# More complex numbers, more trigs

Homework Helper

## Homework Statement

The angles $\theta$ and $\phi$ lie on the interval (-pi/2,pi,2) and

$z=(cos\theta +cos\phi)+i(sin\theta +sin\phi)$

Show that |z|=2cos($\frac{\theta - \phi}{2}$) and find arg(z)

## Homework Equations

If z=x+iy

$$|z|=\sqrt{x^2 +y^2}$$

## The Attempt at a Solution

$$|z|= \sqrt{(cos\theta +cos\phi)^2+(sin\theta +sin\phi)^2}$$

$$=2+2cos(\theta - \phi)$$

I don't know how to get rid of the additional two and how to make the angle halved.

Homework Helper
Solved the first part

2nd part giving trouble

arg(z)=tan-1{Im(z)/Re(z)}

$$tan^{-1}(\frac{sin\theta + sin\phi}{cos\theta + cos\phi})$$

I think I'll need to simplify this further.

I tried multiplying by the conjugate of the denominator but that didn't lead anywhere feasible.

Last edited:
Dick
Homework Helper
(sin(a)+sin(b))/(cos(a)+cos(b))=tan((a+b)/2). I'm going to the wikipedia page of trig identities and fishing around. You might notice this is related to the third problem of your other post if you put phi=0.

Homework Helper
(sin(a)+sin(b))/(cos(a)+cos(b))=tan((a+b)/2). I'm going to the wikipedia page of trig identities and fishing around. You might notice this is related to the third problem of your other post if you put phi=0.

oh my god! I completely forgot factor formula....seems that 1 month away from math has completely worked my brain in the wrong way.... thank you.

Dick
Homework Helper
I don't remember stuff like that either. I just figured that if the problem called for it then something like that must exist.

Homework Helper
I don't remember stuff like that either. I just figured that if the problem called for it then something like that must exist.

Well I knew it had to be arctan of the tan of something, but I thought the something would have been just with theta and phi and not half angled. Was looking to make a tan(A+B) thing.

Dick