Proving e^iθ = cos(θ) + isin(θ) | Mathematics | Page 1

lunar heron Mar 16, 2024, 12:09 PM

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why?

urban ether Mar 16, 2024, 1:42 PM

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Euler's theorem

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https://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-ei-varphi-cos-varphi-i-sin-varphi

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How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\v...

Could you provide a proof of Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?

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several proofs for it here

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Taylor series is probably the most common proof you'll see

glass ridge Mar 16, 2024, 6:14 PM

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A proof that I found which is quite simple (if you know about differential equations)
Is setting f(t)=cos(t)+isin(t) so f’(t)=-sin(t)+icos(t)=i(cos(t)+isin(t)) so f’(t)=if(t) solving the differential equation with f(0)=1 you get f(t)=e^it

#Proving e^iθ = cos(θ) + isin(θ)