How to solve the equation $x + \sin x = A$?

Question

How to solve the equation $x + \sin x = A$?

I have tried Wolfram Alpha, with no success.

I would be satisfied with an answer in special functions.

The context is, I was trying to find an expression for the coordinates $(x,y)$ for a number given with modulus and argument in the second system in MO post #423657.

I also tried to solve Reduce[{(a^2 + b^2)/a == r, 1/4 (ArcTan[b/a] + (a b)/(a^2 + b^2) ) == arg}, {a, b}] in Mathematica, but it took 10 hours after which I manually aborted.

Answer 1

Here is the solution for:

$$\int_0^\phi \cos^2(t) dt=\frac14(2\phi+\sin(2\phi))=\arg(z)$$

Therefore:

$2\phi+\sin(2\phi)=4\arg(z)$

Using

Inverse of $\sin(x)+x$

the result is:

$$\boxed{\phi=\frac12\text{hav}^{-1}\left(\text I^{-1}_{\frac{4\arg(z)}\pi}\left(\frac12,\frac32\right)\right)}$$

if the argument is satisfied for

$0\le \frac{ab}{a^2+b^2}+\text{sgn}(b)\text{Abs}\left(\tan^{-1}(a,b)\right)\le \pi$ for $\arg(z)=\arg(a,b)= \frac14\left(\frac{ab}{a^2+b^2}+\text{sgn}(b)\text{Abs}\left(\tan^{-1}(a,b)\right)\right)$

then the formula

works