Exploring Strange Functions: Deriving and Approximating Transcendental Functions

[Tymick]

I recently saw that the sine function could be approximated greatly by [1-(((2/pi)*x)-1)^2]^(pi/e) for
the range (0,pi) does anyone have any other strange functions like this that may satisfy some of the other transcendentals? (It'd be nice to find out how to derive the above formula too)

 

[Feldoh]

Do you know about maclaurin series?

 

[yaychemistry]

The usual approximation is the Taylor series:
$$f\left(x\right) \approx \sum_n \frac{1}{n!}\left.\frac{d^{n}f}{dx^{n}}\right|_{x=a}\left(x - a\right)^n$$
However, your function (due to the power pi/e) does not look exactly like a Taylor series. My best guess is that it's the Taylor series of something like $$f\left(\sin\left(x\right)\right)$$ and then the function f is inverted. The factor of e makes me want to guess its a logarithm, but I'd have to work it out.

The Maclaurinn Series are specific cases of Taylor series where we set a = 0. If you are very interested in series approximations of functions you can look up Pade approximations and Laurent series on wikipedia as well, as "generalizations" of the Taylor series. These *might* apply to your case, but I doubt it.