Trigonometry $\frac{\pi} 7$

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Labib
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Trigonometry $\frac{\pi} 7$

Unread post by Labib » Wed Jul 13, 2011 1:38 am

This is a plane Euclidean geometry I did not understand... The problem states::

sin$\frac\pi 7\cdot $ sin$\frac{2\pi} 7\cdot $ sin $\frac{3\pi} 7\cdot$ sin $\frac{4\pi} 7 \cdot $ sin$\frac{5\pi} 7 \cdot $sin $\frac{6\pi} 7 \cdot $ sin $\frac{7\pi} 7 $ = $\frac 7 {64}$

(Source ::
Plane Euclidean Geometry . Exercise 2(f) . Problem 32
)

Need help urgently!!
sin$\frac{7\pi} 7 = 0$, so how come this be proved!! :o
Last edited by Labib on Wed Jul 13, 2011 11:59 pm, edited 1 time in total.
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confused
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Re: Trigonometry $\frac{\pi} 7$

Unread post by confused » Wed Jul 13, 2011 12:49 pm

uestion ta ki tik ase :? :?
nafisa

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Labib
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Re: Trigonometry $\frac{\pi} 7$

Unread post by Labib » Wed Jul 13, 2011 11:54 pm

100% @ confused.... I didn't make any mistake stating the problem....

There was a hint as well. I tried using it but didn't get much out of it!!!
I'll post the hint here.

Hint::
Find sin $7\theta$ as a polynomial in terms of sin $\theta$ and then look at the equation sin $7\theta$ / sin $\theta = 0$. ($ \theta \neq n\pi $).
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Labib
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Re: Trigonometry $\frac{\pi} 7$

Unread post by Labib » Thu Jul 14, 2011 12:29 am

I believe I've posted this problem in the wrong Forum....
It should be moved to the Geometry forum....
If any of the admins could please help me out.... :oops: :oops:
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Labib
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Re: Trigonometry $\frac{\pi} 7$

Unread post by Labib » Fri Jul 15, 2011 12:11 am

I've also created an AOPS topic for this problem....
Here's the link:::
http://www.artofproblemsolving.com/Foru ... 3#p2358903
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nisha
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Re: Trigonometry $\frac{\pi} 7$

Unread post by nisha » Tue Jul 19, 2011 11:07 am

maybe its wrong! :?

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leonardo shawon
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Re: Trigonometry $\frac{\pi} 7$

Unread post by leonardo shawon » Tue Jul 19, 2011 3:36 pm

u can use this formula
http://www.artofproblemsolving.com/Foru ... hilit=trig
I also think as Nisha said that the given expression isnot valid. whats ur opinion?
Ibtehaz Shawon
BRAC University.

long way to go .....

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Labib
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Re: Trigonometry $\frac{\pi} 7$

Unread post by Labib » Wed Jul 20, 2011 11:51 pm

This was the theorem I saw::

$n~(\geq 2)$ is a natural number.
$\prod_{k=1}^{n-1}\sin\frac{k\pi}{n}=\frac{n}{2^{n-1}}$

I still believe the problem's invalid! :(
Last edited by Labib on Thu Jul 21, 2011 12:13 am, edited 1 time in total.
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leonardo shawon
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Re: Trigonometry $\frac{\pi} 7$

Unread post by leonardo shawon » Thu Jul 21, 2011 12:03 am

it will be invalid if the sin 7pi/7=0
Ibtehaz Shawon
BRAC University.

long way to go .....

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