# Using trig to find solutions to a quintic

1. Feb 22, 2012

### conorordan

1. The problem statement, all variables and given/known data

Given that

$cos5θ=16cos^{5}θ-20cos^{3}θ+5cosθ$

Find the 3 solutions to

$16x^{5}-20x^{3}+5x+1=0$

2. Relevant equations

3. The attempt at a solution

I have let $x=cosθ$ and $θ=cos^{-1}x$
and I know that $cos5θ=-1$

2. Feb 22, 2012

### tiny-tim

hi conorordan!
that's right!

so what are the 3 solutions to cos5θ = -1 ?

3. Feb 22, 2012

### conorordan

Well there are infinite solutions to it, how am I to know which 3 will satisfy the quintic above?

4. Feb 22, 2012

### tiny-tim

agreed

but don't lose the plot

you're only interested in cosθ (=x), and there aren't infinitely many values of that!

5. Feb 22, 2012

### conorordan

Are you suggesting I plot graphs of cos x and the horrific quintic above? It must be much simpler, this is only a 4 mark question!

6. Feb 22, 2012

### tiny-tim

no

write on one line the infinitely many solutions for θ

write on the next line the values of cosθ for those values of θ

on the next line, cross out any duplicates …

how many are left?

7. Feb 22, 2012

### conorordan

Okay I see now, so x=cosθ for π/5, 5π/3 and π will give me my 3 solutions?

8. Feb 22, 2012

### tiny-tim

(you mean 3π/5 )

yup!

9. Feb 22, 2012

### conorordan

Indeed! Thanks for the help!