高中数学【和差】与【积】互化公式,设y=sin20·sin40·sin60·sin80·(a),求证y=cos30·cos10cos50 cos70 其中cos30= 根3除以2b,由此,求证sin20sin40sin60sin80=3/16
2019-04-19
高中数学【和差】与【积】互化公式,
设y=sin20·sin40·sin60·sin80·
(a),求证y=cos30·cos10cos50 cos70 其中cos30= 根3除以2
b,由此,求证sin20sin40sin60sin80=3/16
优质解答
y=sin20·sin40·sin60·sin80
=sin(90-70)·sin(90-50)·sin(90-30)·sin(90-10)
=cos30·cos10·cos50· cos70
cos10·cos50· cos70 =1/2[cos(10-50)+cos(10+50)]· cos70
=1/2[cos40+cos60]· cos70
=1/2[cos40· cos70+1/2 · cos70 ]
=1/2[1/2(cos30+cos110)+1/2· cos70 ] cos110=-cos70
=1/2*1/2*cos30
=√3/8
=cos30·cos10·cos50· cos70
=√3/2*√3/8
=3/16
y=sin20·sin40·sin60·sin80
=sin(90-70)·sin(90-50)·sin(90-30)·sin(90-10)
=cos30·cos10·cos50· cos70
cos10·cos50· cos70 =1/2[cos(10-50)+cos(10+50)]· cos70
=1/2[cos40+cos60]· cos70
=1/2[cos40· cos70+1/2 · cos70 ]
=1/2[1/2(cos30+cos110)+1/2· cos70 ] cos110=-cos70
=1/2*1/2*cos30
=√3/8
=cos30·cos10·cos50· cos70
=√3/2*√3/8
=3/16