# 1.3 Polar coordinates  (Page 4/16)

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Rewrite the equation $r=\text{sec}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}\theta$ in rectangular coordinates and identify its graph.

$y={x}^{2},$ which is the equation of a parabola opening upward.

We have now seen several examples of drawing graphs of curves defined by polar equations . A summary of some common curves is given in the tables below. In each equation, a and b are arbitrary constants.

A cardioid    is a special case of a limaçon    (pronounced “lee-mah-son”), in which $a=b$ or $a=\text{−}b.$ The rose    is a very interesting curve. Notice that the graph of $r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}2\theta$ has four petals. However, the graph of $r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}3\theta$ has three petals as shown.

If the coefficient of $\theta$ is even, the graph has twice as many petals as the coefficient. If the coefficient of $\theta$ is odd, then the number of petals equals the coefficient. You are encouraged to explore why this happens. Even more interesting graphs emerge when the coefficient of $\theta$ is not an integer. For example, if it is rational, then the curve is closed; that is, it eventually ends where it started ( [link] (a)). However, if the coefficient is irrational, then the curve never closes ( [link] (b)). Although it may appear that the curve is closed, a closer examination reveals that the petals just above the positive x axis are slightly thicker. This is because the petal does not quite match up with the starting point.

Since the curve defined by the graph of $r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\pi \theta \right)$ never closes, the curve depicted in [link] (b) is only a partial depiction. In fact, this is an example of a space-filling curve    . A space-filling curve is one that in fact occupies a two-dimensional subset of the real plane. In this case the curve occupies the circle of radius 3 centered at the origin.

## Chapter opener: describing a spiral

Recall the chambered nautilus introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. [link] shows a spiral in rectangular coordinates. How can we describe this curve mathematically?

As the point P travels around the spiral in a counterclockwise direction, its distance d from the origin increases. Assume that the distance d is a constant multiple k of the angle $\theta$ that the line segment OP makes with the positive x -axis. Therefore $d\left(P,O\right)=k\theta ,$ where $O$ is the origin. Now use the distance formula and some trigonometry:

$\begin{array}{ccc}\hfill d\left(P,O\right)& =\hfill & k\theta \hfill \\ \hfill \sqrt{{\left(x-0\right)}^{2}+{\left(y-0\right)}^{2}}& =\hfill & k\phantom{\rule{0.2em}{0ex}}\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \sqrt{{x}^{2}+{y}^{2}}& =\hfill & k\phantom{\rule{0.2em}{0ex}}\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \text{arctan}\left(\frac{y}{x}\right)& =\hfill & \frac{\sqrt{{x}^{2}+{y}^{2}}}{k}\hfill \\ \hfill y& =\hfill & x\phantom{\rule{0.2em}{0ex}}\text{tan}\left(\frac{\sqrt{{x}^{2}+{y}^{2}}}{k}\right).\hfill \end{array}$

Although this equation describes the spiral, it is not possible to solve it directly for either x or y . However, if we use polar coordinates, the equation becomes much simpler. In particular, $d\left(P,O\right)=r,$ and $\theta$ is the second coordinate. Therefore the equation for the spiral becomes $r=k\theta .$ Note that when $\theta =0$ we also have $r=0,$ so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes $r=a+k\theta$ for arbitrary constants $a$ and $k.$ This is referred to as an Archimedean spiral , after the Greek mathematician Archimedes.

Another type of spiral is the logarithmic spiral, described by the function $r=a·{b}^{\theta }.$ A graph of the function $r=1.2\left({1.25}^{\theta }\right)$ is given in [link] . This spiral describes the shell shape of the chambered nautilus.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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