the function $$ \frac{\theta-\sin(\theta)\cos(\theta)}{\sin(\theta)^2} $$ is monotone increasing diffeomorphism of $(0,\pi)$ onto $(+,+\infty)$.
Is it possible to write an explicit inverse ?
explicit inverse
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$\begingroup$
calculus
inverse-function
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0Probably not... – 2017-01-23
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0Seems unlikely. This can be rewritten as $\frac{2\theta -\sin 2\theta}{1+\cos 2\theta}$. Not that that helps much. – 2017-01-23
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0As it has been said by others, a **rule of thumb** is that it is very unlikely that a function that mixes arc and trigonometric functions of this arc is tractable, whatever you have to do on it. – 2017-01-23
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1This is essentially Kepler's equation (https://en.wikipedia.org/wiki/Kepler's_equation). An explicit inverse can be found through Lagrange's inversion theorem, but it is not a nice series. The inversion of such functions is usually numerically performed through Newton's method. – 2017-01-23