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Sine function linear?

  1. May 17, 2012 #1
    Sine function linear??

    I have a problem concerning trigonometry and calculus but I just need to know my question's answer to solve it.
    I would like to know: can a sine function be construed as a linear function in a very small domain i.e increments of 0.0001??
    Thank you so much in advance and I appreciate all your help :D
     
  2. jcsd
  3. May 17, 2012 #2
    Re: Sine function linear??

    If sine was linear, then ##\sin(x+y) = \sin x + \sin y##, but
    $$
    \sin (x+y) = \sin x\cos y + \sin y\cos x
    $$
    So that would only occur if ##\cos y = \cos x##.

    Therefore, ##y = 2\pi k## and ##x = 2\pi n## where ##k,n\in\mathbb{Z}##.

    The case when ##\sin (x+y) = 0## would be linear.
     
  4. May 17, 2012 #3

    SammyS

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    Re: Sine function linear??

    Depending upon the details of whatever situation is being addressed, it would generally be acceptable to treat the sine function as being linear over a span of 0.0001 of a single period of the sine function.
     
  5. May 17, 2012 #4
    Re: Sine function linear??

    sin(x) is approximately equal to x for "small x," but I don't think this is what you were asking.

    I think your intuition is correct in that most continuous functions can be well approximated and built by small linear increments. This is basically the idea behind Euler's method. http://en.wikipedia.org/wiki/Euler_method#Informal_geometrical_description
     
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