0

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:

  1. Approximate form is also accepted.
  2. I think no prior knowledge to calculus is expected for this question.
  3. 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?)
  4. I heard there are seven answers in total including $0$.
Ѕᴀᴀᴅ
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    AFAIK there are no closed forms for the solutions other than $0$. – Robert Israel Oct 19 '18 at 16:12
  • "Approximate form is also accepted" ... that is where the "calculus" answer may be relevant. Newton's method, for example. When $\sin$ is in radians, Maple gives me solutions $0, 2.785902114, 7.497754811, 7.957321494$ and their negatives. – GEdgar Oct 19 '18 at 16:46
  • You could also graph $x$ versus $8 \sin (x)$ and discover where the graphs intersect (approximately). – bjcolby15 Oct 20 '18 at 09:59

1 Answers1

2

OK in separate steps:

  1. $x=0$ is an obvious solution.

  2. 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 $x >0$.

  3. $|\sin(x)| < 1$, 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.

Andreas
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  • Did you find it by calculus (Newton’s method suggested by GEdgar)? Can you please briefly describe the necessary steps to obtain the solutions? Thanks –  Oct 19 '18 at 17:02
  • Also, I’d like to ask if there’re any other method except calculus that can be applied to achieve the answers as I was told that no prior knowledge to calculus is needed (actually the original question is how many roots are there for the equation). Thank you so much for solving this for me –  Oct 19 '18 at 17:05
  • I merely concentrated on how many solutions there are, not on their actual values (which I simply took from wolframalpha). Their is no analytical solution. A good overview of numerical methods on very similar problems is given in https://math.stackexchange.com/questions/866945/ or https://math.stackexchange.com/questions/1939607/ – Andreas Oct 20 '18 at 16:16