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Math Help - trig wave equations

  1. #1
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    trig wave equations

    well im having quite alot of trouble doing questions like these could someone give me some insight into how they should be done thank you!

    A function f(x) is known to be of the form F(x)=Acos(px+q) + B. It has a local maximum at (6, 8) and the next local minimum is at (10, 2). Find the constants A, p, q and B and hence give the formula for the function.
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  2. #2
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    Hello killer786

    Welcome to Math Help Forum!
    Quote Originally Posted by killer786 View Post
    well im having quite alot of trouble doing questions like these could someone give me some insight into how they should be done thank you!

    A function f(x) is known to be of the form F(x)=Acos(px+q) + B. It has a local maximum at (6, 8) and the next local minimum is at (10, 2). Find the constants A, p, q and B and hence give the formula for the function.
    The function \cos(px+q) has a maximum value +1 when px +q = 0, 2\pi, 4\pi, ... and a minimum value -1 when px+q = \pi, 3\pi, 5\pi, ... Since there are multiple maxima and minima, there will be multiple values of p and q. Below, I've taken the first (and most obvious) values.

    So f(x) varies between B+A and B-A. Since these maximum and minimum values are 8 and 2 respectively, we get:
    B+A = 8

    B-A = 2

    \Rightarrow B = 5, A = 3
    The maximum occurs when x =6. So, from what I said above, the simplest value is:
    6p+q = 0

    \Rightarrow q = -6p
    (1)
    The minimum is when x = 10. So:
    10p+q = \pi

    \Rightarrow 4p = \pi, from
    (1)

    \Rightarrow p = \frac{\pi}{4} and q = -\frac{3\pi}{2}
    So the formula we want is:
    f(x) = 3\cos\left(\frac{\pi}{4}x -\frac{3\pi}{2}\right) + 5
    = 3\cos\frac{\pi}{4}\Big(x-6\Big) + 5
    Grandad
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