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Trigonometric sequence - Find Un out of Un+1 - hard

  1. Sep 18, 2007 #1

    JPC

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

    Un a sequence

    0 < x < pi/2

    U0 = 2cosx

    Un+1 = quroot( 2 + Un)

    1) Calculate U1, U2, and U3 in function of x (simplify maximum)

    2) Show that Un = 2cos( x / 2^n)


    2. Relevant equations



    3. The attempt at a solution

    1)
    U1 = sqroot( 2 + 2cosx) = sqroot( 2 (1 + cosx))
    = ?? (now , how do i simplify)

    because , i should have when i have simplified a maximum :

    U1 = 2cos(x/2)
    = 2 (2cosx² - 1)
    = 4cosx² - 2

    So how do i go from sqroot( 2 + 2cosx) to 4cosx² - 2 ??
     
  2. jcsd
  3. Sep 18, 2007 #2
    Use [tex]\cos 2x = \cos^2 x - \sin^2 x[/tex]
     
  4. Sep 18, 2007 #3

    JPC

    User Avatar

    oh ...

    i realised i did a mistake here :

    this is wrong :
    "U1 = 2cos(x/2)
    = 2 (2cosx² - 1)
    = 4cosx² - 2"

    i cant believe , i confused cos(0.5x) and cos(2x)

    ////

    now that i realised i did it like the other way round :

    U1 = root( 2 + 2cos(x))
    = root(2 + 2(2cos(0.5x)² - 1))
    = root(4cos(0.5x)²)
    = 2cos(0.5x)

    Ok , managed to to do all the rest now , thanks
     
    Last edited: Sep 18, 2007
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