Question

( Tan(pi/9) + Tan(5pi/36) ) / ( 1-Tan( 5pi/36) )

Answer 1

I'm sulpposed to find the exact value

Answer 2

Supposed*

Answer 3

Answer: Consider the equation cos (6x) = 1/2.

The solutions are x = +- pi/18 mod pi/3

Using the Tchebychev polynomial you get

32 y^6 - 48 y^4 + 18 y^2 - 1 = 1/2 where y = cos x.

If you set z = 1/cos^2 x you get the equation

3 z^3 - 36 z^2 + 96 z - 64 = 0.

So for w = tan^2x = z - 1 you get

3 w^3 - 27 w^2 + 33w - 1= 0.

Using the relations between roots and coefficients you get that the roots

w_1, w_2, w_3 add up to 9, but these are

{tan(pi/18)]^2}, {tan(5pi/18)]^2}, {tan(7pi/18)]^2}

Hence the sum of the tangents squared is 9.

In the same way you get 59 and 433 for the sums of powers 4 and 6.

If you raise to any power 2k, you get the values of the sequence

defined by u_0 = 3, u_1 = 9, u_2 = 59 and for k > 2,

u{k+1} = 9 u{k} - 11 u{k-1} +u{k-2}/3

Answer 4

that will explain it

Answer 5

medal ............

Answer 6

That makes no sense to me at all...... I give up with this stuff :(

Answer 7

she copy and paste tht pellet

Answer 8

use the trig identity

tan(a + b) = (tana + tan b) / (1 - tan a tan b)

let a = tan pi/9 and b = tan 5pi/36

tan ( a + b) = tan ( 4pi/36 + 5pi/36) = tan pi/4)

Answer 9

tan pi/4 = 1/1

Answer 10

* correction to my first post
a = pi/9 and b = 5pi/36

Answer 11

have you got any questions about the above?

Answer 12

\[\large\rm \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}\]Hmm I don't see how that identity works here :(\[\large\rm \tan\left(\frac{\pi}{9}+\frac{5\pi}{36}\right)=\frac{\tan\left(\frac{\pi}{9}\right)+\tan\left(\frac{5\pi}{36}\right)}{1-\color{red}{\tan\left(\frac{\pi}{9}\right)}\tan\left(\frac{5\pi}{36}\right)}\]Seems like we're missing part of what we need.

Answer 13

I'm hoping it was a typo in the original question though :))
Your steps make perfect sense then hehe.

Answer 14

The answer to this question doesn't need a very long solution.

Firstly: You need to change all the angles to factors of pi/36
here is an example:
\[\frac{ \pi }{9 }=\frac{ 4\pi }{36 }\]
How does the phrase look like by doing this?

Answer 15

@KathCarl
do it and let me know.

Answer 16

@zepdrix - yes lol - i didnt notice there was a tan pi/9 missing!!!