how to solve sin(x)=.703x?

Question

There are two answers to sin(x)=.703x, one of which is 0, but how do you get the other? I'm drawing a blank here.

Answer 1

Numerically it is
x = 0 and +/- 1.402339505

There are no solutions outside |x| > 1/0.703 = 1.422 since the RHS becomes greater than 1. These are the only solutions.

Answer 2

Actually, there are _three_ solutions, since if x is a solution, so is -x.

There isn't a solution to this in terms of the usual elementary functions. (The answer can be expressed as the inverse sinc of .703, but nobody has sinc or inverse sinc on their calculator.)

One approach is to rewrite this as
sin(x)/.703 = x

which suggests defining
f(x) = sin(x)/.703

and solving for f(x) = x, numerically. Fortunately, this can be done simply by repeatedly applying f. We know f(0) = 0, so pick a nonzero value, such as 1, and calculate:
x0 = 1
x1 = f(x0) = 1.19697
x2 = f(x1) = 1.31005
x3 = f(x2) = 1.37439

...etc. to see if this process will converge on a value such that f(x) = x. In this case, it does, and after a dozen iterations or so you should have a pretty good estimate of a solution.

Answer 3

you can't find exactly but you can find an approximation,
using for example Newton method
http://en.wikipedia.org/wiki/Newton_method