sin²(x)+cos²(x) = 1 isn't 𝑗𝑢𝑠𝑡 Pythagorean theorem on the unit circle It also tells us that a rotation (eⁱˣ) followed by its inverse rotation (e⁻ⁱˣ) is the same as no rotation at allpic.twitter.com/INSZcTxtXr
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The argument is somehow circular (sorry for the pun): you need the Pythagorean identity to establish Euler identity in a geometric way. If, instead, you just use the definition of sin and cos given by the convergent series, how do you identify them with the circular functions?
A complex exponential is taken as a definition, always. You need to define what a complex exponential means first. After that you don’t need Pythagorean theorem to prove Euler’s identity. You can prove it my uniqueness of 2nd order ODE solutions
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ODE arising from the properties of the goniometric functions (the double derivative of the sinus is - the sinus) and then we are back to step 1: Pythagorean theorem.
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No. You define trig functions. You define properties of a complex exponential. Two separate worlds. They satisfy the same differential equations, which has a unique solution .
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Speaking as the devils' advocate (cos(x), sin(x)) is _by definition_ the point on the unit circle such that the oriented "pizza triangle" (0,0) -> ( 1, 0) -> (cos(x), sin(x)) -> (0,0) has area 1/2 x.
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I mean: you solve the ODE for sin and cos, but then you need Pythagorean theorem in order to identify the corresponding solutions with the circular functions (because of uniqueness).
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