Before we convert, please notice that the cosecant function goes to #oo# at #2pi# multiples of 0 and #pi#; this translates into a Cartesian restriction:
#y !=0#
The tangent function has the same problem at #2pi# multiples of #pi/2# and #(3pi)/2#; this translates into a Cartesian restriction:
#x != 0 #
Now, we may proceed.
Substitute, y for #rsin(theta)#:
#y = 2theta - 4tan(theta) - csc(theta)#
Please notice that #tan(theta) = sin(theta)/cos(theta) = (rsin(theta))/(rcos(theta)) = y/x#
Substitute #y/x# for #tan(theta)#, :
#y = 2theta - 4y/x - csc(theta); x!= 0#
Please notice that we must add the restriction for x.
Here is how you find the Cartesian equivalent of #csc(theta)#
#y = rsin(theta)#
#y/sin(theta) = r#
#1/sin(theta) = r/y#
#csc(theta) = r/y#
#csc(theta) = sqrt(x^2 + y^2)/y#
Substitute #sqrt(x^2 + y^2)/y# for #csc(theta)#:
#y = 2theta - 4y/x - sqrt(x^2 + y^2)/y; x!= 0 and y!= 0#
The substitution for #theta# has three forms for three regions:
#theta = tan^-1(y/x); x > 0 and y >0#
#theta = tan^-1(y/x) + pi; x < 0#
#theta = tan^-1(y/x) + 2pi; x > 0 and y <0#
This yields 3 equations corresponding to the three regions:
#y = 2 tan^-1(y/x) - 4y/x - sqrt(x^2 + y^2)/y; x> 0 and y> 0, #
#y = 2(tan^-1(y/x) + pi) - 4y/x - sqrt(x^2 + y^2)/y; x< 0 and y!= 0#
#y = 2(tan^-1(y/x) + 2pi) - 4y/x - sqrt(x^2 + y^2)/y; x> 0 and y< 0#
Undefined elsewhere.
