Find the smallest root of the function $e^{-x} = \sin (x)$

Question

I have the following problem:

Find the smallest root of the function $e^{-x} = \sin (x)$ and focus the root with Newton's method to $8$ decimal accuracy.

Any suggestions?

Answer 1

Some hints.

1. Newton's method is a root-finding algorithm (i.e. given $f(x)$ it finds $x^{*}$ such that $f(x^{*}) = 0$) so you need to find a function $f(x)$ which has a root at the same point that $e^{-x} = \sin x$.
2. Newton's method will find a root close to your initial guess. Where do you think the smallest root may lie?