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Math Help - Solve cos x = x with a calculator

  1. #1
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    Solve cos x = x with a calculator

    Why is it that we can start with a number (say, 1), and press cos x repeatedly, we can get the solution of cos x = x?

    e.g. cos (1) = 0.54
    cos (0.54) = 0.85
    cos (0.85) = 0.65
    cos (0.65) = 0.79
    cos (0.79) = 0.71
    cos (0.71) = 0.76
    cos (0.76) = 0.72
    .....

    until it converges to 0.73
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  2. #2
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    Quote Originally Posted by chopet View Post
    Why is it that we can start with a number (say, 1), and press cos x repeatedly, we can get the solution of cos x = x?

    e.g. cos (1) = 0.54
    cos (0.54) = 0.85
    cos (0.85) = 0.65
    cos (0.65) = 0.79
    cos (0.79) = 0.71
    cos (0.71) = 0.76
    cos (0.76) = 0.72
    .....

    until it converges to 0.73
    Reading this will get you started:

    One Dimensional Dynamical Systems

    Emergence of Chaos from Interactive Mathematics Miscellany and Puzzles
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  3. #3
    GAMMA Mathematics
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    Did you know that you can do the same with sine?

    sin(0)=0
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  4. #4
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    Quote Originally Posted by colby2152 View Post
    Did you know that you can do the same with sine?

    sin(0)=0
    so I realised.
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  5. #5
    GAMMA Mathematics
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    Quote Originally Posted by chopet View Post
    so I realised.
    Well, this is why the trig functions are so beautiful.
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  6. #6
    Super Member wingless's Avatar
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    This is called Picard's Method (to find roots in R).

    It can also be applied to other functions.

    x = e^x-1

    If you start with a positive x value, it'll diverge. But a negative initial value will diverge to zero. You can also change the function to x = \ln (x+1) to make it diverge for positive x values.
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  7. #7
    Super Member angel.white's Avatar
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    Quote Originally Posted by colby2152 View Post
    Did you know that you can do the same with sine?

    sin(0)=0
    That amused me!
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