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        riddles >> medium >> prove
        
(Message started by: tony123 on Jul 19^th, 2007, 9:03am)

Title: prove
Post by tony123 on Jul 19^th, 2007, 9:03am

http://www.mathramz.com/phpbb/latexrender/pictures/3390ffd7b21a77e4996623f1ddec5d29.png

Title: Re: prove
Post by pex on Jul 19^th, 2007, 9:13am

Tony - not so long ago (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1184282061#4), you "were sorry and apologized" for (1) not writing descriptive titles and (2) linking to images which can easily be attached, or even written out.

What happened?

Title: Re: prove
Post by tony123 on Jul 19^th, 2007, 9:44am

why latiks not found any way help me to Attach
and thank :-*

Title: Re: prove
Post by JohanC on Jul 19^th, 2007, 2:34pm

Hi, Barukh,
In my version of Firefox that transparent PNG looks quite ugly, while it renders nicely in IE.
I notice you created an embedded link to that external website, which runs the risk that the image will disappear in a few months.

Title: Re: prove
Post by Barukh on Jul 19^th, 2007, 10:59pm

on 07/19/07 at 14:34:37, JohanC wrote:

I notice you created an embedded link to that external website, which runs the risk that the image will disappear in a few months.

Yes, you are right. It was an instant action... I removed the post.

Title: Re: prove
Post by Barukh on Jul 20^th, 2007, 6:18am

Amazing!

[hide]The whole thing may be derived purely geometrically[/hide].

Title: Re: prove
Post by Barukh on Jul 21^st, 2007, 5:57am

All that is needed is on the attached drawing.

Title: Re: prove
Post by K Sengupta on Jul 21^st, 2007, 10:08am

[hide]From the identity cos 36 = sin 54, we obtain:

1 – 2x^2 = 3x – 4x^3, where x = sin 18
Or, 4x^3 -2x^2 – 3x + 1 = 0
Or, x_1 = 1; x_2,3 = (-1+/-V5)/4

Since sin t is increasing in 0<= t<= 90, we must have sin 18 = (V5 – 1)/4

Thus, cos 36 = 1 – 2*(sin 18)^2 = (V5 + 1)/4

Now, cot 7.5
= cot 15/2
= (1+ cos 15)/sin 15

But sin 15 = sin (45-30) = (V6 – V2)/4 (upon simplification); and:

cos 15 = cos (45 – 30) = (V6 + V2)/4

Thus, cot 7.5
= (4+ V6 + V2)/(V6 – V2)
= (4(V6 + V2) + (8+ 4V3))/4
= V6 + V2 + 2 + V3

Thus, 4 cos 36 + cot 7.5
= (V5 + 1) + (V6 + V2 + 2 + V3)
= V1+ V5 + V6 + V2 + V4 + V3
= V1 + V2 + V3 + V4 + V5 + V6

Q E D [/hide]