From: "Gareth McCaughan" <gareth.mccaughan@pobox.com>
On Wednesday 19 July 2006 03:43, Alec Mihailovs wrote:
sin(x)*(14+cos(x))/(9+6*cos(x)) < x < 8*sin(x)/9/(1+cos(x))+25*sin(x)/(36+9*cos(x))
(I suppose you'd prove them by clearing denominators, expressing everything in terms of sin(kx) and cos(kx), and comparing terms in the Taylor series, or something like that. How do you find them?)
In Maple, I found (the beginning of) Taylor series for sin(x)*(a+b*cos(x))/(c+d*cos(x)), then solved the system of equations, the coefficient at x equals 1 and the coefficients at x^3 and x^5 equal 0. That gave series x - x^7/2100 + O(x^9) That proofs (or can be used to prove) the first inequality. Similarly for the second one, with the resulting series x + x^7/2800 + O(x^9) Alec