求证4sin20cos40＝2sin60+2sin(-20)

来源：百度知道 编辑：UC知道 时间：2024/06/22 20:33:09

因为2sin60+2sin(-20)-4sin20cos40
=2sin(40+20)+2sin(-20)-4sin20cos40
=2sin40cos20+2sin20cos40+2sin(-20)-4sin20cos40
=2sin40cos20-2sin20cos40+2sin(-20)
=2sin(40-20)-2sin20
=2sin20-2sin20
=0
所以4sin20cos40＝2sin60+2sin(-20)

sin60
=sin(40+20)
=sin40cos20+cos40sin20

sin(-20)
=sin(20-40)
=sin20cos40-cos20sin40

所以 2sin60+2sin(-20)
=2sin40cos20+2cos40sin20+2sin20cos40-2cos20sin40
=4sin20cos40

4sin20cos40=4sin20cos(60-20)
=4sin20(cos60cos20+sin60sin20)
=2sin20cos20+2√3(sin20)^2
=sin40+√3(1-cos40)
=sin40-√3cos40+√3
=2sin(40-60)+√3

2sin60+2sin(-20)=倒回去做也可