Given $ \sin (x) = a $ the solutions are given by (with $ A = \arcsin a $) $$ x = A + 2 \pi n$$ $$ x = (\pi - A) + 2 \pi n = - A + (\pi + 2 \pi n) = - A + (2n + 1) \pi $$
The taylor sum is given by:
$$ \sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} $$
How can you quickly see the connection of this factor? Could be trivial, but I don't see it that fast.
And the $ 2n + 1 $ basically represents the following property of sine:
Does this also hold for cosine and the hyperbolic trigonemetric formulas? And even exponential sum (since it is related by the Moivre's Formula?)