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eigenschmeigen

  • 2 years ago

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

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

  2. eigenschmeigen
    • 2 years ago
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    @Callisto

  3. hba
    • 2 years ago
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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.

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

  5. hba
    • 2 years ago
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    lol ^

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

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

  8. mahmit2012
    • 2 years ago
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    |dw:1353765596912:dw|

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

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

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

  12. eigenschmeigen
    • 2 years ago
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    haha

  13. mahmit2012
    • 2 years ago
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    |dw:1353765811446:dw|

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

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

  16. asnaseer
    • 2 years ago
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    |dw:1353766410319:dw|

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

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

  19. asnaseer
    • 2 years ago
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    so x = s/c

  20. asnaseer
    • 2 years ago
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    but s > d

  21. asnaseer
    • 2 years ago
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    and d > b

  22. asnaseer
    • 2 years ago
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    therefore sin(x) <= x

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

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

  25. hba
    • 2 years ago
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    lol

  26. hba
    • 2 years ago
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    He doesn't want geometry or Calculus.

  27. UnkleRhaukus
    • 2 years ago
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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

  28. eigenschmeigen
    • 2 years ago
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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?

  29. eigenschmeigen
    • 2 years ago
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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

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

  31. eigenschmeigen
    • 2 years ago
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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.

  32. mahmit2012
    • 2 years ago
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    But power series related to advance calcules!

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

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

  35. eigenschmeigen
    • 2 years ago
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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.

  36. mahmit2012
    • 2 years ago
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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.

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

  38. eigenschmeigen
    • 2 years ago
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    im fed up. dont worry ill figure it out

  39. CliffSedge
    • 2 years ago
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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*, . . . " ?

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

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

  42. eigenschmeigen
    • 2 years ago
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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

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