Esther Tan
Posted on Wednesday, 05 November, 2003 - 09:33 am:
Prove that tan(p/15) is a root of the equation t4-6Ö3t3+8t2+2Ö3t-1=0 where t=tanq

Please Help!!

Dean Hanafy
Posted on Wednesday, 05 November, 2003 - 02:07 pm:
At first glance this looks similar to a STEP question set a few years back.

I think the approach is to use complex numbers(the trigonometric form) to obtain an expression for sin4x, cos4x and obtain tan4x, obtaining an expression in terms of tanx. From there the substitution above should make it look more managable. However, this is only my intuition. I may well be wrong.

Apologies if i haven't been much help!
Deano
Kerwin Hui
Posted on Wednesday, 05 November, 2003 - 05:43 pm:
A slightly easier method is to work out the expression of tan 5x in terms of tan x. We know tan(5p/15)=Ö3 and there is also one obvious factor (tan x+Ö3) coming from tan(5×2p/3) = tan(p/3), so the other factor of the numerator must have the root tan(p/15), which comes out as the polynomial in question.

Kerwin