이등변삼각형 넓이의 극한

원과 선분AC의 접점을 P라 하면 $\displaystyle \tan \frac{\theta}{2} = \frac{1}{CP}$ 에서 $\displaystyle CP = \frac{1}{\tan \frac{\theta}{2}}$ 이므로 $\displaystyle \cos \theta = \frac{CP}{BC} = \frac{\frac{1}{\tan \frac{\theta}{2}}}{BC}$ $\displaystyle \therefore BC = \frac{1}{\cos \theta \tan \frac{\theta}{2}}$ 점C에서 선분AD의 연장선에 내린 수선의 발을 H라 하면 $\displaystyle CH = AC \sin \theta = 2 CP \sin \theta = \frac{2 \sin \theta}{\tan \frac{\theta}{2}}$ 이므로 $\displaystyle \sin 3 \theta = \frac{CH}{CD}$ 에서 $\displaystyle CD = \frac{CH}{\sin 3 \theta} = \frac{2 \sin \theta}{\sin 3 \theta \tan \frac{\theta}{2}}$ $\displaystyle \therefore S ( \theta ) = \frac{1}{2} \times BC \times CD \times \sin \theta$ $\displaystyle = \frac{1}{2} \times \frac{1}{\cos \theta \tan \frac{\theta}{2}} \times \frac{2 \sin \theta}{\sin 3 \theta \tan \frac{\theta}{2}} \times \sin \theta$ $\displaystyle = \frac{\sin ^{2} \theta}{\sin 3 \theta \cos \theta \tan ^{2} \frac{\theta}{2}}$ $\displaystyle \therefore \lim\limits _{\theta \rightarrow 0 +} {\left\{ \theta \times S ( \theta ) \right\}}$ $\displaystyle = \lim\limits _{\theta \rightarrow 0 +} {\frac{\theta \sin ^{2} \theta}{\sin 3 \theta \cos \theta \tan ^{2} \frac{\theta}{2}}}$ $\displaystyle = \lim\limits _{\theta \rightarrow 0 +} {\left\{ \frac{3 \theta}{\sin 3 \theta} \times \frac{\sin ^{2} \theta}{\theta ^{2}} \times \frac{1}{\cos \theta} \times \left( \frac{\frac{\theta}{2}}{\tan \frac{\theta}{2}} \right) ^{2} \times \frac{4}{3} \right\}}$ $\displaystyle = \frac{4}{3}$