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In this section, you will:
  • Plot points using polar coordinates.
  • Convert from polar coordinates to rectangular coordinates.
  • Convert from rectangular coordinates to polar coordinates.
  • Transform equations between polar and rectangular forms.
  • Identify and graph polar equations by converting to rectangular equations.

Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see [link] ). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.

An illustration of a boat on the polar grid.

Plotting points using polar coordinates

When we think about plotting points in the plane, we usually think of rectangular coordinates ( x , y ) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates    , which are points labeled ( r , θ ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole    , or the origin of the coordinate plane.

The polar grid is scaled as the unit circle with the positive x- axis now viewed as the polar axis    and the origin as the pole. The first coordinate r is the radius or length of the directed line segment from the pole. The angle θ , measured in radians, indicates the direction of r . We move counterclockwise from the polar axis by an angle of θ , and measure a directed line segment the length of r in the direction of θ . Even though we measure θ first and then r , the polar point is written with the r -coordinate first. For example, to plot the point ( 2 , π 4 ) , we would move π 4 units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in [link] .

Polar grid with point (2, pi/4) plotted.

Plotting a point on the polar grid

Plot the point ( 3 , π 2 ) on the polar grid.

The angle π 2 is found by sweeping in a counterclockwise direction 90° from the polar axis. The point is located at a length of 3 units from the pole in the π 2 direction, as shown in [link] .

Polar grid with point (3, pi/2) plotted.

Plot the point ( 2 , π 3 ) in the polar grid .

Polar grid with point (2, pi/3) plotted.

Plotting a point in the polar coordinate system with a negative component

Plot the point ( 2 , π 6 ) on the polar grid.

We know that π 6 is located in the first quadrant. However, r = −2. We can approach plotting a point with a negative r in two ways:

  1. Plot the point ( 2 , π 6 ) by moving π 6 in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;
  2. Move π 6 in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.

See [link] (a). Compare this to the graph of the polar coordinate ( 2 , π 6 ) shown in [link] (b).

Two polar grids. Points (2, pi/6) and (-2, pi/6) are plotted. They are reflections across the origin in Q1 and Q3.

Plot the points ( 3 , π 6 ) and ( 2 , 9 π 4 ) on the same polar grid.

Points (2, 9pi/4) and (3, -pi/6) are plotted in the polar grid.

Converting from polar coordinates to rectangular coordinates

When given a set of polar coordinates    , we may need to convert them to rectangular coordinates . To do so, we can recall the relationships that exist among the variables x , y , r , and θ .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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