Rectangular form of z: z=x + y*i
x is the real part <=> x= Re(z)
y is the imaginary part <=> y=Im(z)
i is the imaginary unit, i^2=-1.
The trigonometric (polar) form is:
z=r(cos t + i sin t), where r = |z|
r = sqrt(x^2 + y^2)
r is the modulus of the complex number z.
If x and y are the coordinates of a point M(x,y), r is the position vector of the point M.
The formula for r = sqrt(x^2 + y^2) results from Pythagorean theorem, where r is hypotenuse.
cos t = x/r=>x=r cos t
sin t = y/r=>y=r sin t
z = r cos t + ir sin t
tan t = y/x => t = arctan y/x
z = r( cos t+i sin t) - trigonometric/polar form of the complex number z.