Using trig to find solutions to a quintic

[conorordan]

Homework Statement

Given that

cos5θ=16cos^{5}θ-20cos^{3}θ+5cosθ

Find the 3 solutions to

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

Homework Equations

The Attempt at a Solution

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

 

[tiny-tim]

hi conorordan!

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

that's right!

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

 

[conorordan]

hi conorordan!


that's right!

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

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

 

[tiny-tim]

Well there are infinite solutions to it

agreed

but don't lose the plot …

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

 

[conorordan]

agreed

but don't lose the plot …

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

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!

 

[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? ​

 

[conorordan]

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? ​

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

 

[tiny-tim]

(you mean 3π/5 )

yup!

 

[conorordan]

(you mean 3π/5 )

Indeed! Thanks for the help!