Consider this diagram. There are three shapes drawn about the unit circle and the angle POA = x[br][br]1. Area of Triangle OBA = [math]\frac{1}{2}[/math]base x height - now base = 1 and tan(x) = [math]\frac{AB}{OA}[/math] since OA = 1 height = tan(x)[br] Area = [math]\frac{1}{2}[/math]tan(x)[br][br]2. Area of Triangle OPA = [math]\frac{1}{2}[/math]ab sinC - now a and b = 1 (since it is the unit circle)[br] Area = [math]\frac{1}{2}[/math]sin(x)[br][br]3. Area of the sector = the fraction of the circle x the area of the circle.[math]\frac{x}{2\pi}\times\pi r^2[/math]. And since r =1[br] Area = [math]\frac{x}{2}[/math]

The area of the sector is obviously between the areas of the triangles. Therefore[br][br][math]\frac{1}{2}sin\left(x\right)\le\frac{x}{2}\le\frac{1}{2}tan\left(x\right)[/math] We can multiply everything by 2[br][br][math]sin\left(x\right)\le x\le tan\left(x\right)[/math] Now tan (x) is sin (x) divided by cos (x)[br][br][math]sin\left(x\right)\le x\le\frac{sin\left(x\right)}{cos\left(x\right)}[/math] Now as x tends towards zero. cos(x) tends towards 1. So we get[br][br][math]sin\left(x\right)\le x\le sin\left(x\right)[/math] Therefore using this squeeze principle as x tends toward zero sin(x) = x