I was taught the following proof in high school.
By constructing triangles with 0<θ<π/2 and a circle with radius r and by comparing the areas, we have
12r2sinθcosθ≤12r2θ≤12r2tanθ
Hence
cosθ≤θsinθ≤1cosθ
Then by squeeze theorem, we have the result.
My question is, the middle term in the above inequality comes from the fact that the area of the circle is πr2, which in my textbooks, is later proved by integration. But the integration requires results in calculus which comes from the fact that
lim
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