## anonymous 3 years ago is there a method for proving the following without calculus or arguing by a diagram?

1. anonymous

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

2. anonymous

@Callisto

3. hba

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

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

5. hba

lol ^

6. Callisto

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

7. anonymous

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

8. anonymous

|dw:1353765596912:dw|

9. anonymous

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

10. anonymous

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

11. amistre64

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

12. anonymous

haha

13. anonymous

|dw:1353765811446:dw|

14. anonymous

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

15. anonymous

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

16. asnaseer

|dw:1353766410319:dw|

17. asnaseer

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

18. asnaseer

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

19. asnaseer

so x = s/c

20. asnaseer

but s > d

21. asnaseer

and d > b

22. asnaseer

therefore sin(x) <= x

23. asnaseer

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

24. asnaseer

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

25. hba

lol

26. hba

He doesn't want geometry or Calculus.

27. UnkleRhaukus

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. anonymous

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. anonymous

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

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

31. anonymous

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. anonymous

But power series related to advance calcules!

33. anonymous

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

34. anonymous
35. anonymous

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. anonymous

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. anonymous

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

38. anonymous

im fed up. dont worry ill figure it out

39. anonymous

@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

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

41. anonymous

Yeah, why wouldn't someone want a diagram?

42. anonymous

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