MAT - Paper 2010 - Q3
Flashcards
You want
\[x\cos x < \sin x < x\]
and you have
\[sin x < x < tan x\]
How can you convert between the two (long)?
If you multiply by $\cos x$ you get
\[\sin x \cos x < x\cos x < \tan x \cos x\]The $x\cos x$ is right there in the middle, and $\tan x \cos x$ is $\sin x$, so you have:
\[\sin x \cos x < x\cos x < \sin x\]You know from the original inequality that $\sin x < x$, so you now have:
\[x\cos x < \sin x < x\]