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Sinx, and tanx

  1. Jan 13, 2008 #1
    i am trying to figure out 2 quesitons below:

    (a) sin x < x, for all x > 0;
    (b) x < tan x, for all x 2 (0;Pi/2).

    at the first, i solved by the graph, but it seemed too tedious.

    can someone tell me how to solve them in kinda simple ways

    thx
     
  2. jcsd
  3. Jan 13, 2008 #2

    HallsofIvy

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    It's not clear what the question is! If you want to prove "sin x< x for all x> 0", you can't- it's not true. If you want to determine for what values of x, x>0 sin x< x, then you first look at the equation sin(x)= x since those points will separate "<" from ">".

    It's not all that tedious to graph y= sin(x)- you should know its shape already. And y= x is certainly not difficult to graph. Unfortunately, since both these equations involve a transcendental function (sin(x) and tan(x)) with a linear function (x), there is no "kinda simple" way to solve them. I would think that graphing on a graphing calculator would be the simples. Another way would be to use the "Newton-Raphson" numerical method.
     
  4. Jan 13, 2008 #3

    mathman

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    You can use elementary calculus.

    For (a) let f(x)=x-sinx, then f'(x)=1-cosx which is always non-negative. Therefore x-sinx is never decreasing and starts out =0 at x=0, increasing immediately.

    For(b) let f(x)=tanx-x, then f'(x)=sec2x-1, which is also non-negative. Same argument as for (a).
     
  5. Jan 13, 2008 #4

    Integral

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    Since the OP did not state what class he is taking I am going to assume that this is a precalc problem.
     
  6. Jan 13, 2008 #5

    Gib Z

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    [​IMG]

    I've uploaded a scanned extract of page 48 from Courant, Volume 1. As it already shows you from the diagram on the page, if [tex] 0 < x < \pi/2[/tex], then [tex] 1< \frac{x}{\sin x} < \frac{1}{\cos x}[/tex].

    Since we are dealing with all positive quantities here, finding the reciprocal changes each inequality sign: [tex] 1 > \frac{\sin x}{x} > \cos x[/tex]. Im not going to go any further because I have some other work to do.

    In fact I think I've already given you too much! You can do the rest =]
     
  7. Jan 14, 2008 #6

    HallsofIvy

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    The problem I have is that the OP said
    but there are no questions there!
     
  8. Jan 14, 2008 #7

    Gib Z

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    I'm sure he just forgot the critical word, "show" or "prove" etc etc lol
     
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