How do you find the x-intercepts of #1/2 x - sin(x) = 0#?

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Answer 1 · 2018-07-17T16:56:43

#0 and +- 1.8955#., for 5 significant digits (5-sd)

Solutions of 1/2x - sin x = 0 are

x-intercepts of # y = x - 2 sin x#.

Obvious solution is x = 0. The other two located near

#+- 2# by the first graph are transcendental.

Numerical iterative methods could give very high precision

approximations.Trial-and-error root-bracketing graphical method

gives 5-sd values.

The 1st graph locates two transcendental roots near #+- 2#.

graph{y-x/2+sin x=0}
.
Simple or faster Newton-Raphson numerical iterative methods

would generate about (at the least )17-sd solutions, in long

precision. The starters are #x_0 = +_ 2#, respectively..

Here, root-bracketing graphical method gives 5-sd solutions.

The graphs below gives 5-sd x-intercept #+-1.8955#.

Of course, they are not equidistant from O.

graph{y-x+2 sin x = 0[-1.896 -1.895 -.001 .001]}

graph{y-x+2 sin x = 0[1.8954 1.8956 -.001 .001]}