email@example.com writes: > True or False? : > > There exists a polynomial P such that: > > | P(t) - cos(t) | <= 10^-6.
Presumably missing "for all t"?
> I said false. Because P(t) could be very large.
Indeed. When you go to very large |t| (large positive or negative t), only the term with the largest power of t will matter. This mean that P will go off towards plus or minus infinity for sufficiently large |t|. The only exception to this are constant polynomials like P(t)=1, which obviously won't work.
So, there is no such P.
> cos(t) = 1 - x^2/2! + x^4/4! - x^6/6! + ... > > and this is valid for all values of t, so perhaps there could be a > polynomial (I'm not sure what that would be).
The Taylor series is infinite, and has to be so to work. (My argument above won't work, because there is no largest-power term)
Polynomials are defined to have only a finite number of terms.
If you have some easy way of plotting graphs you might want to make a graph of the sum up the fifth term or so of the Taylor series. It will show a graph that follows cos(t) well near t=0 but breaks off when |t| gets larger.
On the other hand, one often isn't interested in _all_ t, but only in t in an interval like e.g. -pi < t <= pi, or even 0 <= t <= pi/2. (think about it)
Then there _are_ good solutions for P(t). The first terms of the Taylor series is one possibility, but there are others, slightly changing the terms to make the difference a bit smaller. (I don't know how many terms are needed to get below 10^-6)