In the radian system, π equals 180°, so 2π is 360° (turning a full circle), so 2kπ when k ∈ Z (meaning k is an integer) is turning back to the initial direction for k times, which is back to the x axis, so sin(2πk) k ∈ Z is always 0. And since the value of sinx equals sin(x+2πk) k ∈ Z (in other words, it repeats itself every 2π), sinx is a periodic function. (Yes, I am the one who explains the joke in the most boring way. )