What is solution the of $\sin x=x$ where $ x>0$

Question

What is solution the of $\sin x=x$ where $x>0$? For $\cos x=x$ is $0.739085.... $

If the solution not exist why?

Answer 1

Draw graphs of both $\sin x$ and $x$.They meet at $0$, right? Find derivatives at $0$ whether $y=x$ would be go above or below it? What can you conclude?

Answer 2

Even before computing derivatives you should know that $$ |\sin x| \le |x| \qquad \forall x\in \mathbb R $$ From the geometric point of view this relation is explained by noticing that the chord is shorter than the arc. Equality holds only for $x=0$ when the chord and arc are both zero.