Before we convert, please notice that the cosecant function goes to oo∞ at 2pi2π multiples of 0 and piπ; this translates into a Cartesian restriction:
y !=0y≠0
The tangent function has the same problem at 2pi2π multiples of pi/2π2 and (3pi)/23π2; this translates into a Cartesian restriction:
x != 0 x≠0
Now, we may proceed.
Substitute, y for rsin(theta)rsin(θ):
y = 2theta - 4tan(theta) - csc(theta)y=2θ−4tan(θ)−csc(θ)
Please notice that tan(theta) = sin(theta)/cos(theta) = (rsin(theta))/(rcos(theta)) = y/xtan(θ)=sin(θ)cos(θ)=rsin(θ)rcos(θ)=yx
Substitute y/xyx for tan(theta)tan(θ), :
y = 2theta - 4y/x - csc(theta); x!= 0y=2θ−4yx−csc(θ);x≠0
Please notice that we must add the restriction for x.
Here is how you find the Cartesian equivalent of csc(theta)csc(θ)
y = rsin(theta)y=rsin(θ)
y/sin(theta) = rysin(θ)=r
1/sin(theta) = r/y1sin(θ)=ry
csc(theta) = r/ycsc(θ)=ry
csc(theta) = sqrt(x^2 + y^2)/ycsc(θ)=√x2+y2y
Substitute sqrt(x^2 + y^2)/y√x2+y2y for csc(theta)csc(θ):
y = 2theta - 4y/x - sqrt(x^2 + y^2)/y; x!= 0 and y!= 0y=2θ−4yx−√x2+y2y;x≠0andy≠0
The substitution for thetaθ has three forms for three regions:
theta = tan^-1(y/x); x > 0 and y >0θ=tan−1(yx);x>0andy>0
theta = tan^-1(y/x) + pi; x < 0θ=tan−1(yx)+π;x<0
theta = tan^-1(y/x) + 2pi; x > 0 and y <0θ=tan−1(yx)+2π;x>0andy<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=2tan−1(yx)−4yx−√x2+y2y;x>0andy>0,
y = 2(tan^-1(y/x) + pi) - 4y/x - sqrt(x^2 + y^2)/y; x< 0 and y!= 0y=2(tan−1(yx)+π)−4yx−√x2+y2y;x<0andy≠0
y = 2(tan^-1(y/x) + 2pi) - 4y/x - sqrt(x^2 + y^2)/y; x> 0 and y< 0y=2(tan−1(yx)+2π)−4yx−√x2+y2y;x>0andy<0
Undefined elsewhere.
