Understanding Engineering Mathematics

Another two-dimensional coordinate system in common use is thepolar coordinate
system, especially convenient when dealing with engineering problems relating to circular
motion for example. In polar coordinates an origin and an ‘initial line’Oxradiating from
this origin are chosen. The position of any pointP is then referred to by specifying the
distance,OP, fromOto the point along a ‘radius vector’ whose angle,θ, with the initial
line is also specified. Then the position of a point is specified bypolar coordinates(r, θ)
(Figure 7.1).

x

r

0 x

q

y

y

P

Figure 7.1Cartesian (x,y) and polar (r,θ) coordinates in a plane.

In Figure 7.1 the polar coordinates are shown superimposed on a rectangular coordinate
system where the positivexaxis coincides with the initial line. With this arrangement the
relation between the coordinates is seen to be (175

➤

):

x=rcosθy=rsinθ

Notice that in polar coordinatesθ is measured anticlockwise and its value is restricted
to 0≤θ< 2 πor sometimes−π<θ≤π(175

➤
). Also, the origin is somewhat special
(‘singular’ is the technical term) in polar coordinates – itsrcoordinate isr=0, but itsθ
coordinate is not defined.

Example

To plot the curve with polar equation

r= 1 +2cosθ

take some appropriate values ofθ, and calculate 1+2cosθ,asbelow.

θ 0

π

6

π

4

π

3

π

2

2 π

3

3 π

4

5 π

6

π

2cosθ 2 1.732 1.414 1 0 − 1 −1.414 −1.732 − 2

1 +2cosθ 3 2.732 2.414 2 1 0 −0.414 −0.732 − 1

These values are plotted in Figure 7.2. Note that negative values ofrare plotted in the
opposite direction to the positive values, so for example(− 1 ,π)is plotted as (1, 0) – in
the opposite direction toπ. Now for any angleθ,cos(−θ)=cosθ, therefore the same
values ofrwill be obtained for negative values ofθ, and we have only to reflect the curve
in the initial line to obtain the complete plot.