Introduction
The name polar coordinates comes from
thinking of the origin of the plane at (0,0) as being the pole
of the coordinate system. The standard way of finding points on the
plane is do give the x and y coordinate so that, for example, (1,2) is
one units to the right of the origin and two units above it. Polar
coordinates give a direction and a distance from the origin instead of
an x and y coordinate. The point (1,2) is in a direction about 63.4
degrees counterclockwise of the positive x axis and at a
distance of the square root of five (about 2.236) units from the
origin, as shown below.
Since angle repeat every 360 degrees, each point can be specified
in many ways in polar coordinates. The point with radius 3 and angle
60 degrees is the same as the point with radius 3 and angle 420
degrees. To keep things simple we always keep the angle in the range
from 0360 degrees.
Functions in polar coordinates.
Consider a circle of radius 4 centered at the origin.
The set of points that make up this circle satisfy the equation
but this set of points isn't a function y=f(x) at all: it fails the
vertical line test to be a function. In polar coordinates we make the
radius of function of the angle and the equation of the circle of
radius four is just r=4. The points at a radius of four in
every direction from the origin do form a circle of radius four. In
polar coordinates the vertical line test is for points moving out from
the origin and the circle passes the test.
Polar coordinates are not better or worse than the standard xy
coordinates. Instead there are some problems that are easier to solve
in polar coordinates, such as adding vectors.
Converting back and forth between the two sorts of coordinates isn't
too hard. Consider the following diagram:
Using trigonometry, this diagram implies that
Which we can solve to get equations for converting back and forth
in both directions.
Some common sense is needed to use these equations. If, for
example, x=0 then the angle is vertical  either 90 or 270 degrees.
Likewise, the arctangent function returns results in the range from
90 to 90 degrees. This means that you may need to correct the angle
by adding 180 degrees if the result is in the riegon from 90 to 270
degrees. To see an interesting application of polar coordinates, read
the article on petal curves.
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