Math

Given that f(x) = 2sinx - cosx

- Write f(x) in the form Asin(x+a)

- Find the maximum value of 2sinx - cosx

- Find the lowest positive value of p such that f(x) = f(x + p)

Thankyou. All help is appreciated. I'm really struggling on this assignment.

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  1. how about expanding Asin(x+a)
    = Asinxcosa + Acosxsina

    so Asinxcosa + Acosxsina = 2sinx - cosx

    This must now be an identity, that is, it must be true for all values of x
    (I will work in degrees, if you need your answers in radians either convert these values or repeat the following steps with your calculator set to radians)

    let x = 0
    Asin0cosa + Acos0sina = 2sin0 - cos0
    0 + Asina = 0 - 1
    Asina = -1 (#1)

    let x=90º
    then Asin90cosa + Acos90sina = 2sin90 - cos90
    Acosa + 0 = 2 - 0
    Acosa = 2 (#2)

    divide equation #1 by #2 to get
    sina/cosa = -1/2
    tana = -1/2
    so a = 153.4º or a = -26.6º,...there are more

    using a=153.4 back in #1 gives us A=-2.236
    using a=-26.6 back in #1 gives us
    A=+2.236

    so we could write
    f(x) = -2.236sin(x+153.4º) or
    f(x) = 2.236sin(x-26.6º)

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    posted by Reiny
  2. We could just trust my answer to the first part, and simply say that since │A│ is the amplitude of our function we found, so the maximum value must be 2.236

    or

    we could Calculus.

    the f'(x) = 2cosx + sinx
    = 0 for a max/min of f(x)

    sinx = -2cosx
    sinx/cosx = -2
    tanx = -2
    x = 116.4º or x=296.6º

    going back to our function
    f(116.4) = 2.236 and f(296.6)=-2.236

    confirming our answer to the first part.

    so the max value of f(x) is 2.236

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    posted by Reiny
  3. recall from above that we concluded that

    f(x) = -2.236sin(x+153.4º) or f(x) = 2.236sin(x-26.6º)

    the period of either one is 360º, so a phase shift of 360º either to the left or right would result in the same graph

    But the wanted the smallest possible value in (x+p)

    so if we added 360 to the existing phase shifts, we would only get a value larger than what we already have in

    f(x) = -2.236sin(x+153.4º) so the smallest value of p is 153.4º

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    posted by Reiny
  4. Just wanted to express my gratitude. Thankyou very much for your help.

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    posted by sqleung

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