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I Sin x < x

  1. Nov 6, 2017 #1
    Given that 0 < sin x < x is true for 0 < x < π/2.
    From the above, can we conclude that 0 < sin (x/2) < x/2? How about 0 < sin (x/5) < x/5? Why?
    How about 0<sin 3x < 3x ? Why?
     
  2. jcsd
  3. Nov 6, 2017 #2

    mfb

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    For suitable ranges of x, sure.

    If you replace x by x/2 everywhere consistently, you don't change anything, you just replaced your variable. An analogy would be to replace all "x" by "y".

    0 < sin y < y is true for 0 < y < π/2
    Now define y=x/2.
     
  4. Nov 6, 2017 #3
    Thank you very much for the explanation.
     
  5. Nov 6, 2017 #4

    PeroK

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    Yes, but you also have to change your range:

    ##0 < \sin x < x ## for ##0 < x < \pi/2##

    Is equivalent to:

    ##0 < \sin x/2 < x/2 ## for ##0 < x < \pi##
     
  6. Nov 6, 2017 #5
    :ok::thumbup:
     
  7. Nov 11, 2017 #6

    WWGD

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    Essentially, as long as 0<x/2< ##\pi/2## (although this is not an iff condition) , same for 3x; you want 3x to fall within an interval where the property holds. This is essentially a change of variable.
     
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