TITLE: How to rearrange $y=\sin(x)+2x$ with $x$ as the subject?
QUESTION [1 upvotes]: How to rearrange
$$y=\sin(x)+2x$$
with $x$ as the subject?
Is this possible?
I'm struggling to work out how I should go about this.
I also tried putting it into wolfram alpha to check how it was worked out but it doesn't return a result.

REPLY [1 votes]: As seen here, it is generally impossible to solve for $x$ given a particular $y$, so I doubt you would have much luck solving for $x$ for general $y\in\mathbb Q$.  The only closed form solution for rational $y$ occurs at $(0,0)$...
One may, however, approximate the solution using Newton's method:
Let us define $x_n$ as follows:
$$x_{n+1}=x_n-\frac{\sin(x_n)+2x_n-y}{\cos(x_n)+2}$$
Then as $n\to\infty$, $x_n\to x$ is our solution.  For example, with $y=1$ and $x_0=0$, we get the following:
$x_1=0-\frac{\sin(0)+0-1}{\cos(0)+2}=\frac12$
$x_2=\frac12-\frac{\sin(1/2)+1-1}{\cos(1/2)+2}=0.335417790$
$x_3=0.335418032$
$x_4=0.335418032$
And thus we have our solution:
$$1\approx\sin(0.335418032)+2(0.335418032)$$
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