Calculate alpha from $\alpha + \sin(\alpha)$ = K Sorry for the dumb question, but I'm not involved in math.
I need to reverse the following formula, to calculate $\alpha$:
$$a = b(\alpha + \sin \alpha)/c$$
So I have:
$$(\alpha + \sin \alpha)=ac/b = K$$
Since $a$, $b$, $c$ are constant, I put equal to $K$.
$\alpha$ is measured in radians. I need to find the value of $\alpha$ (in radians or degree).
Thanks to all!!
 A: There is no "closed-form" solution to this equation.  You can use numerical methods to solve it for any given value of $K$.  If $K$ is small,  you can use a series:
$$\alpha = {\frac {1}{2}}K+{\frac {1}{96}}{K}^{3}+{\frac {1}{1920}}{K}^{5}+{\frac {43}{1290240}}{K}^{7}+{\frac {223}{92897280}}{K}^{9} + \ldots $$
The error in the approximation using the terms above is less than about $2 \times 10^{-7}$ for $-1 \le K \le 1$.
A: These are what we call Transcendental Equations. 
Solving these involve graphical or numerical analysis, both of which yield approximate results. 
In graphical analysis, you first rearrange the equation as:
$\sin\alpha = K - \alpha $
Now, on a graph, plot the curve $y=\sin\alpha$. On the same graph, plot the straight line $y=K-\alpha$
The point at which the two intersect is basically the solution of the equation. From the $y$ coordinate of the intersection point, you can then easily calculate the value of $\alpha$ using $y=K-\alpha$
Numeric solutions involve methods like the Newton Raphson Method, Bisection Method, etc.
Here is a nice wikipedia article enlisting all such methods.
A: A useful link with a useful terminology as well 
http://en.wikipedia.org/wiki/Lagrange_inversion_theorem 
This equation is often present in the mathematical problems of celestial mechanics 
A: You are trying to find the root of the equation $f(\alpha) = \sin(\alpha) + \alpha - K$. Just use Newton-Raphson to get to the solution. If your $K$ is fairly small, then initializing Newton-Raphson with $\frac{K}{2}$ should be good and if its quite large, then initializing $\alpha$ with $K$ should do.
