complex numbers, find the absolute valueof z

[green_tea]

Hi!
I have to find the absolute value of z when z=1+sin(a) + i cos(a)
The answer is IzI = 2 Icos(a/2)I

I can solve this if I draw w=sin(a) + icos(a) in the complex plane and then add 1 by drawing, use that IwI=1, look at the angels and the triangels and use that cos(a/2)= (IzI /2 )/1 = IzI/2

Now my question is: how do you solve this without the drawing part???

 

[royhaas]

Use the fact that \(\displaystyle |z|^2 = Real(z)^2 + Imaginary(z)^2\). This is the same as \(\displaystyle z \bar z\), where \(\displaystyle \bar z\) is the conjugate of \(\displaystyle z\).