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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

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Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
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Damian
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
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SUYASH Reply
for screen printed electrodes ?
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s. Reply
of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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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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