Question

What is the quotient theorem of complex numbers in polar form?

Answer 1

Given two complex numbers in polar coordinates by their modulo `r` and polar angle `phi`:
`z_1=r_1*(cos phi_1+i*sin phi_1)` and
`z_2=r_2*(cos phi_2+i*sin phi_2)` where `r_2 != 0`,

Then the result of the division of the first by the second is equal to
`z_1/z_2=(r_1/r_2)*[cos (phi_1-phi_2)+i*sin (phi_1-phi_2)]`
This is the quotient theorem.

The trigonometric proof of this theorem is easier to start with a product theorem:

`z_1*z_2=(r_1*r_2)[cos (phi_1+phi_2)+i*sin (phi_1+phi_2)]`

The proof:

`r_1(cos phi_1+i*sin phi_1) * r_2(cos phi_2+i*sin phi_2) =`
`= r_1*r_2*(cos phi_1*cos phi_2 + i^2*sin phi_1*sin phi_2 + i*sin phi_1*cos phi_2 + i*cos phi_1*sin phi_2) =`
`= r_1*r_2*[(cos phi_1*cos phi_2 - sin phi_1*sin phi_2) + i*(sin phi_1*cos phi_2 + cos phi_1*sin phi_2)] =`
`=r_1*r_2*[cos(phi_1+phi_2)+i*sin(phi_1+phi_2)]`

Verbally, the product theorem states that, to multiply two complex numbers defined in polar form via modulo and a polar angle, modulo of one complex number is multiplied by another and the angles are added together.

For complex numbers with modulo `1`, geometrically, multiplication is a rotation of a vector representing the first complex number counterclockwise by the angle of the second number.

Back to the division of complex numbers in polar form.
To prove the quotation theorem mentioned above, all we have to prove is that `z_1/z_2` in the form we presented, multiplied by `z_2`, produces `z_1`.

Indeed, using the product theorem,
`(z_1/z_2)*z_2 = {(r_1/r_2)[cos (phi_1-phi_2)+i*sin (phi_1-phi_2)]}*r_2(cos phi_2+i*sin phi_2)=`
`=(r_1/r_2)*r_2*[cos(phi_1-phi_2+phi_2)+i*sin (phi_1-phi_2+phi_2)]=`
`= r_1*(cos phi_1+i*sin phi_1)`

End of proof.

Verbally, the quotient theorem states that, to divide one complex number by another, modulo of the first complex number is divided by another and the angle of the second one is subtracted from the first.

Geometrically, on the unit circle, it represents a clockwise rotation of the first vector by an angle of the second.