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Math Help - trignometric equations help

  1. #1
    Super Member
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    trignometric equations help

    (b) The diagram shows part of the curve with equation y=cos(px−q) , where p and q are positive constants and q<180. The curve cuts the x-axis at points A, B and C, as shown.
    Given that the coordinates of A and B are (100, 0) and (220, 0) respectively:
    (i) Write down the coordinates of C.
    (ii) Find the value of p and the value of q.
    I have guessed that the coordinates of C is (340,0), which is correct, but is there any 'formal' way of working out the coordinates of C?

    Also can someone show me how to work out the vales of p and q?

    thank you.
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  2. #2
    MHF Contributor
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    Hi

    \cos (px - q) = 0 \implies px - q = 90 + 180 k

    A, B and C are consecutive zeros of cos(px-q) therefore

    px_A - q = 90 + 180 k

    px_B - q = 90 + 180 (k+1)

    px_C - q = 90 + 180 (k+2)

    Subtracting gives
    p(x_B - x_A) = 180
    p(x_C - x_B) = 180

    Then p(x_C - x_B) = p(x_B - x_A) or x_C = 2x_B - x_A = 340

    And p = \frac{180}{x_B - x_A} = \frac32
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