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ParthKohli
 one year ago
@ganeshie8 @Miracrown @dan815 @UsukiDoll
ParthKohli
 one year ago
@ganeshie8 @Miracrown @dan815 @UsukiDoll

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ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4\[\cot(\pi/15)  4\sin(\pi/15) = \sqrt{15}\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4My work so far. This expression is equivalent to \(\cot(16\pi/15) + 4\sin(16\pi/15)\). \[x := e^{2\pi i /11}\]Then:\[\cot(16\pi/15) = \frac{\cos(16\pi/15)}{\sin(16\pi/15) } = \dfrac{(x^4 + \overline{x^4})/2}{(x^4  \overline{x^4})/2i}\]\[\Rightarrow i\cot(16\pi/15) = \dfrac{x^{11}+ x^4}{x^{11}  x^4} = \dfrac{x^7+1}{x^7  1}\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Similarly,\[4i \sin(16\pi/15) = 2(x^4  x^{11})\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0I quit... what is this?!

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0like complex analysis?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4If we add those two, we get\[i \cot(16\pi/15) + 4i \sin(16\pi/15) = \frac{x^7+1}{x^7  1} + 2(x^4  x^{11})\]Somehow, I want to make that fraction a polynomial using properties of roots of unity. Any ideas?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4@UsukiDoll No, I'm just trying to prove a trigonometric identity using complex numbers. Specifically, I mean roots of unity. Is that complex analysis? I don't know.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0I'm still a noob when it comes to proofs D:

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Also, we have to show that the righthand side is \(i\sqrt{15}\) and we'll be done.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4I know I can write \(1\) as \(x^{15}\) or \(x^{30}\) or so on. Which exponent should I choose so as to make it divisible by \(x^7 + 1\)?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4\(x^7  1\) actually, which means that we're also looking at a seventh root of unity.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4@ganeshie8 Speak out. :P

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433849058768:dw

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4\[x^7  1 \mid x^{15n}+x^7\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0im still going through... you meant \(x := e^{2\pi i /15}\) right

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Yes, that's what I meant. Whoops.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0let me grab pen and paper

skullpatrol
 one year ago
Best ResponseYou've already chosen the best response.0A pencil is preferable :P

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433850807361:dw pacman

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Dan just made me realise that this problem is lame and we shouldn't be working on it as much as we are right now.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4This problem is also really hard.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0if it's hard then why are we viewing it? rofl xD

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4To help me solve it.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0but we're not pros at this D:

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Then why are you viewing it?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Know what, this problem was asked at MSE, and is currently unanswered.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433851540716:dw

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4This isn't researchlevel.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Oh, it was answered. Lookielookie. http://math.stackexchange.com/questions/1317960/howprovethiscotpi154sinpi15sqrt15

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4OK, so he used the 30th cyclotomic polynomial. Cool.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Nah, that's not me.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4I still want to use complex numbers for this  that should improve my currently zero understanding of them.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4It's known that\[\sin(\pi/5) = \dfrac{\sqrt{10  2\sqrt 5}}{4} = 3\sin^3 (\pi/15)  4\sin(\pi/15)\]We can simply solve for trigratios of \(\pi/15\) like this, but it's stupid and timeconsuming.

skullpatrol
 one year ago
Best ResponseYou've already chosen the best response.0math consumes time

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0il get back to this later, my head is spinning right nw

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Same, I'm sleepy even though I woke up at 1 PM.

Miracrown
 one year ago
Best ResponseYou've already chosen the best response.0I need sleep. G'night

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Dan kisses you good night.
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