Been thinking about this for quite a while, I know $0$ is one of the answers but I just cannot figure out how can I find the others (not by plotting the graphs but working it out steps by steps), please someone help me.
Edit:
- Approximate form is also accepted.
- I think no prior knowledge to calculus is expected for this question.
- How can calculus be used to find the answer? (Sorry for asking this as I have only shallow understanding about calculus, is it that we differentiate it once? And what next?)
- I heard there are seven answers in total including $0$.
OK in separate steps:
$x=0$ is an obvious solution.
x and $\sin(x)$ are both odd functions, so any solution $x = 8\sin(x)$ will also lead to the solution $-x = 8\sin(-x)$. So we only need to consider $x >0$.
$|\sin(x)| < 1$, so for $x >8$ there can be no solution. Hence, with (2.), consider $8>x >0$. In this regime, we have 3 solutions $x≈2.7859$ and $x≈7.49775$ and $x≈7.95732$. [Are you fine with numerical solutions?] With (2.), the negative values are also solutions. Formally, since $f(x) = x - 8\sin(x)$ is convex at the solutions, there will be no other solutions which you have "missed".
So in total, you have 7 solutions and you need look no further.