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 April 7th, 2019, 03:15 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 540 Thanks: 82 Prove inequality Prove inequality : $\displaystyle \underbrace{\sin \sin ... \sin }_{N-times} (N)\leq \sin \frac{1}{N}\;$ , $\displaystyle N\in \mathbb{N}$. To write it better in short-terms : $\displaystyle \sin_{N} (N)\leq\sin \frac{1}{N}$.
 April 7th, 2019, 01:36 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,467 Thanks: 1342 if this is a product chain it should be written as $\sin^N(N) \leq \sin\left(\dfrac 1 N \right),~N \in \mathbb{N}$ if it's a function composition chain I've seen it written as $\underset{\text{N times}}{\underbrace{\sin\circ \sin \circ \dots \sin(N)} }=\sin^{\circ N}(N) \leq \sin\left(\dfrac 1 N \right)$ which is it? Thanks from idontknow
 April 7th, 2019, 01:38 PM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 540 Thanks: 82 The composition N-times of sin function.
 April 7th, 2019, 02:00 PM #4 Senior Member   Joined: Aug 2012 Posts: 2,329 Thanks: 720 Wolfram Alpha gives $\sin(\sin(2)) \approx 0.78907 \dots > \frac{1}{2}$. Of course, that's in radians. However, it's $< \frac{1}{2}$ for $2$ degrees. Just wanted to note that. OP must mean degrees. Thanks from topsquark and idontknow Last edited by skipjack; April 7th, 2019 at 03:21 PM.
 April 7th, 2019, 02:01 PM #5 Global Moderator   Joined: May 2007 Posts: 6,770 Thanks: 700 For $x \gt 0$ $\sin(x)\lt x$. Therefore $\sin(\sin(1/N))\lt \sin(1/N)$..Repeat N times to get what you want. Thanks from topsquark and idontknow Last edited by skipjack; April 7th, 2019 at 03:14 PM.
 April 7th, 2019, 02:18 PM #6 Senior Member   Joined: Dec 2015 From: somewhere Posts: 540 Thanks: 82 Tried all methods posted above but got no result .
April 8th, 2019, 01:18 PM   #7
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 Originally Posted by idontknow Tried all methods posted above but got no result .

April 8th, 2019, 01:19 PM   #8
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Quote:
 Originally Posted by mathman Did you understand my reply?
No.

April 9th, 2019, 12:11 PM   #9
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Quote:
 Originally Posted by idontknow No.

 April 9th, 2019, 06:06 PM #10 Senior Member   Joined: Sep 2016 From: USA Posts: 624 Thanks: 396 Math Focus: Dynamical systems, analytic function theory, numerics Use induction and the fact that $\sin$ is entire and its Taylor expansion can be written as $\sin(N) = \sin(N-1) + \frac{1}{2} \cos(N-1) - \frac{1}{6}\sin(N-1) + \dotsc$. Thanks from idontknow

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