# Approximating Trig Functions with Polynomials

I was thinking about the graphs of different trig functions and noticed that most of them are of a similar shape to some different types of polynomials. For example:

• Higher degree polynomials create a wave like sin or cos
• $x^3$ looks like one repetition of tan, and could be flipped and shifted to look like cot
• Each repetition of sec and csc looks like two quadratic parabolas

While obviously the polynomials aren't going to be an exact approximation, are there a set of coefficients that create a reasonably close (to a few decimal places) approximation of one period of the trig functions?

If so, is this useful? Or are there other, better, post Pre Calculus approximations of the trig functions?

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You will learn about Taylor series, which can be used to approximate functions using polynomials in higher calculus courses –  user130512 Feb 25 '14 at 18:30

Seems to me that you are getting ready for Taylor series of trig functions. I would suggest to google this and you are getting lots of answers

http://en.wikipedia.org/wiki/Taylor_series would do but there are many many other great sites.

As far as usefullness, that can't be even described in one sentence. I appreciate you being inquisitive. That approach is very good, therefore (+1)

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My teacher mentioned those when I asked her. Am I right in saying that it is a kind of Fourier transform with polynomials instead of waves? It tells you what coefficients you need for the best fitting polynomial of a given degree? –  Linuxios Feb 25 '14 at 18:35
Fourier used an (infinite) sum of trig terms to describe certain types of waves. That's not quite what Taylor series is about, but the idea is similar in the sense that a sum of infinite terms "models" a particular curve –  imranfat Feb 25 '14 at 19:43
That's more what I meant. Awesome! Thanks for the quick answer, and I finally understand why precalc is spending so much time on curves (polynomials, e^x, log, ln, sin, etc.). –  Linuxios Feb 25 '14 at 19:44
@Linuxios If you intend to go into the Calculus sequence, consider your precalc course as its foundation. Make it strong. Lots of hard calc problems are considered "hard" because people are not solid in their algebra that is so much needed. Notwithstanding the fact that Calc isn't to be taken easy, having a solid precalc background really, REALLY puts you at an advantage. Good luck –  imranfat Feb 25 '14 at 20:16
Thanks. I'm just excited. I love math :). –  Linuxios Feb 25 '14 at 20:16