삼각함수 점의 주기

$a _{n} = 3 + ( - 1 ) ^{n}$에서 $a _{1} = 2$, $a _{2} = 4$, $a _{3} = 2$, $a _{4} = 4$, $\cdots$ $\displaystyle P _{1} ( 2 \cos \frac{2 \pi}{3} , 2 \sin \frac{2 \pi}{3} )$ 즉, $\displaystyle ( - 1 , \sqrt{3} )$ $\displaystyle P _{2} ( 4 \cos \frac{4 \pi}{3} , 4 \sin \frac{4 \pi}{3} )$ 즉, $\displaystyle ( - 2 , - 2 \sqrt{3} )$ $\displaystyle P _{3} ( 2 \cos \frac{6 \pi}{3} , 2 \sin \frac{6 \pi}{3} )$ 즉, $( 2 , 0 )$ $\displaystyle P _{4} ( 4 \cos \frac{8 \pi}{3} , 4 \sin \frac{8 \pi}{3} )$ 즉, $\displaystyle ( - 2 , 2 \sqrt{3} )$ $\displaystyle P _{5} ( 2 \cos \frac{10 \pi}{3} , 2 \sin \frac{10 \pi}{3} )$ 즉, $\displaystyle ( - 1 , - \sqrt{3} )$ $\displaystyle P _{6} ( 4 \cos \frac{12 \pi}{3} , 4 \sin \frac{12 \pi}{3} )$ 즉, $( 4 , 0 )$ $\displaystyle P _{7} ( 2 \cos \frac{14 \pi}{3} , 2 \sin \frac{14 \pi}{3} )$ 즉, $\displaystyle ( - 1 , \sqrt{3} )$ $\cdots \cdots$ $2009 = 6 \times 334 + 5$이므로 점 $P _{2009}$와 같은 점은 $P _{5}$이다.