Monday, 25 January 2016

trigonometry - have trouble with this limit question



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a) By considering the areas of the triangle OAD, the sector OAC and the triangle OBC,
show that
(cosθ)(sinθ)<θ<sinθcosθ
I find out:
Area of OAD=12ODADsinθ
Area of OAC=12OC2θ
Area of OBC=12OCBCsinθ
Now I'm stuck at how to apply this to prove
How to prove?




(b) Use (a) and the Squeeze Theorem to show that
limθ0+sinθθ=1


Answer



Hint: WORK IN RADIANS!
a) Area of ΔOAD=12ODADArea of sector OAC=θ360π(OA)2Area of ΔOBC=12OCBC


See that
Area of ΔOAD<Area of sector OAC<Area of ΔOBC12cosθsinθ<θ360π(OA)2<121BC

Now,
DC=1cosθBC=tanθ



b) Then, after doing a), use the fact that
12sinθcosθ<θ/2θ/2<12tanθ


Then use the squeeze theorem. The limit follows.


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