Before we begin the conversion, please observe that the cosecant function and the tangent function have division by zero issues at integer multiples of pi offset by 0 and pi/2, respectively; this translates into the Cartesian restrictions x !=0 and y!=0.
The derivation for the substitution for csc(theta) is as follows:
y = rsin(theta)
1/sin(theta) = r/y
csc(theta) = r/y
csc(theta) = sqrt(x^2+y^2)/y
Allowing y to equal zero would be trouble.
The derivation for the substitution for tan(theta) is as follows:
y = rsin(theta) and x = rcos(theta)
y/x = (rsin(theta))/(rcos(theta)) = sin(theta)/cos(theta) = tan(theta)
Allowing x to equal zero is a division by zero issue.
The substitution for r^2sin(theta):
r^2sin(theta) = (rsin(theta))r = ysqrt(x^2 + y^2)
I am saving the best for last so let's write the equation with these 3 substitutions:
ysqrt(x^2 + y^2) = 2theta - 4y/x -sqrt(x^2+y^2)/y
The substitution for theta is:
theta = tan^-1(y/x); x > 0 and y > 0
theta = tan^-1(y/x) + pi; x < 0, and y !=0
theta = tan^-1(y/x) + 2pi; x > 0, and y <0
This creates the 3 following equations:
ysqrt(x^2 + y^2) = 2tan^-1(y/x) - 4y/x -sqrt(x^2+y^2)/y; x > 0 and y > 0
ysqrt(x^2 + y^2) = 2(tan^-1(y/x) + pi) - 4y/x -sqrt(x^2+y^2)/y; x < 0 and y != 0
ysqrt(x^2 + y^2) = 2(tan^-1(y/x) + 2pi) - 4y/x -sqrt(x^2+y^2)/y; x > 0 and y < 0
Undefined elsewhere.
