Maths Problem 1: Trignometry and Calculus

[Dave]

At last, the first problem! Your posts will not appear until the competition winner has been announced.

 

Please read the rules thread before posting a solution in this thread.

 

This week, there are two problems (as a bonus ) for you to pit your wits against. If I can't see any correct proofs or people are struggling a lot within a decent amount of time, then I will post a couple of hints here.

 

Problem 1: Trigonometry Values

 

(Angles stated are in degrees.)

 

Calculate the exact value of cos(72), and hence find the value of cos(36)-cos(72).

 

This is a fairly simple question, but a proof is required. It's aimed at the majority of users here.

 

Problem 2: Calculus

 

This is quite a tough calculus question. If you don't like integrations, then don't bother with this one

 

Find :lint: 1/(1+x^4) dx

 

(that is one divided by (x^4 + 1) for anyone that can't interpret the brackets).

 

Have fun! The winners will be announced at 7:00pm GMT on 18th April 2003.

[Dave]

Amendment: there must be a proof behind the answer given. A stated answer will not count. Sorry for the misunderstanding.

 

Also, the answer does not have to be in degrees for the second part of the challenge.

[Dave]

Clues

 

Since there's been little in the way of proofs, I thought I'd post some clues here to kickstart it a bit.

 

Problem 1: Try considering an isoceles triangle with angles 72, 72 and 36.

 

Problem 2: The secret to integrating this beast lies in splitting the fraction up.

[blike]

sh, i havn't had time yet. stop giving hints

[Dave]

sorry

[blike]

bump

[Dave]

Oops, I'm a bit late posting the solutions, but I only remembered at 7 and had to write the solution up.

 

I'm sorry to say that we didn't have any correct entries for either question, with the exception of faf who answered both problems but didn't provide any proof

 

I'm going to post a solution to the first question now, and the second question a bit later (since the proof is rather long and I don't have a copy of it yet that I can copy, I'll have to work through it again).

 

Question 1

 

Calculate the exact value of cos(72), and hence find the value of cos(36)-cos(72).

 

Perhaps this was a little misleading. My solution is outlined in the zip'd file which contains a Microsoft Word document with the images in.

 

I should be posting more problems tomorrow night. Have fun with this until then.

maths challenge 1.doc.zip

[Guest mrleao]

I can't open the file. There is a problem in the file...

[Dave]

Wow, this is a blast from the past.

 

The file works fine for me.

[bloodhound]

nope. the file seems to be corrupted. working on the integral rite now..