Before we begin. Please notice that thetaθ must not be a 2npi2nπ multiple of 0 or pi0orπ, because the cosecant function is undefined at the these points. This converts to the Cartesian restriction y != 0y≠0
Eventually, we will substitute tan^-1(y/x)tan−1(yx) for thetaθ, therefore, we must look at the pristine function, r = (2cos(theta) - csc(theta))/(theta + 1)r=2cos(θ)−csc(θ)θ+1, and see what happens at theta = pi/2 and (3pi)/2θ=π2and3π2
r = -2/(pi + 2) and r = -2/(3pi + 2)r=−2π+2andr=−23π+2
These are the y coordinates at x = 0.
y = -2/(pi + 2) and y = -2/(3pi + 2)y=−2π+2andy=−23π+2
Now, we may proceed with the conversion.
For csc(theta)csc(θ):
y = rsin(theta)y=rsin(θ)
1/sin(theta) = r/y1sin(θ)=ry
csc(theta) = r/ycsc(θ)=ry
csc(theta) = sqrt(x^2 + y^2)/ycsc(θ)=√x2+y2y
r = (2cos(theta) - sqrt(x^2 + y^2)/y)/(theta + 1)r=2cos(θ)−√x2+y2yθ+1
Multiply both sides by r:
r^2 = (2rcos(theta) - (x^2 + y^2)/y)/(theta + 1)r2=2rcos(θ)−x2+y2yθ+1
Substitute x for rcos(theta)rcos(θ)
r^2 = (2x - (x^2 + y^2)/y)/(theta + 1)r2=2x−x2+y2yθ+1
Substitute x^2 + y^2x2+y2 for r^2r2
(x^2 + y^2) = (2x - (x^2 + y^2)/y)/((theta + 1)(x2+y2)=2x−x2+y2y(θ+1)
Multiply the right side by y/yyy:
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(theta + 1))(x2+y2)=2xy−x2−y2y(θ+1)
The substitution for thetaθ breaks the equation into 3 equations plus the 2 for x = 0 and y = 0:
Undefined for y = 0
y = -2/(pi + 2) and y = -2/(3pi + 2); x = 0y=−2π+2andy=−23π+2;x=0
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(tan^-1(y/x) + 1)); x > 0 and y > 0(x2+y2)=2xy−x2−y2y(tan−1(yx)+1);x>0andy>0
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(tan^-1(y/x) + pi + 1)); x < 0 and y != 0(x2+y2)=2xy−x2−y2y(tan−1(yx)+π+1);x<0andy≠0
(x^2 + y^2) = (2xy - x^2 - y^2)/(y(tan^-1(y/x) + 2pi + 1)); x > 0 and y < 0(x2+y2)=2xy−x2−y2y(tan−1(yx)+2π+1);x>0andy<0
