beginner and trigonometry doubt

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where i can find the proof of the simple trigonometric formulas like
sin = opposite / hypotenuse

in the Right-angled triangles.
are based on the base unit circle?
i searched on wiky but i don't find the proof
i would start study from base

thanks

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I believe that there are conceptially several different approaches here. Often the relations that you gave are taken as definitions of the functions. The extensions of angles to the whole set of real numbers can be done e.g. with the concept of unit circle. This is a geometrical approach.

An analytical approach would be to define the functions via e.g. series expansion. One could first define the exponential function as exp(x) = sum_(n=0)^inf x^n/(n!). Then define sin(x) = (exp(ix)-exp(-ix))/(2i). The connection to geometry can be established by representing a triangle in a coordinate system in (Euclidean) space. In this approach one would actually have to prove the relation sin = opposite / hypotenuse, as the sine function was defined analytically.

A vector algebraic proof that I could come up with would be as follows. Consider a Cartesian coordinate system (x,y) in R^2. Vector a points in direction (0,1) and has length |a| and vector b points in (1,0) and has length |b|. They are orthogonal i.e. dot(a,b) = 0. They constitute a right-angled triangle, where the hypotenuse is represented by a vector c=b-a. Now

0 = dot(-a,b) = dot(-a,c+a) = dot(-a,c) + dot(-a,a) = cos(-a,c)|a||c| - |a|^2

This yields cos(-a,c) = |a|/|c|. The formula for the sine can be obtained by using the analytical properties of sin and cos. The connection of the cosine of angle between a and b, i.e., cos(a,b) to the analytical definition of cos can be done in the coordinate representation of the dot product, but I believe that this proof gets more involved.

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