Problems from dhk div 2010!!

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Labib
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Problems from dhk div 2010!!

Unread post by Labib » Thu Dec 30, 2010 12:08 am

$1$)
If $x$ is very very very small $sin x\approx x$. An operator $S_n$ is defined such that
$S_n(x)= sin\cdot sin\cdot sin\cdot ....sin x $(a total of $n, sin$ operators are included here). For
sufficiently large $n, S_n(x) \approx S_{n-1}(x)$. In that case, express cos $(S_n(x))$ as the
nearest rational value.

Please help me understand and solve the prob.
Last edited by Labib on Thu Dec 30, 2010 10:34 pm, edited 2 times in total.
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tushar7
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Re: Problems from dhk div 2010!!

Unread post by tushar7 » Thu Dec 30, 2010 12:31 am

yap , i didnt even understand this problem .... i wasted 14 mins

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Tahmid Hasan
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Re: Problems from dhk div 2010!!

Unread post by Tahmid Hasan » Thu Dec 30, 2010 8:23 pm

for the small x,the value of $$x=0 $$ and $$S_n(x)=0$$
that's all i can say
বড় ভালবাসি তোমায়,মা

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Avik Roy
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Re: Problems from dhk div 2010!!

Unread post by Avik Roy » Thu Dec 30, 2010 8:42 pm

Tahmid, you have reached the right answer, but in a wrong way...
The question doesn't require x to be small or 0. The result is 1 and it needs quite jumpy travels to reach the answer rigorously. However, the question was meant to judge your intution
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

atiqur_jhe
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Re: Problems from dhk div 2010!!

Unread post by atiqur_jhe » Fri Dec 31, 2010 8:53 pm

I didnt understood that nSn(x) Sn-1(x). My ans is like that sin sin sin sin(x)=x(because x is so small so sin x =x) so Sn(x)=x and cos Sn (x)=cos x=1

atiqur_jhe
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Re: Problems from dhk div 2010!!

Unread post by atiqur_jhe » Fri Dec 31, 2010 8:55 pm

Amar ta ki vul holo.confused

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Avik Roy
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Re: Problems from dhk div 2010!!

Unread post by Avik Roy » Sat Jan 01, 2011 10:06 pm

The problem never said that $x$ has to be small. So you have to think in a generalized manner. You have reached the correct answer. But your understanding is faulty.
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

kamrul2010
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Re: Problems from dhk div 2010!!

Unread post by kamrul2010 » Wed Jan 05, 2011 6:46 pm

আরে বাবা, আর কিছু না পারলে x এর একটা মান ধইরা দশ-বারোটা চেষ্টা কইরা দেখো!
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sm.joty
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Re: Problems from dhk div 2010!!

Unread post by sm.joty » Sat Dec 24, 2011 12:26 am

আমার সমাধান এরকম,
আমারা দেখতে পাচ্ছি যে,$S_n(x)\approx S_{n-1}(x)$
তাহলে উলটা দিক থেকে দেখলে। আমারা $S_1$,$S_2$,....... এভাবে বের করে করে $S_n$ পর্যন্ত যাব। যখন n বেশ বড়। একবার অপারেশন করলে $x$ এর যেকোনো মানের জন্য $sinx$ এর মান ১ এর চেয়ে কম হবে। তাই প্রতিবার অপারেশন করার পর আমরা যে মান পাব সেটা ক্রমেই ছোট হতে থাকবে। তাই $n$ যথেষ্ট বড় হলে $S_n$ অনেক অনেক ছোট হবে ।
আবার আমরা জানি, $x$ অনেক ছোট হলে $cos(x)\approx 1$ হয়।
তাহলে $Cos(S_n)\approx 1$
কাজেই উত্তর $1$ (বলা হইছে কাছাকাছি মুলদ সংখ্যা। কাজেই এটা আসন্ন মানই হবে)। :D
কোথাও কোনো ভুল করলাম কি ?
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

Dipika
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Re: Problems from dhk div 2010!!

Unread post by Dipika » Sat Dec 24, 2011 10:21 am

A article was published about this article on 'Gonit School' of The prothom-alo.....

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