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There are many different approaches to trigonometric functions, and it's clear that they share many properties. However, i don't know what guarantees nongeometric definition and geometric definition are the same. That is, how do i know if there is a property that 'geometrically defined trigonometric functions' have, but 'nongeometrically defined trigonometric functions' don't have?

For example, it is a theorem that "Every ordered field with least-upper-bound-property is homomorphic to each other"

This gurantees me that "real number i intuitively know" is actually the same thing as $\mathbb{R}$ and there would be nothing againt my intuition.

I think it really doesn't make any sense to say which definition is more precise than the other, but at least there should be something that gurantees these two definitions are equivalent in some sense.

Please give me some brief explanation and thank you in advance!

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    How can one know the "real number you intuitively know"? It's by trying to make your intuition precise that you come to appreciate the technical details of the definitios, axioms, etc.2012-12-19

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The guarantee is given by corresponding proofs, which base on inerpreting $x$ as an arc length in cartesian coordinates and then showing that "$\sin x=\sin x$"

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    The standard definition of $\pi$ in analysis is "the first positive zero of $\sin$", so it's automatic there as well.2012-12-19