Trig identity in complex multiplication

[Shaybay92]

Just wondering how this is simplified to the third line:

If w, z are complex numbers

wz = rs( cos\alpha + isin \alpha ) (cos \varphi + isin \varphi)

wz = rs(cos\alpha cos \varphi - sin \alphasin\varphi) + i(sin \alphacos\varphi + cos \alpha sin \varphi))

wz = rs(cos (\alpha +\varphi) + i sin(\alpha +\varphi))

What sort of trigonometric identity is used here between the 2nd and 3rd lines?

 

[Tedjn]

Exactly those as written:

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

 

[Shaybay92]

Thanks I hadn't seen these identities before

 

[mathman]

Thanks I hadn't seen these identities before

These are basic identities, which are taught in the first course of trigonometry.

 

[Shaybay92]

You would think so but apparently my school doesn't see the importance in teaching this stuff. The only identity we were taught was

sin^2(x) + cos^2(x) = 1

not even all the half angle ones which I'm finding out about now... How helpful for me!

 

[mathman]

It looks like your school must teach trigonometry for a couple of weeks within a broader math course. When I was in high school, we had a one semester course for trig.