Apply Newton's Method to approximate the $x$-coordinates of all intersections: $$y=1=e^x\sin(x)\qquad 0<x<\pi$$
Let $f(x)=e^x\sin(x)-1$ and $$x_{n+1}=x_n +\frac{1-e^x\sin x_n}{e^x(\cos x_n+\sin x_n)}$$ Now I am to propose an initial approximation $x_0$ so that I am able to guarantee convergence, and this is where I am stumped.
How can I guess $x_0$ without knowing their graph? Because sometimes it is very difficult to sketch the graph and approximate $x_0$. Any hints or solution will be appreciated.