How do you prove that the geometric definition of sine and cosine as the ratio of the two perpendicular sides of an orthogonal triangle to its hypotenuse is identical to the analytic definition of the two corresponding Maclaurin series? And how do you prove analytically Euler's formula that $\exp(i\pi)+1=0$, even if you know the MacLaurin series of the $\exp(x)$ function?
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1Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Mar 08 '25 at 08:00
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Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Ved Patel Mar 08 '25 at 08:13
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1The problem is that you first have to rigorously define what an angle is (without using sine and cosine, of course). Calculus/Analysis books (i.e. the place where people learn about sine and cosine) don't do that. I think your best hope is to check out (elementary, but rigorous,) books on Euclidean Geometry. – Stefan Mar 08 '25 at 08:30
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You can obtain the second one from Euler's formula. https://en.wikipedia.org/wiki/Euler%27s_formula – Supernerd411 Mar 08 '25 at 15:05
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One of its proofs uses the Maclaurin series of $\exp (x)$ – Supernerd411 Mar 08 '25 at 15:06
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This is an interesting question (The first part, anyway). It doesn't deserve to be closed. – Supernerd411 Mar 08 '25 at 15:34
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The answers to this question look promising: https://math.stackexchange.com/questions/4682867/rigorous-and-purely-geometric-definition-of-sine-and-cosine – Stefan Mar 08 '25 at 20:04