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.
