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If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions.

I know that what defines a function are it's values, for example, if you assign to each point on the number line a value it doesn't matter what process you use to compute it, as long as at the end you get the same values for each number.

But people usually argue that two definitions are equivalent by showing that they have "the same basic properties", from this point of view it's not clear to me, for example, why the trig definition of sine is equivalent to the calculus definition of sine by power series, to me, the fact that they have some properties in common doesn't guarantee that all of the blue they don't start to disagree on their values.

I've tried to interpret these "common basic properties" as axioms and these definitions as "isomorphic" but it didn't work for me. I don't see how these "basic properties" guarantee that they values will be the same for every argument.

tengan rieu
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It depends on exactly what you mean by "basic properties." There are collections of conditions that do imply equality of two functions: For example,

  • If we know that $f$ and $g$ are continuous on the same domain, and they can be shown to agree on a dense subset of that domain, then $f = g$ everywhere on the domain.

  • If we know that $f' = g'$ everywhere, and $f$ and $g$ agree at a single point, then $f = g$ on the domain.

and so on. Alternatively, there are occasions where one can prove that two functions are equal because they satisfy the same functional equation, or because they have some fixed property that enforces a lot of structure. For example, as a somewhat trivial example, the only $n$-homogeneous functions on $\mathbb{R}^+$ are power functions of degree $n$ - once we know that $f(\lambda x) = \lambda^n f(x)$ for all $x, \lambda > 0$, this determines the function almost uniquely.

Regarding your specific example of trig functions, one way to prove that the series definition and the circle / triangle definition is to show that they both satisfy the differential equation

$$\left\{\begin{array}{cc} y'' + y &= 0 \\ y(0) &= 0 \\ y'(0) &= 1\end{array}\right.$$

Once you know this, you can invoke some big existence-uniqueness theorem (Picard-Lindelof comes to mind) to conclude that they actually are the same function.

But telling when some basic set of properties uniquely determines a function is hard in general.