반원 삼각형 넓이 극한

[출제의도] 도형의 성질을 이용하여 삼각함수의 극한에 대한 문제를 해결한다.

두 직각삼각형 $\mathrm{PCE}$와 $\mathrm{ADE}$는 닮음이므로 $\displaystyle \mathrm{\overline{EP}} : {\overline{EA}} = {\overline{EC}} : {\overline{ED}}$에서 $\displaystyle \mathrm{\overline{EP}} \times {\overline{ED}} = {\overline{EA}} \times {\overline{EC}}$ $\displaystyle \mathrm{\angle} DEP = \frac{\pi}{2} + 2 \theta$이므로 $S ( \theta )$$\displaystyle = \frac{\mathrm{1}}{2} \times {\overline{EP}} \times {\overline{ED}} \times \sin \left( \frac{\pi}{2} + 2 \theta \right)$$\displaystyle = \frac{\mathrm{1}}{2} \times {\overline{EA}} \times {\overline{EC}} \times \cos 2 \theta$ 직각삼각형 $\mathrm{APB}$에서 $\displaystyle {\overline{\mathrm{AP}}} = 2 \cos \theta$ 삼각형 $\mathrm{ACP}$에서 $\mathrm{\angle} ACP = \pi - 3 \theta$이므로 사인법칙에 의하여 $\displaystyle \frac{{\overline{\mathrm{AC}}}}{\sin 2 \theta} = \frac{{\overline{\mathrm{AP}}}}{\sin \left( \pi - 3 \theta \right)}$에서 $\displaystyle \mathrm{\overline{AC}} = \frac{2 \sin 2 \theta \cos \theta}{\sin 3 \theta}$ 삼각형 $\mathrm{ACE}$에서 $\displaystyle \mathrm{\angle} ACE = \frac{\pi}{2} - 3 \theta$, $\displaystyle \mathrm{\angle} CEA = \frac{\pi}{2} + 2 \theta$이고 사인법칙에 의하여 $\displaystyle \frac{{\overline{\mathrm{EC}}}}{\sin \theta} = \frac{{\overline{\mathrm{EA}}}}{\sin \left( \frac{\pi}{2} - 3 \theta \right)} = \frac{{\overline{\mathrm{AC}}}}{\sin \left( \frac{\pi}{2} + 2 \theta \right)}$이므로 $\displaystyle {\overline{\mathrm{EC}}} = \frac{{\overline{\mathrm{AC}}} \sin \theta}{\sin \left( \frac{\pi}{2} + 2 \theta \right)} = \frac{2 \sin 2 \theta \sin \theta \cos \theta}{\sin 3 \theta \cos 2 \theta}$ $\displaystyle {\overline{\mathrm{EA}}} = \frac{{\overline{\mathrm{AC}}} \sin \left( \frac{\pi}{2} - 3 \theta \right)}{\sin \left( \frac{\pi}{2} + 2 \theta \right)} = \frac{2 \sin 2 \theta \cos \theta \cos 3 \theta}{\sin 3 \theta \cos 2 \theta}$ $\displaystyle S \left( \theta \right) = \frac{2 \sin ^{2} 2 \theta \sin \theta \cos ^{2} \theta \cos 3 \theta}{\sin ^{2} 3 \theta \cos 2 \theta}$ $\displaystyle \lim\limits _{\theta \rightarrow 0 +} {\frac{S ( \theta )}{\theta}}$$\displaystyle = \mathrm{\lim\limits} _{\theta \rightarrow 0 +} {\frac{2 \sin ^{2} 2 \theta \sin \theta \cos ^{2} \theta \cos 3 \theta}{\theta \sin ^{2} 3 \theta \cos 2 \theta}}$ $\displaystyle = \frac{8}{9} \mathrm{\lim\limits} _{\theta \rightarrow 0 +} {\left( \frac{\sin 2 \theta}{2 \theta} \right) ^{2} \left( \frac{\sin \theta}{\theta} \right) ^{2} \left( \frac{3 \theta}{\sin 3 \theta} \right) ^{2} \left( \frac{\cos ^{2} \theta \cos 3 \theta}{\cos 2 \theta} \right)}$ $\displaystyle = \frac{8}{9}$