# Finding modulus and argument of a complex number

1. ### choob

32
1. The problem statement, all variables and given/known data
Find the modulus and argument of z=1-cos(a)-i*sin(a)

2. Relevant equations
mod(z)=sqrt(a^2+b^2)

3. The attempt at a solution
mod(z)=sqrt((1-cos(a))^2+(-sin(a))^2)
=sqrt(2-2cos(a))

arg(z)=arctan((-sin(a))/(1-cos(a)))

This is as far as I can get, I have asked my math teacher but he is not very familiar with thise.

my textbook gives mod(z) as 2sin(2a) and arg(z) as (a-pi)2

BTW this question is from the IB math textbook, there exists a solutions manual but I do not have it... the question is 11.2, 19 a)

2. ### tiny-tim

26,055
Hi choob!

You need to learn your trigonometric identities …

in this case, sin = 2 sin1/2 cos1/2

and 1 - cos = 2 sin21/2

3. ### choob

32
i can get arg(z) to arctan(-cos(a/2)/(sin a/2)), how do i finish this?

4. ### rock.freak667

6,219
$$tan(\frac{\pi}{2}-\theta)= \frac{1}{tan\theta}$$