math

Solve the equation cos3x=cos2x for 0¡Üx¡Ü360

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asked by Sarah
  1. cos3x = cos2x
    we need 3(x+2m*pi) = 2(x+2n*pi) for some m and n. Some playing around will get you

    x = 0,2/5 pi, 4/5 pi
    or, in degrees,
    x = 0,144,288

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    posted by Steve
  2. ?? but the ans is 0, 72 ,144,216,288,360 degree

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    posted by Sarah
  3. In terms of cosx
    cos 3x = 4 cos^3 x - 3cosx , and cos 2x = 2 cos^2 x - 1

    so cos 3x = cos 2x
    4 cos^3 x - 3cosx - 2cos^2 x + 1 = 0
    let cosx = y
    so we need
    4y^3 - 2y^2 - 3y + 1 = 0

    I tried y = 1 and sure enough that was a solution.
    then by synthetic division ...
    4y^3 - 2y^2 - 3y + 1 = 0
    (y-1)((4y^2 + 2y - 1) = 0

    y = 1 or y = .30902 or y = -.80902 , using the formula

    if y = 1, cosx = 1
    x = 0 , 360

    if y = .30902 , cosx = .30902
    x = 72°, 288

    if y = -.80902 , cosx = -.80902
    x = 144° , 216°

    x = 0, 72, 144, 216, 288, and 360°

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    posted by Reiny
  4. Good catch, Reiny - I was a bit sloppy.

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    posted by Steve

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