Part 1：3的倍数

${sin0^\circ=0，sin30^\circ=\frac{1}{2}，sin45^\circ=\frac{\sqrt{2}}{2}，sin60^\circ=\frac{\sqrt{3}}{2}，sin90^\circ=1；}$

${sin15^\circ=\frac{\sqrt{6}-\sqrt{2}}{4}=\frac{\sqrt{2-\sqrt{3}}}{2}，sin75^\circ=\frac{\sqrt{6}+\sqrt{2}}{4}=\frac{\sqrt{2+\sqrt{3}}}{2}；}$

${sin18^\circ=\frac{-1+\sqrt{5}}{4}，sin36^\circ=\frac{\sqrt{10-2\sqrt{5}}}{4}，sin54^\circ=\frac{1+\sqrt{5}}{4}，sin72^\circ=\frac{\sqrt{10+2\sqrt{5}}}{4}；}$

${sin9^\circ=\frac{\sqrt{10}+\sqrt{2}-2\sqrt{5-\sqrt{5}}}{8}=\frac{\sqrt{8-2\sqrt{10+2\sqrt{5}}}}{4}，sin27^\circ=\frac{-\sqrt{10}+\sqrt{2}+2\sqrt{5+\sqrt{5}}}{8}=\frac{\sqrt{8-2\sqrt{10-2\sqrt{5}}}}{4}，}$

${sin63^\circ=\frac{\sqrt{10}-\sqrt{2}+2\sqrt{5+\sqrt{5}}}{8}=\frac{\sqrt{8+2\sqrt{10-2\sqrt{5}}}}{4}，sin81^\circ=\frac{\sqrt{10}+\sqrt{2}+2\sqrt{5-\sqrt{5}}}{8}=\frac{\sqrt{8+2\sqrt{10+2\sqrt{5}}}}{4}；}$

${sin6^\circ=\frac{-1-\sqrt{5}+\sqrt{30-6\sqrt{5}}}{8}，sin42^\circ=\frac{1-\sqrt{5}+\sqrt{30-6\sqrt{5}}}{8}，}$

${sin66^\circ=\frac{1+\sqrt{5}+\sqrt{30-6\sqrt{5}}}{8}，sin78^\circ=\frac{-1+\sqrt{5}+\sqrt{30+6\sqrt{5}}}{8}；}$

${sin12^\circ=\frac{\sqrt{10+2\sqrt{5}}-\sqrt{15}+\sqrt{3}}{8}=\frac{\sqrt{7-\sqrt{5}-\sqrt{30-6\sqrt{5}}}}{4}，sin24^\circ=\frac{-\sqrt{10+2\sqrt{5}}+\sqrt{15}+\sqrt{3}}{8}=\frac{\sqrt{7+\sqrt{5}-\sqrt{30-6\sqrt{5}}}}{4}，}$

${sin48^\circ=\frac{\sqrt{10+2\sqrt{5}}+\sqrt{15}-\sqrt{3}}{8}=\frac{\sqrt{7-\sqrt{5}+\sqrt{30-6\sqrt{5}}}}{4}，sin84^\circ=\frac{\sqrt{10+2\sqrt{5}}+\sqrt{15}+\sqrt{3}}{8}=\frac{\sqrt{7+\sqrt{5}+\sqrt{30-6\sqrt{5}}}}{4}；}$

${sin3^\circ=\frac{\sqrt{8-2\sqrt{7+\sqrt{5}+\sqrt{30-6\sqrt{5}}}}}{4}，sin21^\circ=\frac{\sqrt{8-2\sqrt{7-\sqrt{5}+\sqrt{30-6\sqrt{5}}}}}{4}，}$

${sin33^\circ=\frac{\sqrt{8-2\sqrt{7+\sqrt{5}-\sqrt{30-6\sqrt{5}}}}}{4}，sin39^\circ=\frac{\sqrt{8-2\sqrt{7-\sqrt{5}-\sqrt{30-6\sqrt{5}}}}}{4}，}$

${sin51^\circ=\frac{\sqrt{8+2\sqrt{7-\sqrt{5}-\sqrt{30-6\sqrt{5}}}}}{4}，sin57^\circ=\frac{\sqrt{8+2\sqrt{7+\sqrt{5}-\sqrt{30-6\sqrt{5}}}}}{4}，}$

${sin69^\circ=\frac{\sqrt{8+2\sqrt{7-\sqrt{5}+\sqrt{30-6\sqrt{5}}}}}{4}，sin87^\circ=\frac{\sqrt{8+2\sqrt{7+\sqrt{5}+\sqrt{30-6\sqrt{5}}}}}{4}，}$

2024.5.31

Part 2：5的倍数而非3的倍数