Solving 10 Sin x = -x: Tips & Ideas

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The equation 10 sin x = -x is a transcendental equation, which typically requires numerical methods for solutions. The obvious root is x = 0, while two additional roots can be estimated using Newton's method, which involves calculus for refining guesses. The method corrects initial estimates based on the function's behavior, but requires knowledge of derivatives. For those unfamiliar with calculus, trial and error can also be a viable approach. Ultimately, the discussion highlights the need for numerical techniques to find all solutions effectively.
kelvintc
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10 sin x = -x
can anyone give me tips how to solve this equation?
i run out of ideas even to transform this equation to another form.. sigh sigh

thanks...
 
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It seems you have a transcendental equation0--i don't think those are solvable, unless you know how to find intercepts between trig and non-vertical/non-horizontal functions.
 
x=0

Or do you want more solutions? Then look up Newton's method.
 
there are 3 roots in this equation. 0 is the obvious one, the other two needs to estimate by computer.
 
oh ... thanks...
p/s : (1) what's Newton method?
(2) find intercepts between trig and non-vertical/non-horizontal functions ?

i not yet learn both also and my textbook is doesnot contain Newton's method also.. i will try to find it out...
my friend told me use 'trials and errors' method, is it applicable ?
thanks
 
Newton's method is a clever way of correcting your guess by using the deviation. But you need to know some calculus, and I don't think you know any. In any case, if x_n is your n^th trial, then you find your next trial as follows:
x_{n+1}=x_n-{10\sin x_n+x_n\over 10\cos x_n +1}
Of corse, with more than one root to 10sin x+x=0, you'll need an appropriate initial guess or you'll just keep going back to a known solution like x=0.
 
here are the 5 solutions to 50 digits...
0
+/- 3.49906381990775821737274462727340586713040476067
+/- 5.67920779631440367042018458778408009777615397061
 
oh .. thanks a lot ... :)
 

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