SOLUTION: How do I find the exact value of {{{y = cos(4pi/5)}}} ?

Let A = 



Then  must be one of the solutions to:



Use identity 



Use identity  on 
Use identity  on 



Use identity  on 
Use identity  on 



Simplify:









Factor sin²(A) out of the last two terms on the left:



Use the identity  on sin²(A)



FOIL out the last term on the left:









Let cos(A)=x





Try a root of 1

1 | 16   0  -20   0   5  -1
  |     16   16  -4  -4   1
    16  16   -4  -4   1   0

So it factors as



We try factoring  as
the product of two quadratics.  In fact we might
expect that it is the square of a quadratic because
the leading term  is a square and
also the constant term 1 is a square.

So we try to see if there is a quadratic (4x²+kx-1)
which when squared gives  

 



Equating coefficients of x³:

8k = 16
  k = 2

Equating coefficients of x²:

k²-8 = -4
   k² = 4
    k = ±2

So far, k=2 works

Equating  coefficients of x

-2k=-4
   k=2

So indeed 



factors as



So we have solutions

x-1 = 0    and   
  x = 1           
                  
                  
                 
                 
                 
                 
                 

So the solutions are

,  , .

Since  is in Q2, its cosine must be negative.  The
only solution which is negative is

.

thus

.

Edwin