anonymous
  • anonymous
is there a method for proving the following without calculus or arguing by a diagram?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
\[\forall x \ge 0 \text{ }\sin(x) \le x \]
anonymous
  • anonymous
@Callisto
hba
  • 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.

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amistre64
  • amistre64
yelling, shouting, and just being overall obnoxius tend to be used by kids today to prove their point :)
hba
  • hba
lol ^
Callisto
  • Callisto
I don't know if this works lol -1
anonymous
  • anonymous
that works for 1=
anonymous
  • anonymous
|dw:1353765596912:dw|
anonymous
  • anonymous
yeah see that's what i meant by arguing from a diagram
anonymous
  • anonymous
maybe using the cosine rule we can convert the diagram proof into something more formal
amistre64
  • amistre64
arent most things proved by being vague and nondescript? or is it just my number theory text that does that?
anonymous
  • anonymous
haha
anonymous
  • anonymous
|dw:1353765811446:dw|
anonymous
  • anonymous
hmm are you using the area of a sector there? doesn't that require calculus?
anonymous
  • anonymous
Yes it doesn't. It just uses compare between to area. And first one compare two line and curve.
asnaseer
  • asnaseer
|dw:1353766410319:dw|
asnaseer
  • asnaseer
from that diagram you can see that: sin(x) = b/c
asnaseer
  • asnaseer
also, from arc length, we know: cx = s (assuming x is measured in radians)
asnaseer
  • asnaseer
so x = s/c
asnaseer
  • asnaseer
but s > d
asnaseer
  • asnaseer
and d > b
asnaseer
  • asnaseer
therefore sin(x) <= x
asnaseer
  • asnaseer
I am starting the proof off from the basic definition of what sin(x) represents. so you need a diagram.
asnaseer
  • asnaseer
@eigenschmeigen - what "methods" are you looking for if no diagrams can be used?
hba
  • hba
lol
hba
  • hba
He doesn't want geometry or Calculus.
UnkleRhaukus
  • 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
anonymous
  • 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
anonymous
  • anonymous
when i stated it without proof my tutor wrote "why?" next to it which means i should have given some form of proof
asnaseer
  • asnaseer
aren't you contradicting yourself here - the power series is derived using calculus and you stated "no calculus allowed"?
anonymous
  • 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.
anonymous
  • anonymous
But power series related to advance calcules!
anonymous
  • anonymous
x-sinx=x3/3!-x5/5!+-...>0 for all x>0
anonymous
  • anonymous
http://www.maths.ox.ac.uk/system/files/coursematerial/2012/2644/22/12MT-AnalysisI-extsyn13.pdf
anonymous
  • 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.
anonymous
  • 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.
anonymous
  • anonymous
its given as sinz but thats the general case of sinx...
anonymous
  • anonymous
im fed up. dont worry ill figure it out
anonymous
  • 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*, . . . " ?
UnkleRhaukus
  • UnkleRhaukus
whoops,and yeah @CliffSedge , its kinda hard when you dont draw it
anonymous
  • anonymous
Yeah, why wouldn't someone want a diagram?
anonymous
  • anonymous
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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