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Expansion of Sine and Cosine

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With the Pythagoras derived by rotational dynamics from the last post, it is natural to think of sines and cosines as they are both related to rotation more so directly.   1 Definitions A big difference about sines and cosines is that they depend on angles rather than side lengths. So what is an angle? Well it is nothing but the length of the arc swiped counterclockwise by a line segment that equals to 1,  here O A rotates about the origin O then become O C , and the length of arc A C is the definition of an angle(in radian), let us call this angle α . Now in a 2-D coordinate system, an angle is defined w.r.t. the positive x-axis. For example, let us put this model in a 2-D coordinate, where  O is the origin, A is at ( 1 , 0 ) , then O A would be at a angle 0 as it overlaps with the positive x-axis and O C would be at an angle α . Now to define sine and cosine,  sin α is defined to be the y-coordinate of C (or vertical component of  O C ) and cos α is defined to be the x-coord
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Two visualizing proofs of the Pythagorean theorem

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This is my first ever blog post. I had some ideas on proving the Pythagorean theorem in a direct and dynamical manner, and recently I find a lot of motivation to write them down and share them on a blog(hopefully I can keep this going), so here we go.  I have also written a paper version(the same content as this blog) which can be found at   https://vixra.org/abs/2305.0047 . The Pythagorean theorem is one of the most proved theorem of all time, most of the proofs use manipulation of areas to prove that the square of the hypotenuse is indeed the sum of the squares of the two legs. I often felt disconnected with the proofs, it was one of the situations where I could prove something, but could not quite see why. So here let me present two methods of proving the Pythagorean theorem that are hopefully direct, dynamical and visualizing.   1 The question   The problem to solve here is quite straight forward: given two side lengths  a and b , let them form a right-angled triangle, with a and
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