求证1/sin50°+4sin10°=2

更新时间2019-01-30 20:22:51

1/sin50°+4sin10°

=(1+4sin10°sin50°)/sin50°

=[1+2cos(50°-10°)-2cos(50°+10°)]/sin50°

=2cos40°/sin50°

=2sin50°/sin50°

=2

2sinαsinβ=cos(α-β)-cos(α+β)

公式推导：cos(α-β)=cosαcosβ+sinαsinβ  ①

cos(α+β)=cosαcosβ-sinαsinβ  ②

①-②：cos(α-β)-cos(α+β)=2sinαsinβ

sin50°=sin（90°-40°）=cos40°

sin10°=sin（90°-80°）=cos80°=2cos²40°-1

原式=1/（cos40°+8cos²40°-4）=2

解得：cos40°=0.775

实际上cos40°=0.766