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Trig Functions Analysis Proof

  1. May 24, 2013 #1
    1. The problem statement, all variables and given/known data

    This isn't really a homework question, just working through Rudin and got caught up on something. [itex]C(x)[/itex] and [itex]S(x)[/itex] refer to [itex]cos(x)[/itex] and [itex]sin(x)[/itex] respectively.

    Here is the section in question:


    2. Relevant equations

    3. The attempt at a solution

    Well the part I'm having trouble understanding is the claim: "Hence, if [itex]0≤x≤y[/itex], we have [itex]S(x)(y-x)<\int^{y}_{x}{S(t)}dt = C(x)-C(y)≤2[/itex]"

    In particular, the inequality [itex]S(x)(y-x)<\int^{y}_{x}{S(t)}dt[/itex] is not clear to me. I reviewed a number of integration theorems but couldn't come up with anything that states this. Any help understanding how this inequality is derived would be much appreciated!

    EDIT: OK, this might be really obvious. Is this simply true by the definition of the Riemann integral?
    Last edited: May 24, 2013
  2. jcsd
  3. May 25, 2013 #2


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    It follows from the observation that S(t) is strictly increasing on the interval, so within the interval S(x) < S(t). Then integrate both sides over the interval.
  4. May 25, 2013 #3
    Ah, of course. Got it. Thanks!
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