Trigonometric function

We can define trigonometric function mathematically as follow:

Definition

• $\pi :=2{\int }_{-1}^1\sqrt{1-x^2}dx$
• For $-1\le x\le 1$, $A(x):=\frac{x}{2}\sqrt{1-x^2}+{\int }_x^1\sqrt{1-t^2}dt$
• For $x\in [0,\pi ]$, $\cos x$ is the unique number in $[-1,1]$ such that $A(\cos x)=\frac{x}{2}$ and $\sin x=\sqrt{1-\cos x^2}$