SOLUTION: Is there a way to prove that {{{e^(i*theta) = cos(theta) + i sin(theta)}}} without using Taylor series?

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Question 619152: Is there a way to prove that e%5E%28i%2Atheta%29+=+cos%28theta%29+%2B+i+sin%28theta%29 without using Taylor series?
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!

Yes

Since all complex numbers can be expressed in polar coordinates, for some real numbers and depending on



Since

Differentiate both sides using the chain rule in the RHS:



Substitute for



Equating the real and imaginary parts in the above gives and

Since we have the initial values of and , hence and and therefore:



John

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