The following steps are helpful in tracing a polar curve r = ƒ(θ).
(i) The curve is symmetrical about the initial line if its equation remains unchanged when θ is replaced by – θ.
(ii) The curve is symmetrical about the line through the pole perpendicular to the initial line if its equation remains unchanged when θ is replaced by π – θ.
(iii) The curve is symmetrical about the pole if its equation remains unchanged when θ is replaced by π + θ.
The curves passes through the pole if r = 0 for a value of θ.
3. Solving the equation
Solve the equation of r in terms of θ and see how r varies as θ varies. Make a table of corresponding values of θ and r. Many polar equations involve periodic functions such as sin θ, cos θ etc. In such cases we consider values of θ from 0 to 2π alone.
Find out the asymptotes of the curve, if any.
Find out the region on the plane in which no part of the curve lies.
Remark: Sometimes a polar curve easily traced by converting it into a Cartesian curve by means of the transformation x = r cos θ and y = r sin θ.
For example: Consider a polar curve r cos θ = a cos 2θ.
On multiplying both sides by r2, we get
r cos θ. r2 = ar2 (cos2 θ – sin2 θ)
x (x2 + y2) = a (x2 – y2).
which can be easily traced.