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POLAR COORDINATES So far we have located a point in a plane by giving the
distances of the point from two perpendicular lines. We can define the location
of a point equally well by noting its distance and bearing. This method is
commonly used aboard ship to show the position of another ship or target. Thus,
3 miles at 35 locates the position of a ship relative to the course of the
ship making the reading. We can use this method to develop curves and bring out
their properties. Assume a fixed direction OX and a fixed point O on the line
in figure 219. The position of any point, P, is fully determined if we know
the directed distance from O to P and the angle that the line OP makes with
reference line OX. The line OP is called the radius vector and the
angle POX is called the polar angle. The radius vector is denoted by
Q, while 0 denotes the polar angle. Point O is the pole or origin.
As in conventional trigonometry, the polar angle is positive when
measured counterclockwise and negative when measured clockwise. However,
unlike the convention established in trigonometry, the
radius vector for polar coordinates is positive only when it is laid off on the
terminal side of the angle. When the radius vector is laid off on the
terminal side of the ray produced beyond the pole (the given angle plus 180'), nates. a
negative value is assigned the radius vector. For this reason, more than one equation may be used in polar
coordinates to describe a given locus. It is sufficient that you remember that
the radius vector can be negative. In this course, however, the radius vector, Q, will always be positive.
Figure 219.Defining the polar coordi TRANSFORMATION FROM CARTESIAN TO POLAR COORDINATES At times you will find working with the equation of a
curve in polar coordinates will be easier than working in Cartesian
coordinates. Therefore, you need to know how to change from one system to the
other. Sometimes both forms are useful, for some properties of the curve may be
more apparent from one form of the equation. We can make transformations by applying the following
equations, which can be derived from figure 220:
Figure 220.Cartesian and polar relationship.
EXAMPLE: Change the equation y=x^{2} from
rectangular to polar coordinates. SOL UTION: Substitute _{Q }cos e for x and Q sin 0 for
y so that we have
or
EXAMPLE: Express the equation of the following circle with its center at (a,0)
and with radius a, as shown in figure 221, in polar coordinates:
SOLUTION: First, expanding this equation gives us _{ }^{} Rearranging
terms, we have
The
use of equation (2.13) gives us
Figure
221.Circle with center (a,0). and
applying the value of x given by equation (2.12), results in
Dividing
both sides by Q, we have the equation of a circle with its center at (a,0) and
radius a in polar coordinates
