Writing A Trig Expression as an Algebraic Expression

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

    Write the Trigonometric Expression as an algebraic expression.

    cos(2arccos 2x)

    2. Relevant equations

    Probably the inverse properties, I'm not sure.

    3. The attempt at a solution

    I know I can rewrite this equation as.

    u= arccos 2x

    cos(2cos u=2x)

    I can also say that the adjacent leg is 2x units long and the hypotenuse is 1 unit long. Then using the pythagorean theorm I can figure the opposite leg to be sqrt(1-4x2)

    I'm not sure If this is necessary though can someone point me in the right direction? The 2 in front of the arccos is throwing me off because if that wasn't there I would just use the inverse property and cos(arccos 2x) would equal 2x.
     
  2. jcsd
  3. rock.freak667

    rock.freak667 6,232
    Homework Helper

    If u=cos-1(2x) then you want to find cos(2u).

    cos(2u)=cos2u-sin2u=2cos2u-1 = 1-2sin2u

    and cos2u = (cosu)2
     
  4. I'm not sure I understand why you'd want to find cos(2u)

    The answer is supposed to be 8x2-1 and thats the answer listed in the back of the book.
     
  5. rock.freak667

    rock.freak667 6,232
    Homework Helper

    cos(2cos-1(2x))

    if you put u = cos-1(2x), wouldn't cos(2cos-1(2x)) become cos(2u)?
     
  6. Thanks, now I understand.

    Once you have simplified it to 2cos2u-1 all you have to do is simplify it with the u in place

    cos2(arcsin 2x)=2x

    2(2x)2-1

    8x2-1

    Thanks!
     
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