Before we begin the conversion, please observe that the secant function has a division by zero issue at theta = pi/2 + npi. The same is true for the cosecant function and theta = npi. This translates into the Cartesian as a restriction of x !=0 and y != 0
For csc(theta), begin with:
y = rsin(theta)
1/sin(theta) = r/y
1/sin(theta) = sqrt(x^2 + y^2)/y
csc(theta) = sqrt(x^2 + y^2)/y
A similar substitution exists for the secant function:
sec(theta) = sqrt(x^2 + y^2)/x
Substitute x^2 + y^2 for r^2 and sqrt(x^2 + y^2) 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!=0
The substitution for theta 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>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!=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<0
Undefined elsewhere.
