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eigenschmeigen
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is there a method for proving the following without calculus or arguing by a diagram?
 one year ago
 one year ago
eigenschmeigen Group Title
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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eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
\[\forall x \ge 0 \text{ }\sin(x) \le x \]
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
@Callisto
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.0
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

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
yelling, shouting, and just being overall obnoxius tend to be used by kids today to prove their point :)
 one year ago

Callisto Group TitleBest ResponseYou've already chosen the best response.0
I don't know if this works lol 1<sinx <1 x < xsinx < x xsinx < x < see this?!
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
that works for 1=<x but i dont think it helps particularly for the harder case 0=<x=<1
 one year ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.0
dw:1353765596912:dw
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
yeah see that's what i meant by arguing from a diagram
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
maybe using the cosine rule we can convert the diagram proof into something more formal
 one year ago

amistre64 Group TitleBest ResponseYou've already chosen the best response.0
arent most things proved by being vague and nondescript? or is it just my number theory text that does that?
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
haha
 one year ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.0
dw:1353765811446:dw
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
hmm are you using the area of a sector there? doesn't that require calculus?
 one year ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.0
Yes it doesn't. It just uses compare between to area. And first one compare two line and curve.
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
dw:1353766410319:dw
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
from that diagram you can see that: sin(x) = b/c
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
also, from arc length, we know: cx = s (assuming x is measured in radians)
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
so x = s/c
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
but s > d
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
and d > b
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
therefore sin(x) <= x
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
I am starting the proof off from the basic definition of what sin(x) represents. so you need a diagram.
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
@eigenschmeigen  what "methods" are you looking for if no diagrams can be used?
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.0
He doesn't want geometry or Calculus.
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
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

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
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

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
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

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
aren't you contradicting yourself here  the power series is derived using calculus and you stated "no calculus allowed"?
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
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

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.0
But power series related to advance calcules!
 one year ago

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.0
xsinx=x3/3!x5/5!+...>0 for all x>0
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
http://www.maths.ox.ac.uk/system/files/coursematerial/2012/2644/22/12MTAnalysisIextsyn13.pdf
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
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

mahmit2012 Group TitleBest ResponseYou've already chosen the best response.0
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

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
its given as sinz but thats the general case of sinx...
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
im fed up. dont worry ill figure it out
 one year ago

CliffSedge Group TitleBest ResponseYou've already chosen the best response.0
@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

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
whoops,and yeah @CliffSedge , its kinda hard when you dont draw it
 one year ago

CliffSedge Group TitleBest ResponseYou've already chosen the best response.0
Yeah, why wouldn't someone want a diagram?
 one year ago

eigenschmeigen Group TitleBest ResponseYou've already chosen the best response.0
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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