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eigenschmeigen

is there a method for proving the following without calculus or arguing by a diagram?

  • one year ago
  • one year ago

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  1. eigenschmeigen
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    \[\forall x \ge 0 \text{ }\sin(x) \le x \]

    • one year ago
  2. eigenschmeigen
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    @Callisto

    • one year ago
  3. hba
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    Actually calculus is a very powerful tool to solve problems.So Most of the problems can be solved my Calculus,Therfore,they maybe other ways to solve questions without the use of calculus.

    • one year ago
  4. amistre64
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    yelling, shouting, and just being overall obnoxius tend to be used by kids today to prove their point :)

    • one year ago
  5. hba
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    lol ^

    • one year ago
  6. Callisto
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    I don't know if this works lol -1<sinx <1 -x < xsinx < x xsinx < x <- see this?!

    • one year ago
  7. eigenschmeigen
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    that works for 1=<x but i dont think it helps particularly for the harder case 0=<x=<1

    • one year ago
  8. mahmit2012
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    |dw:1353765596912:dw|

    • one year ago
  9. eigenschmeigen
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    yeah see that's what i meant by arguing from a diagram

    • one year ago
  10. eigenschmeigen
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    maybe using the cosine rule we can convert the diagram proof into something more formal

    • one year ago
  11. amistre64
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    arent most things proved by being vague and nondescript? or is it just my number theory text that does that?

    • one year ago
  12. eigenschmeigen
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    haha

    • one year ago
  13. mahmit2012
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    |dw:1353765811446:dw|

    • one year ago
  14. eigenschmeigen
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    hmm are you using the area of a sector there? doesn't that require calculus?

    • one year ago
  15. mahmit2012
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    Yes it doesn't. It just uses compare between to area. And first one compare two line and curve.

    • one year ago
  16. asnaseer
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    |dw:1353766410319:dw|

    • one year ago
  17. asnaseer
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    from that diagram you can see that: sin(x) = b/c

    • one year ago
  18. asnaseer
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    also, from arc length, we know: cx = s (assuming x is measured in radians)

    • one year ago
  19. asnaseer
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    so x = s/c

    • one year ago
  20. asnaseer
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    but s > d

    • one year ago
  21. asnaseer
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    and d > b

    • one year ago
  22. asnaseer
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    therefore sin(x) <= x

    • one year ago
  23. asnaseer
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    I am starting the proof off from the basic definition of what sin(x) represents. so you need a diagram.

    • one year ago
  24. asnaseer
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    @eigenschmeigen - what "methods" are you looking for if no diagrams can be used?

    • one year ago
  25. hba
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    lol

    • one year ago
  26. hba
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    He doesn't want geometry or Calculus.

    • one year ago
  27. UnkleRhaukus
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    the sine of an angle in a triangle (with the opposite side equal to one,) is the length of the hypotenuse, the hypotenuse is the longest side of a triangle the angle \(x\) is a measure of the arc, the arc is longer than the opposite side

    • one year ago
  28. eigenschmeigen
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    my situation is that im using this fact as part of a solution to a problem on my analysis sheet, last time i thought i could state it without giveing a proof (at the top of the sheet it said we may assume basic properties of trig functions) in the course we have only defined sinx in terms of its power series. should i just show that for 0<x<1 the power series converges to a limit less than x?

    • one year ago
  29. eigenschmeigen
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    when i stated it without proof my tutor wrote "why?" next to it which means i should have given some form of proof

    • one year ago
  30. asnaseer
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    aren't you contradicting yourself here - the power series is derived using calculus and you stated "no calculus allowed"?

    • one year ago
  31. eigenschmeigen
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    no, in our course we are _defining_ the functions sin, cos, e^x as their power series, we are not deriving them through the maclauren.

    • one year ago
  32. mahmit2012
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    But power series related to advance calcules!

    • one year ago
  33. mahmit2012
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    x-sinx=x3/3!-x5/5!+-...>0 for all x>0

    • one year ago
  34. eigenschmeigen
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    http://www.maths.ox.ac.uk/system/files/coursematerial/2012/2644/22/12MT-AnalysisI-extsyn13.pdf

    • one year ago
  35. eigenschmeigen
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    of course i know its related. the way we are doing it (which is perfectly valid) is by defining the functions as the power series and deriving properties from there.

    • one year ago
  36. mahmit2012
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    The link you have already suggested was about complex function, and we know there is no comparing in complex number. And also sinZ is not bounded.

    • one year ago
  37. eigenschmeigen
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    its given as sinz but thats the general case of sinx...

    • one year ago
  38. eigenschmeigen
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    im fed up. dont worry ill figure it out

    • one year ago
  39. CliffSedge
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    @UnkleRhaukus , did you mean to say, "the sine of an angle in a triangle (with the *hypotenuse* equal to one,) is the length of the *opposite side*, . . . " ?

    • one year ago
  40. UnkleRhaukus
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    whoops,and yeah @CliffSedge , its kinda hard when you dont draw it

    • one year ago
  41. CliffSedge
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    Yeah, why wouldn't someone want a diagram?

    • one year ago
  42. eigenschmeigen
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    you cant argue from a diagram in formal mathematics. i cant exactly draw a diagram and hand it in to my analysis tutor

    • one year ago
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