겹친 정사각형 삼각형 넓이 극한

$\displaystyle \frac{\overline{\mathrm{E}} F}{\overline{\mathrm{K}} E} = \cos \theta$이므로 $\displaystyle {\overline{\mathrm{K}} E} = \frac{1}{\cos \theta}$ ∴ $\displaystyle {\overline{\mathrm{K}} B} = \frac{1}{\cos \theta} - 1$ 또, $\displaystyle \angle \mathrm{B} KL = \frac{\pi}{2} - \theta$에서 $\displaystyle \frac{\overline{\mathrm{B}} L}{\overline{\mathrm{K}} B} = \tan \left( \frac{\pi}{2} - \theta \right) = \cot \theta$ ∴ $\displaystyle \overline{\mathrm{B}} L = \overline{KB} \cdot \cot \theta$ $S ( \theta )$$\displaystyle = \frac{1}{2} \cdot \overline{\mathrm{K}} B \cdot \overline{BL}$$\displaystyle = \frac{1}{2} \left( \frac{1}{\cos \theta} - 1 \right) \times \left( \frac{1}{\cos \theta} - 1 \right) \cdot \cot \theta$ $\displaystyle = \frac{1}{2} \frac{( 1 - \cos \theta ) ^{2}}{\cos ^{2} \theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{( 1 - \cos \theta ) ^{2}}{2 \sin \theta \cos \theta}$ ∴ $\displaystyle \lim\limits _{\theta \rightarrow 0} \frac{S ( \theta )}{\theta ^{3}}$$\displaystyle = \lim\limits _{\theta \rightarrow 0} \frac{{( 1 - \cos \theta ) ^{2}}}{2 \sin \theta \cos \theta \cdot \theta ^{3}}$ $\displaystyle = \lim\limits _{\theta \rightarrow 0} {} \frac{( 1 - \cos \theta ) ^{2} ( 1 + \cos \theta ) ^{2}}{2 \sin \theta \cos \theta \cdot \theta ^{3} ( 1 + \cos \theta ) ^{2}}$ $\displaystyle = \lim\limits _{\theta \rightarrow 0} \frac{\sin ^{4} \theta}{2 \sin \theta \cos \theta \cdot \theta ^{3} ( 1 + \cos \theta ) ^{2}}$ $\displaystyle = \frac{1}{2 \cdot 1 \cdot 2 ^{2}} = \frac{1}{8}$ ∴ $p ^{2} + q ^{2} = 65$