y = tan x k y=\tan\dfrac{x}{k} y = tan k x k π k\pi k π
직선 y = p y=p y = p A A A B B B x x x α \alpha α β \beta β ( α < β ) (\alpha<\beta) ( α < β )
β = α + k π
\beta=\alpha+k\pi
β = α + k π 점 P P P ( 0 , p ) (0,p) ( 0 , p ) P A ‾ = α \overline{PA}=\alpha P A = α A B ‾ = β − α = k π \overline{AB}=\beta-\alpha=k\pi A B = β − α = k π A B ‾ = 3 P A ‾ \overline{AB}=3\overline{PA} A B = 3 P A
k π = 3 α α = k π 3
\begin{aligned}
k\pi&=3\alpha \\
\alpha&=\dfrac{k\pi}{3}
\end{aligned}
k π α = 3 α = 3 k π 점 A A A
p = f ( k π 3 ) = tan ( 1 k ⋅ k π 3 ) = tan π 3 = 3
\begin{aligned}
p&=f\!\left(\dfrac{k\pi}{3}\right) \\
&=\tan\!\left(\dfrac{1}{k}\cdot\dfrac{k\pi}{3}\right) \\
&=\tan\dfrac{\pi}{3} \\
&=\sqrt{3}
\end{aligned}
p = f ( 3 k π ) = tan ( k 1 ⋅ 3 k π ) = tan 3 π = 3 직선 y = − p = − 3 y=-p=-\sqrt{3} y = − p = − 3 C C C x x x γ \gamma γ tan γ k = − 3 \tan\dfrac{\gamma}{k}=-\sqrt{3} tan k γ = − 3 0 ≤ x < 3 k π 2 0\le x<\dfrac{3k\pi}{2} 0 ≤ x < 2 3 k π 0 ≤ x k < 3 π 2 0\le\dfrac{x}{k}<\dfrac{3\pi}{2} 0 ≤ k x < 2 3 π
γ k = 2 π 3 ⇒ γ = 2 k π 3
\dfrac{\gamma}{k}=\dfrac{2\pi}{3}
\quad\Rightarrow\quad
\gamma=\dfrac{2k\pi}{3}
k γ = 3 2 π ⇒ γ = 3 2 k π 또 β = α + k π = 4 k π 3 \beta=\alpha+k\pi=\dfrac{4k\pi}{3} β = α + k π = 3 4 k π B ( 4 k π 3 , 3 ) B\!\left(\dfrac{4k\pi}{3},\ \sqrt{3}\right) B ( 3 4 k π , 3 ) C ( 2 k π 3 , − 3 ) C\!\left(\dfrac{2k\pi}{3},\ -\sqrt{3}\right) C ( 3 2 k π , − 3 )
y = f ( x ) y=f(x) y = f ( x ) ( k π , 0 ) (k\pi,0) ( k π , 0 ) D D D △ O C B \triangle OCB △ O C B △ O B D \triangle OBD △ O B D △ O C D \triangle OCD △ O C D
△ O B C = △ O B D + △ O C D = 1 2 × k π × 3 + 1 2 × k π × 3 = 3 k π
\begin{aligned}
\triangle OBC
&=\triangle OBD+\triangle OCD \\
&=\dfrac{1}{2}\times k\pi\times\sqrt{3}+\dfrac{1}{2}\times k\pi\times\sqrt{3} \\
&=\sqrt{3}\,k\pi
\end{aligned}
△ O B C = △ O B D + △ O C D = 2 1 × k π × 3 + 2 1 × k π × 3 = 3 k π 조건에서 3 k π = 5 π 3 \sqrt{3}\,k\pi=\dfrac{5\pi}{3} 3 k π = 3 5 π
k = 5 3 3 = 5 3 9
\begin{aligned}
k&=\dfrac{5}{3\sqrt{3}} \\
&=\dfrac{5\sqrt{3}}{9}
\end{aligned}
k = 3 3 5 = 9 5 3 따라서
k + p = 5 3 9 + 3 = 14 3 9
\begin{aligned}
k+p
&=\dfrac{5\sqrt{3}}{9}+\sqrt{3} \\
&=\dfrac{14\sqrt{3}}{9}
\end{aligned}
k + p = 9 5 3 + 3 = 9 14 3 따라서 정답은 ③ 예요.
