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how does one define sin without Pythagorean theorem? E. g. need to prove that it only depends on angle and not triangle size.

You don't need to prove anything to define sin.

the definition of function in mathematics requires that same input always gives same output for all valid inputs.

So yeah, to define sin as a function one absolute must prove that it only depends on angle and not triangle size.

This proof is done via similar triangle properties (same angles => same proptions of sides) btw.