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University Problem Solving Thread

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Mao:

--- Quote from: Ahmad on March 28, 2008, 09:39:34 pm ---
--- Quote from: dcc on March 28, 2008, 03:12:58 pm ---show that:




--- End quote ---

Use the above result to show that,



--- End quote ---








looking at that



which yields an 0/0 indeterminant form, so we use l'Hospital's rule ;D









and for dcc:





if we let

then u approaches 0 as n approaches infinity:








sooo....





QED

*ANOTHER VICTORY*

dcc:

--- Quote from: Ahmad on March 28, 2008, 09:39:34 pm ---
--- Quote from: dcc on March 28, 2008, 03:12:58 pm ---show that:


--- End quote ---


--- End quote ---


Consider:




so we can rewrite the limit as:



So:
so from this, we get:



Now, consider





now expanding this out:



now we know from our handy double angle formulae:


rearranging:



similiarly, the same thing can be done with the rest of the cos terms:

Mao:

--- Quote from: dcc on March 28, 2008, 10:29:07 pm ---
similiarly, the same thing can be done with the rest of the cos terms:


--- End quote ---

i'll anti-cbf for u:

with our handy cosine double angle formula:



let k be a natural number







using the above identity, we can expand to get:



substituting that back in and we'll get



doing this repeatedly will yield:

, k-2 radicals

, k-1 radicals

:D now the last step makes sense :D

Ahmad:
Find .

dcc:
man Ahmad i did some wonderful things trying to figure out this problem.  I managed to solve 2 cubics which gave 6 complex roots then matched them up and did all sorts of wonderful things with complex numbers and all i managed to show was that sin(10) = cos(80). ... quite definitively though, as my proof was quite comprehensive.

anyway i figured it out with algebra, no geometry required!

Consider firstly (as we know sin 30 already)
(look, they are doubles!)

we know also (by manipulating the double angle formulae)


so our original product:



applying the double angle formula again:



and again:



But we know sin(180 - x) = sin(x), so:



Now, from this, we can ascertain:

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