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# Trig Equation watch

1. Require assistance with a question from one of TeeEm's IYGB Special Papers:

4cos(x)cos(2x)cos(5x)+1=0.

First I tried using the Prosthaphaeresis Formulae to get cos(2x)+cos(4x)+cos(6x)+cos(8x)+ 1=0, but that didn't seem to be any more solvable. Then I wrote it as a polynomial in cos(x): it ended up being a quartic in cos^2x that does factorise according to WolframAlpha, but gives surds as answers that are then impossible to turn into the required values of x without a calculator. Thus any assistance would be appreciated.
2. Have another look at cos(2x)+cos(4x)+cos(6x)+cos(8x)+ 1=0. Does it remind you of anything?
3. Maybe double angle formulae: cos(2x)+2cos^2(2x)-1+cos(6x)+4cos^2(2x)-2+1=0?
4. (Original post by constellarknight)
Maybe double angle formulae: cos(2x)+2cos^2(2x)-1+cos(6x)+4cos^2(2x)-2+1=0?
This question is from the IYGB, Special paper B and it requires tricks and manipulations
5. I know it is, I was doing the paper and got stuck on it. So since the solution isn't on your website yet, care to give me a hint about the "tricks and manipulations" required?
6. (Original post by constellarknight)
I know it is, I was doing the paper and got stuck on it. So since the solution isn't on your website yet, care to give me a hint about the "tricks and manipulations" required?
I'd guess he's looking for a complex approach. Are you familiar with De Moivre's theorem and that kind of stuff?
7. Yes, but I don't see how it could be applied as there are no imaginary terms.
8. (Original post by constellarknight)
Yes, but I don't see how it could be applied as there are no imaginary terms.
BTW, you ought to reply to my post, else I won't see any response till I look at the thread.

You have to create the imaginary terms yourself, in a useful sort of way. Start by considering a sum of complex exponentials that look like what you already have. Then consider real and imaginary parts.
9. Well, the LHS is 1+Re(cis(2x)+cis(4x)+cis(6x)+cis (8x))... I really have no idea here...
10. (Original post by constellarknight)
I know it is, I was doing the paper and got stuck on it. So since the solution isn't on your website yet, care to give me a hint about the "tricks and manipulations" required?
it does not require complex numbers.
here it is
Attached Images
11. trig.pdf (269.5 KB, 75 views)
12. Thanks for the solution. But how did you think to multiply by sin x in the first place? Was it just "intuition" or is this a general method for equations with a product of sin or cos terms?
13. (Original post by TeeEm)
it does not require complex numbers.
here it is
Brilliant!
14. (Original post by Zacken)
Brilliant!
did you like it ...?
an evil question ... which reminds me I need to write solutions to this paper.
15. (Original post by TeeEm)
did you like it ...?
an evil question ... which reminds me I need to write solutions to this paper.
Very much so!

Kind of wished I'd given it a try before opening your solutions, eurgh.
16. Yes, you've been rather slow with IYGB stuff recently: indeed, after the question paper was attached to the forum thread, it wasn't put on your own website for about a week.
17. (Original post by constellarknight)
Yes, you've been rather slow with IYGB stuff recently: indeed, after the question paper was attached to the forum thread, it wasn't put on your own website for about a week.
18. What is that image supposed to be of?
19. Just out of interest, roughly what level of question is this? I'm studying A level maths independently, with a view to sitting all 6 exams (C1-4, S1, D1) this year. I'm a little behind schedule, just about to get going on C3/4 for the first time, but was still fairly confident I've got enough time (especially after recently discovering examsolutions.net). I'm hoping this question is way above C3/4 level, as if not I'm pretty sure I'm in more trouble than I may have realised...
20. It is indeed above ordinary C3/C4 level: the IYGB SP papers are for exceptional people. Hence why despite me having got an A* in A-Level Maths when I was 10 and being a member of MENSA, I still couldn't do this question - although obviously I have been able to do most of the other ones from IYGB SPs.

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