Induction (1 Viewer)

[Shady01]

Using induction can anyone here prove DE Movire's theorem?
Cosn0+ isinn0= Cis^n0 ??????

 

[Trev]

Prove cos(nx)+isin(nx)=(cosx+isinx)ⁿ
(cosx+isinx)^k = cos(kx)+isin(kx).
Consider n=k+1:
(cosx+isinx)^k+1
=(cosx+isinx).(cosx+isinx)^k
=(cosx+isinx).(cos(kx)+isin(kx))
Multiplying out we get:
cos(kx).cos(x) - sin(kx).sin(x) + i(sin(kx).cos(x) + cos(kx).sin(x))
[Noting that:
sin(x + y) = sin(x) cos(y) + sin(y) cos(x)
cos(x + y) = cos(x) cos(y) - sin(x) sin(y)]
It is the same as cos[(k+1)x]+isin[(k+1)x]

Just format that to how you're supposed to for induction questions.

 

[_ShiFTy_]

Just remember that you need to also let n=0, not just n=1

 

[Shady01]

cheers fellas. I had a rough idea except my teachers wasn't sure(at the time) whether i could use additions theroems to solve it but thanx fer that

 

[Trev]

Your teacher sux.

 

[who_loves_maths]

but just remember that induction on its own is not sufficient for a proof of the generalised DeMoivre's Theorem. (in other words, at the HSC level, you can only restrict the use of DMT to {k: k is a non-negative integer})

although the case for all rationals is simple, the case for all reals is not. (assuming one is not allowed to use Euler's equation, which itself deserves a proof before use)

 

[Riviet]

although the case for all rationals is simple, the case for all reals is not. (assuming one is not allowed to use Euler's equation, which itself deserves a proof before use)

Fortunately for us, this is not examined in the HSC, but I wouldn't mind studying it in uni.

P.S who loves maths, I didn't know you were a benefactor! When did you become one?

 

[icycloud]

although the case for all rationals is simple, the case for all reals is not. (assuming one is not allowed to use Euler's equation, which itself deserves a proof before use)

They really should teach Euler's formula in HSC, it makes things so much easier. But I guess the proof of the formula involves complex analysis so that integration/differentiation is defined over C, although taking "z" as a normal variable and using HSC calculus methods works, it's not mathematically sound without establishing the fundamentals of complex analysis? (e.g. z=c+is, dz=iz d@, 1/z=i@, e^i@=z=c+is)