If (a,b) is a are the coordinates of a point in Cartesian Plane, u is its magnitude and alpha is its angle then (a,b) in Polar Form is written as (u,alpha).
Magnitude of a cartesian coordinates (a,b) is given bysqrt(a^2+b^2) and its angle is given by tan^-1(b/a)
Let r be the magnitude of (3sqrt3,-3) and theta be its angle.
Magnitude of (3sqrt3,-3)=sqrt((3sqrt3)^2+(-3)^2)=sqrt(27+9)=sqrt36=6=r
Angle of (3sqrt3,-3)=Tan^-1((-3)/(3sqrt3))=Tan^-1(-1/sqrt3)=-pi/6
implies Angle of (3sqrt3,-3)=-pi/6
This is the angle in clockwise direction.
But since the point is in fourth quadrant so we have to add 2pi which will give us the angle in anti-clockwise direction.
implies Angle of (3sqrt3,-3)=-pi/6+2pi=(-pi+12pi)/6=(11pi)/6
implies Angle of (3sqrt3,-3)=(11pi)/6=theta
implies (3sqrt3,-3)=(r,theta)=(6,(11pi)/6)
implies (3sqrt3,-3)=(6,(11pi)/6)
Note that the angle is given in radian measure.
Also the answer (3sqrt3,-3)=(6,-pi/6) is also correct.
