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How to solve this?

  1. Sep 10, 2005 #1
    Two simple waves but on is a Y function and the other a X function, on do you find the intersecting point?

    y=cos(x)
    x=cos(y)

    cos^-1(x)=cos(x)
     
  2. jcsd
  3. Sep 10, 2005 #2

    VietDao29

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    Homework Helper

    You can use Newton's method to solve this:
    [tex]x_n = \frac{f(x_{n - 1})}{f'(x_{n - 1})}[/tex]
    And the solution x is:
    [tex]x = \lim_{n \rightarrow \infty} x_n[/tex]
    Newton's method
    You can change the equation a bit so it's easier to take the dirivative of the function:
    Since the graph of Arccos(x) is the reflection of the graph Cos(x) across the line y = x.
    So the intersection of the two graph Arccos(x) and Cos(x) is right on the line y = x. So the equation can be changed to:
    [tex]\cos(x) = x \Leftrightarrow \cos(x) - x = 0[/tex]
    Let f(x) = cos(x) - x.
    So f'(x) = -sin(x) - 1.
    Using the formula, you have:
    [tex]x_n = \frac{\cos (x_{n - 1}) - x_{n - 1}}{-\sin (x_{n - 1}) - 1}[/tex]
    Since the two graph cos(x) and x will cut each other at some x lies between 0, and pi. So you just choose [tex]x_0 \in [0, \ \pi][/tex], eg: x_0 = 0.5,...
    Viet Dao,
     
    Last edited: Sep 10, 2005
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