부채꼴 접선

[출제의도] 삼각함수의 덧셈정리를 활용하여 문제해결하기 $\displaystyle \mathrm{\overline{PC}} = 6 \tan \theta$, $\displaystyle \mathrm{\overline{CQ}} = 6 \tan \left( \frac{\pi}{2} - \theta \right)$ $\displaystyle \mathrm{\overline{PQ}} \mathrm{=} {\overline{PC}} + \mathrm{\overline{CQ}} = 15$ $\displaystyle 6 \tan \left( \frac{\pi}{2} - \theta \right) + 6 \tan \theta = 15$ $\displaystyle \frac{6}{\tan \theta} + 6 \tan \theta = 15$ $2 \tan ^{2} \theta - 5 \tan \theta + 2 = 0$ $\left( 2 \tan \theta - 1 \right) \left( \tan \theta - 2 \right) = 0$ $\displaystyle \tan \theta = \frac{1}{2}$ 또는 $\tan \theta = 2$ $\displaystyle 0 < \theta < \frac{\pi}{4}$이므로 $\displaystyle \tan \theta = \frac{1}{2}$ 따라서 $\displaystyle \tan 2 \theta = \frac{2 \tan \theta}{1 - \tan ^{2} \theta} = \frac{4}{3}$