Before we begin the conversion, please observe that the secant function has a division by zero issue at theta = pi/2 + npiθ=π2+nπ. The same is true for the cosecant function and theta = npiθ=nπ. This translates into the Cartesian as a restriction of x !=0 and y != 0x≠0andy≠0
For csc(theta)csc(θ), begin with:
y = rsin(theta)y=rsin(θ)
1/sin(theta) = r/y1sin(θ)=ry
1/sin(theta) = sqrt(x^2 + y^2)/y1sin(θ)=√x2+y2y
csc(theta) = sqrt(x^2 + y^2)/ycsc(θ)=√x2+y2y
A similar substitution exists for the secant function:
sec(theta) = sqrt(x^2 + y^2)/xsec(θ)=√x2+y2x
Substitute x^2 + y^2x2+y2 for r^2r2 and sqrt(x^2 + y^2)√x2+y2 for r:
x^2 + y^2 + sqrt(x^2 + y^2) = 2theta - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x != 0 and y!=0x2+y2+√x2+y2=2θ−2√x2+y2x−√x2+y2y;x≠0andy≠0
The substitution for thetaθ breaks the equation into 3 equations:
x^2 + y^2 + sqrt(x^2 + y^2) = 2tan^-1(y/x) - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x > 0 and y>0x2+y2+√x2+y2=2tan−1(yx)−2√x2+y2x−√x2+y2y;x>0andy>0
x^2 + y^2 + sqrt(x^2 + y^2) = 2(tan^-1(y/x) + pi) - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x < 0 and y!=0x2+y2+√x2+y2=2(tan−1(yx)+π)−2√x2+y2x−√x2+y2y;x<0andy≠0
x^2 + y^2 + sqrt(x^2 + y^2) = 2(tan^-1(y/x) + 2pi) - 2sqrt(x^2 + y^2)/x - sqrt(x^2 + y^2)/y; x > 0 and y<0x2+y2+√x2+y2=2(tan−1(yx)+2π)−2√x2+y2x−√x2+y2y;x>0andy<0
Undefined elsewhere.
