Sine and Cosine
parametrize the unit circle with constant speed 1
TaylorSine(x) = x - (x^3)/3! + (x^5)/5! -+...
sin(3x) = P3(sin(x)), with P3(y) = 3*y - 4*y^3
P3(P3(Taylor5(x/9))) is a good approximation
on [0, 3*pi]
Given two functions f(t), g(t) with f(t)^2 + g(t)^2 = 1.
Then [f(t),g(t)] is orthogonal to [f'(t),g'(t)].
If this circle parametrization has constant speed = 1,
f'(t)^2 + g'(t)^2 = 1, then [f''(t),g''(t)] is also orthogonal
to [f'(t),g'(t)], or [f''(t),g''(t)] = c*[f(t),g(t)].
Differentiate f^2 + g^2 = 1 to get f*f' + g*g' = 0 and again:
gives f'^2 + g'^2 + f*f'' + g*g'' = 0 or [f,g]'' = - [f,g].
These are the ODEs which define sine and cosine.
cos and sin' are solutions of f" = -f with f(0)=1, f'(0)=0,
therefore sin' = cos, hence cos' = sin" = -sin.
Note: While the values of polynomials can be computed exactly,
nobody can exactly compute the values of the sine function.
Only approximations are available, approximations however
which can be made as precise as needed.