Prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of this equation.

  • #1

Homework Statement



When cos(4θ)=cos(3θ). prove that θ=0, 2∏/7, 4∏/7, 6∏/7

Hence prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of 8x3+4x2-4x-1

Homework Equations





The Attempt at a Solution



I can do the first part, but i have some difficulty in solving the second part. For the second part, I start by letting x= cosθ and try to solve the equation, however, i notice that i couldn't simplify the equation... so it is correct to let x=cosθ? How is the second part related to the first part? Thanks in advance.
 

Answers and Replies

  • #2
Dick
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Homework Statement



When cos(4θ)=cos(3θ). prove that θ=0, 2∏/7, 4∏/7, 6∏/7

Hence prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of 8x3+4x2-4x-1

Homework Equations





The Attempt at a Solution



I can do the first part, but i have some difficulty in solving the second part. For the second part, I start by letting x= cosθ and try to solve the equation, however, i notice that i couldn't simplify the equation... so it is correct to let x=cosθ? How is the second part related to the first part? Thanks in advance.
Look at the equation cos(4θ)-cos(3θ)=0 and expand the multiple angles in terms of cos(θ). Yes, put x=cos(θ). Then try to factor it.
 
  • #3
haruspex
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so it is correct to let x=cosθ?
Yes. Do you know how to expand cos(a+b)?
 

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