Example 19.1. Reduce 1-cos α + i sin α to the modulus amplitude form

1.	Example 19.1. Reduce 1-cos α + i sin α to the modulus amplitude form.
Answer» let us keep −π<α<π let the point o=0,a=1, u=cosα+isinα and the point z=1+u observe that \|u\|=1 and thetriangle oaz is an isosceles Triangle so the angle aoz=α2 and the side \|z\|=2sin(α/2) Therefore in polar formz=2sin(α/2)eiα/2 if 0≤α<πz=2sin(−α/2)eiα/2 if −π≤α<0

Example 19.1. Reduce 1-cos α + i sin α to the modulus amplitude form.

Answer»

let us keep −π<α<π

let the point o=0,a=1, u=cosα+isinα and the point z=1+u observe that |u|=1 and thetriangle oaz is an isosceles

Triangle so the angle aoz=α2 and the side |z|=2sin(α/2)

Therefore in polar formz=2sin(α/2)eiα/2 if 0≤α<πz=2sin(−α/2)eiα/2 if −π≤α<0