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 $$\alpha$$sin$$\varphi$$) + i(sin $$\alpha$$cos$$\varphi$$ + 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.