If we know the Taylor expansion for the $\cos(x)$ function around $0$, how can we use it to derive the Taylor expansion of a similar function ($\cos(x+π/4)$) around $0$?

I do know how to get the Taylor series with formula $T(x) = f(a) + f'(a)(x-a) + \ldots$, but want to learn some tricks on getting expansions of similar functions using the ones we already know.

  • $\begingroup$ It seems difficult to do this if you only know the first $n$ terms of the first Taylor expansion. $\endgroup$ – Brian Tung Jun 16 '16 at 18:04
  • $\begingroup$ @MathematicsStudent1122: OP wants another Maclaurin series (centered at zero), not a mere shift, which presumably they already know how to do. $\endgroup$ – Brian Tung Jun 16 '16 at 18:05
  • $\begingroup$ Ah, I see, thanks. $\endgroup$ – MathematicsStudent1122 Jun 16 '16 at 18:06

In general, it is not possible to obtain a Taylor series for $f(x+a)$ from a Taylor series for $f(x)$. Even theoretically, it is only possible if the Taylor series for $f(x)$ converges against $f$ at least in a neighbourhood of $a$.

Nevertheless, you can rewrite $\cos(x+a)$ as $b\cos x+c\sin x$ with suitable constants $b,c$ by using the addition theorem. Now you can just combine the Taylor series for sine and cosine appropriately.


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