Is it possible to prove e^(i x) = Cos(x)+i Sin(x) without infinite series?

Question

Euler's formula was originally derived by means of infinite series of trigonometric and exponential functions. Can it be proven without expanding functions, and without using DeMoivre's formula, which would make it a circular argument? Why necessarily e^(i x) a vector on the complex plane where x is its angle from the real axis?

Hey, not bad, Jim!

It's true, it does say that the tangent of the function z is at right angles, which would make z a vector describing a circle in the complex plane for 0

Answer 1

From the mathforum:

Calculus: Start with

z = cos(x) + sin(x) i
and notice that when x = 0, z = 1. Then differentiate,


dz/dx = -sin(x) + cos(x) i
dz/dx = sin(x) i2 + cos(x) i
dz/dx = [cos(x) + sin(x) i]i
dz/dx = zi
(1/z)dz/dx = i
ln(z) = xi + C
for some constant C, by indefinite integration. Now use the fact that when x = 0, z = 1, to conclude that C = 0. Thus


ln(z) = xi
z = e^xi
e^xi = cos(x) + sin(x) i

Answer 2

Analytically, cos(x) is defined by [ e^(ix) + e^(-ix) ] / 2, whilst sin(x) is defined by [ e^(ix) - e^(-ix) ] / (2i). Using these definitions it is not hard to verify the Euler identity.

Answer 3

L.H.S = a million+sinx-cosx/a million+sinx+cosx = a million+sinx-cosx(a million+sinx+cosx)/a million+sinx+cosx(a million+s... =(a million+sinx)^2-(cosx)^2/(a million+sinx+cosx)^2 =2sinx+2sin^2x/2+2sinx+cosx+2sinxcosx =2sinx(a million+sinx)/2(a million+sinx)+2cosx(a million+sinx) =sinx/2cosx =tanx/2

Answer 4

i would say everything is possible:D