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=12OD⋅AD⋅sinθ
Area of OAC=12OC2θ
Area of OBC=12OC⋅BC⋅sinθ
Now I'm stuck at how to apply this to prove
How to prove?
(b) Use (a) and the Squeeze Theorem to show that
limθ→0+sinθθ=1
Answer
Hint: WORK IN RADIANS!
a) Area of ΔOAD=12⋅OD⋅ADArea of sector OAC=θ360π(OA)2Area of ΔOBC=12⋅OC⋅BC
See that
Area of ΔOAD<Area of sector OAC<Area of ΔOBC⟹12⋅cosθ⋅sinθ<θ360π(OA)2<12⋅1⋅BC
Now,
DC=1−cosθBC=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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