Compare the polar equation of the circle r 2 with its equat

Compare the polar equation of the circle r = 2 with its equation in rectangular coordinates. In which coordinate system is the equation simpler? Do the same for the equation of the four leafed rose r = sin 2. Which coordinate system would you choose to study these curves?

Solution

r= 2

We know in rectangular coordinate system x = rcostheta

y = rsintheta

sqrt(x^2 +y^2) = r

So, x^2 +y^2 = r^2 =4

x^2 +y^2 = 4   ( equation of circle with centre (0.0)

It is easier to study these cirves in rectangular coordinate system

r = sin2(theta) = 2sintheta*costheta

r^3 = 2rsintheta*rcostheta

substituting x = rcostheta

y = rsintheta

(x^2 +y)^3/2 = 2xy

So, this becomes a complex polynomial in x -y plane

It is easier to study in polar coordinate system