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The question pretty much says it.

I need to solve $t$ in this equation: $$ x = t - \sin{(t)} $$ Either I've forgotten how to do it, or I am just blind, etc. Anyway, I'm completely stuck at this.

Actually, I need to solve a vector:

$$(\; x \; , \; y \;) = (\; t - \sin{(t)} \; , \; 1 - \cos{(t)} \;)$$

Inverse of $y$ is trivial: $t = \cos^{-1}{(1 - y)}$. But that doesn't help me much further on.

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    You're not going to get a nice analytic solution.2012-06-07
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    You have not forgotten how to do it, nor are you blind. As far as I know, there is no way to invert $t-\sin t$ with (a finite number of) elementary functions. Instead (for your vector problem) you should use the $y$ coordinate to find $t$ up to a factor of $2\pi$, and then find this factor of $2\pi$ involving the $x$ coordinate. In your homework do $x$ and $y$ have particular values?2012-06-07
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    @anon no, they don't. $t$ is limited: $t \in [0,2 \pi]$, other than that, I should find the length of the line in this interval.2012-06-07
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    If you need to find the length of the curve, you do not need to solve for $t$.2012-06-07
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    If you are trying to find the area under an arch of the cycloid, or the arclength of part of the cycloid, the parametric equation is nice to work with directly.2012-06-07
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    The assignment just says, that I should find the length of it, doesn't say a thing about area.2012-06-07
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    Side remark: This equation appears in study of the two-body problem as "Kepler's equation", $\omega t = \psi - e \sin \psi$. Quoting from Goldstein's "Classical mechanics", at the end of Section 3-8: "The solution of the transcendental Kepler's equation to give the value of $\psi$ corresponding to a given time is a problem that has attracted the attention of many famous mathematicians ever since Kepler posed the question early in the seventeenth century. [...]2012-06-07
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    (cont.) Indeed, it can be claimed that the practical need to solve Kepler's equation to accuracies of a second of arc over the whole range of eccentricity fathered many of the developments in numerical mathematics in the eighteenth and nineteenth centuries. A few of the more than 100 methods of solution developed in the pre-computer era are considered in the exercises to this chapter."2012-06-07

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