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I've been battling with parametric equations recently and one of the situations I came across was a system of parametric equations that graph the cycloid of circle with radius $a$, which looks as follows:

$y=a(1-\cos\theta)$, $x=a(\theta-\sin\theta)$

Now I'm aware that it's possible to remove the parameter in this system (the parameter being either a or $\theta$), but is it possible to remove both? And also, when expressing $\theta$ in terms of x, is this process correct:

$\frac{x}{a}=\theta-\sin\theta$

$\sin^{-1}(\frac{x}{a}) = 2\theta$

$\frac{\sin^{-1}(\frac{x}{a})}{2}=\theta$

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  • $\begingroup$ $$\cos\theta=?$$ $\endgroup$
    – lab bhattacharjee
    Jul 31, 2017 at 18:46
  • $\begingroup$ In this instance, $cos\theta$ has no defined value as $\theta$ is the parameter variable, meaning $\theta$ can be anywhere between 0 and 89, and the result can be anywhere between 0 and 0.99999... @lab bhattacharjee $\endgroup$
    – joshuaheckroodt
    Jul 31, 2017 at 18:58

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This is a variation on Kepler's equation (https://en.wikipedia.org/wiki/Kepler%27s_equation) which can not be solved algebraically.

It can only be solved numerically.

However, by using the power series for $\sin$, you could write $y =a\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}x^{2n+1}}{(2n+1)!} =ax\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}(x^2)^{n}}{(2n+1)!} $ and invert the resulting power series to as many terms as you want.

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