Esther Tan

Posted on Wednesday, 05 November, 2003 - 09:33 am:

Prove that

\tan (π / 15)

is a root of the equation

t^{4} - 6 \sqrt{3} t^{3} + 8 t^{2} + 2 \sqrt{3} t - 1 = 0

where

t = \tan θ

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 5 x

in terms of

\tan x

. We know

\tan (5 π / 15) = \sqrt{3}

and there is also one obvious factor (

\tan x + \sqrt{3}

) coming from

\tan (5 \times 2 π / 3) = \tan (π / 3)

, so the other factor of the numerator must have the root

\tan (π / 15)

, which comes out as the polynomial in question.

Kerwin