Question

9.化简sin三分之nπ

Answer 1

n是整数，根据n的值，分类化简。正弦sinx的周期是2π，nπ/3的周期是2π，nπ的周期是6π，
n分6中情况讨论：n=6k，n=6k+1，n=6k+2，n=6k+3，n=6k+4，n=6k+5，其中k为整数。
n=6k，sin（nπ/3)=sin（6kπ/3)=sin(2kπ)=0;
n=6k+1，sin（nπ/3)=sin（(6k+1)π/3)=sin(2kπ+π/3)=sin(π/3)=✓3/2;
n=6k+2，sin（nπ/3)=sin（(6k+2)π/3)=sin(2kπ+2π/3)=sin(2π/3)=✓3/2;
n=6k+3，sin（nπ/3)=sin（(6k+3)π/3)=sin(2kπ+π=)=sin(π)=0;
n=6k+4，sin（nπ/3)=sin（(6k+4)π/3)=sin(2kπ+4π/3)=-sin(π/3)=-✓3/2;
n=6k+5，sin（nπ/3)=sin（(6k+5)π/3)=sin(2kπ+5π/3)=sin(2π-π/3)=-✓3/2;