Recreational Problems (SM level)

[ryley]

I had a go at it, but I had to fudge my working a bit, pretty sure I f**cked up somewhere, could someone show me where?

[kamil9876]

seems ok good. However remember that if then but the converse is not true. e.g: however so you may have missed some solutions. Although the idea to use the geometric series is spot on

/0 has a habit of recycling questions in different threads Ive noticed . this one is slightly similair:
http://vcenotes.com/forum/index.php/topic,13271.msg149423.html#msg149423

It's also quite interesting how complex numbers can be used to do trigonometry that can also be done geometrically. (a formula for cosx+cos2x...cosnx can be derived both with complex numbers and with just geometry by treating it as a vector sum(sum of the horizontal components of unit vectors))

[ryley]

So to clarify, if I have exp(ix)=some real number, I cannot assume that cos(x)=that real number?

EDIT: Thanks for clarifying, I just misunderstood what you meant, you shouldn't be scaring me like that this time of the year 

[kamil9876]

that statement is true. since "exp(ix)=some real number" implies that the imaginary part is zero and so cosx=that real number.  However all I meant to say is that say for some real number x, if cosx=a, then it's not neccesary that exp(ix)=a since the imaginary part may happen to be non-zero. But I just realised your method does NOT suffer from that mistake! since you used exp(ix)+exp(-ix)=2cosx. LOL sorry it was too late at night and didn't realise that and just thought you solved exp(ix)+exp(2ix)....=0 instead.

[/0]

Awesome solution. Another method would be:



Or perhaps the more highschooley method could be to use De Moivre's to derive expansions for and and to use that to construct a polynomial in .

[kamil9876]

For interest's sake, there's also a purely geometric way to find the formula for cosx+cos2x...+cosnx. It comes from this diagram:

[TrueTears]

For interest's sake, there's also a purely geometric way to find the formula for cosx+cos2x...+cosnx. It comes from this diagram:

Rofl I remember a question like this in an exam about working out the perimeter the geometric way.

[TrueTears]

Here's a fun one:

Prove that

where

[/0]

For interest's sake, there's also a purely geometric way to find the formula for cosx+cos2x...+cosnx. It comes from this diagram:

Rofl I remember a question like this in an exam about working out the perimeter the geometric way.

Lol that was a NSW Exam I think... pretty wack stuff

[TrueTears]

For interest's sake, there's also a purely geometric way to find the formula for cosx+cos2x...+cosnx. It comes from this diagram:

Rofl I remember a question like this in an exam about working out the perimeter the geometric way.

Lol that was a NSW Exam I think... pretty wack stuff

lol hsc ftw

[kamil9876]

lol wtf the looks so reduntant. Might as well have been or any function with positive range.

[dcc]

Here's a fun one:

Prove that

where

Since as and , the above limit exists and is zero.

(what was the purpose of this question?)

[TrueTears]

Here's a fun one:

Prove that

where

Since as and , the above limit exists and is zero.

(what was the purpose of this question?)

Must there be a purpose?

(what was the purpose of your answer?)

[dcc]

Must there be a purpose?

(what was the purpose of your answer?)

The purpose of my answer was to illustrate this question's lack of a purpose.

[kamil9876]

for more rigour i would sandwhich the motherfucker:

The purpose of my answer was to illustrate this question's lack of a purpose.

lol true the looked so redundant to me.