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Question:

pavelpimen
pavelpimen
Student
High School - 10th Grade

Prove the identity

cos (pi/5)=[(sqrt 5) + 1]/4

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Posted by pavelpimen on Thursday October 29, 2009 at 1:28 AM and tagged with identity, science.

Answers:

  1. neela
    neela Teacher
    Graduate School

    eNotes Editor

    We know that  cos [2* (Pi/5)] = -cos [Pi - 2* (Pi/5)] =

    = - cos [3* (Pi/2)]

    Expanding both sides and putting cos(Pi/5) = x we get:

    2x^2-1= -( 4x^3-3x) or

    4x^3+2x^2-3x-1=0

    (x+1)(4x^2-2x-1)=0

    4x^2-2x-1 =0 gives: x= (2+sqrt20)/(2*4) or x= (2-sqrt20)/8 or x=-1,

    But Pi/5 is an acute angle (equal t0 36 degree). So co 36 is a positive angle and x= (2+sqrt20)/8 or (1+sqrt5)/4 only valid.

    So x = (1+sqrt5)/4 or

    cos(Pi/5) = (1+sqrt5)/4

     

     

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    Posted by neela on Thursday October 29, 2009 at 6:31 AM