## eigenschmeigen Group Title is there a method for proving the following without calculus or arguing by a diagram? one year ago one year ago

1. eigenschmeigen Group Title

$\forall x \ge 0 \text{ }\sin(x) \le x$

2. eigenschmeigen Group Title

@Callisto

3. hba Group Title

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 Group Title

yelling, shouting, and just being overall obnoxius tend to be used by kids today to prove their point :)

5. hba Group Title

lol ^

6. Callisto Group Title

I don't know if this works lol -1<sinx <1 -x < xsinx < x xsinx < x <- see this?!

7. eigenschmeigen Group Title

that works for 1=<x but i dont think it helps particularly for the harder case 0=<x=<1

8. mahmit2012 Group Title

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9. eigenschmeigen Group Title

yeah see that's what i meant by arguing from a diagram

10. eigenschmeigen Group Title

maybe using the cosine rule we can convert the diagram proof into something more formal

11. amistre64 Group Title

arent most things proved by being vague and nondescript? or is it just my number theory text that does that?

12. eigenschmeigen Group Title

haha

13. mahmit2012 Group Title

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14. eigenschmeigen Group Title

hmm are you using the area of a sector there? doesn't that require calculus?

15. mahmit2012 Group Title

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

16. asnaseer Group Title

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17. asnaseer Group Title

from that diagram you can see that: sin(x) = b/c

18. asnaseer Group Title

also, from arc length, we know: cx = s (assuming x is measured in radians)

19. asnaseer Group Title

so x = s/c

20. asnaseer Group Title

but s > d

21. asnaseer Group Title

and d > b

22. asnaseer Group Title

therefore sin(x) <= x

23. asnaseer Group Title

I am starting the proof off from the basic definition of what sin(x) represents. so you need a diagram.

24. asnaseer Group Title

@eigenschmeigen - what "methods" are you looking for if no diagrams can be used?

25. hba Group Title

lol

26. hba Group Title

He doesn't want geometry or Calculus.

27. UnkleRhaukus Group Title

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 Group Title

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 Group Title

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 Group Title

aren't you contradicting yourself here - the power series is derived using calculus and you stated "no calculus allowed"?

31. eigenschmeigen Group Title

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 Group Title

But power series related to advance calcules!

33. mahmit2012 Group Title

x-sinx=x3/3!-x5/5!+-...>0 for all x>0

34. eigenschmeigen Group Title
35. eigenschmeigen Group Title

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 Group Title

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 Group Title

its given as sinz but thats the general case of sinx...

38. eigenschmeigen Group Title

im fed up. dont worry ill figure it out

39. CliffSedge Group Title

@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 Group Title

whoops,and yeah @CliffSedge , its kinda hard when you dont draw it

41. CliffSedge Group Title

Yeah, why wouldn't someone want a diagram?

42. eigenschmeigen Group Title

you cant argue from a diagram in formal mathematics. i cant exactly draw a diagram and hand it in to my analysis tutor