## mahmit2012 3 years ago Q9:suppose that 0<x1<x2<...<xn<pi/2 then show that : tanx1<(sinx1+sinx2+...+sinxn)/(cosx1+cosx2+...+cosxn)<tanxn

1. experimentX

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2. slaaibak

not sure how to do this, but I would use a comparison test.. maybe showing that first: tan xn = sin xn/cos xn = nsinxn / ncosxn then showing: [sin x is increasing on x e (0, pi/2) ] (n-1)(sin x(n-1)) + sin xn < nsinxn [cos x is decreasing on x e (0, pi/2) ] and (n-1)(cos x(n-1)) + cos xn > ncosxn Then [(n-1)(sin x(n-1)) + sin xn]/[(n-1)(cos x(n-1)) + cos xn] < n sin xn/ n cos xn Similar reasoning to show it's > tan x1. but ye, Just a suggestion, it's probably wrong. but ye use if you can..

3. mahmit2012

what is a (n-1)sinx(n-1)?and how you use that?

4. slaaibak

it's $\sin x_{n-1}$ n-1 times... so $\sin x_{n-1} +\sin x_{n-1} +\sin x_{n-1} +\sin x_{n-1} + ... +\sin x_{n-1}$

5. mahmit2012

I know !bot I want to solve: tanx1<(sinx1+sinx2+...+sinxn)/(cosx1+cosx2+...+cosxn)<tanxn

6. slaaibak

well, you can't solve that. I'm saying you can use a comparison to prove it... Read my first message thoroughly

7. mahmit2012

I see your first commend. But choosing those two terms in num and den confusing. However I accept your opinion for sin and cos.

8. mahmit2012

$[n.\sin x _{1} \le \sin x _{1}+\sin x _{2}+...\sin x _{n}\le n.\sin x _{n}\$ n.\cos x _{n} \le \cos x _{1}+\cos x _{2}+...\cos x _{n}\le n.\cos x _{1}\]\] as a known a<b<c ,A<B<C => a/C<b/B<c/A so $\tan x _{1} \le (\sin x _{1}+\sin x _{2}+...\sin x _{n})/(\cos x _{1}+\cos x _{2}+...\cos x _{n}) \le \tan x _{n}$