sin(x) + 2 sin(2x)= √(x).
There are no standard algebraic or trigonometric methods for solving an
equation like this one which contains a variable which is both
1. part of a trigonometric function, as in the term sin(x)
and also
2. not part of a trigonometric function, as in the term √(x).
They can be solved by iterative (guess and check) methods. That's the method of
computers and graphing calculators. At lightning speed they guess an
approximate solution and then successively improve on it, getting closer and
closer, and eventually getting as close as is desired to the actual solution.
Occasionally a solution will be obvious. Such is the case in this one, as an
obvious solution is x = 0.
In addition to the x = 0 solution, there are four other solutions. I have
approximated them by a computer program below:
x ~ 0.0400728749475875660160835459576477048399766237746285782270624983...
x ~ 1.51275945812645026464843139037785711835717362190264861068799171...
x ~ 7.00697814299500935029704040024283479348882650864271753095319661...
x ~ 7.25027785575978077317786523805533381477772319549846961135371664...
where ~ means "is approximately equal to"
Edwin