I am reviewing complex numbers and how to convert between rectangular, trigonometric, and polar forms. Easy stuff...
What I am trying to figure out is why the angles are simply added in trig and polar forms, so I went back to some basics.
(a+ib)(c+id) is the same as (cos[a]+isin[a])(cos+isin).
Going through the algebra there, one gets:
(cos[a]cos + i(cos[a]sin+sin[a]cos) - sin[a]sin)
where is the leap from the above to the following:
cos[a+b] + i(sin[a+b])
I assume there is an identity that one "just remembers', but I tend to be anal and would like to derive it from first principles. Or is it that the derivation is so insidious and one better choose to "just remember"?
What I am trying to figure out is why the angles are simply added in trig and polar forms, so I went back to some basics.
(a+ib)(c+id) is the same as (cos[a]+isin[a])(cos+isin).
Going through the algebra there, one gets:
(cos[a]cos + i(cos[a]sin+sin[a]cos) - sin[a]sin)
where is the leap from the above to the following:
cos[a+b] + i(sin[a+b])
I assume there is an identity that one "just remembers', but I tend to be anal and would like to derive it from first principles. Or is it that the derivation is so insidious and one better choose to "just remember"?