start Assignment
	10 Sociology 10 Global Inequality MCQ By OpenStax Start Quiz
	General Chemistry I MCQ By Joanna Smithback Start Quiz
	9 Neuroanatomy 09 The Auditory System By Stephen Voron Start Quiz
	8 Arts Society: Theater 8 By Jonathan Long Start Quiz
	7 Neuroanatomy 07 The Visual System By Stephen Voron Start Quiz
	7 Microbiology Unit 3 By Madison Christian Start Test
	Macroeconomics By OpenStax Read Online Course
©flickr: Urko	Music Reviewer MCQ By Edgar Delgado Start Quiz
©flickr:	Math for Economists MCQ By Tony Pizur Start Quiz