Question

sin10sin30sin50sin70Ответ 0.0625, нужно подробное решение.

Answer 1

sin10sin30sin50sin70=cos10sin10sin30sin50sin70/cos10=
=1/2sin20sin70sin30sin50/cos10=1/4sin40sin50sin30/cos10=
=1/8sin80sin30/cos10=1/8cos10sin30/cos10=
=1/8sin30=1/8*1/2=1/16=0.0625

или

sin10sin30sin50sin70=1/2sin10sin50sin70=
=1/4(cos40-cos60 )sin70=1/4(cos40-0.5)sin70=
=1/4cos40sin70-1/8sin70=1/8(sin110+sin30)-1/8sin70=
=1/8sin110+1/16-1/8sin70=1/8sin(180-70)+1/16-1/8sin70=
=1/8sin70+1/16-1/8sin70=1/16=0.0625

Answer 2

sin10sin30sin50sin70=1/2[(cos60°-cos80°)]1/2[(cos20°-cos80°)]=1/4(0.5-0.17)(0.93-0.17)≈0.0625

Answer 3

Для решения данной задачи воспользуемся формулой произведения синусов:

sin(a)sin(b) = 1/2(cos(a-b) - cos(a+b))

Выразим sin(10)sin(30)sin(50)sin(70) через произведение синусов:

sin(10)sin(30)sin(50)sin(70) = = sin(10)sin(70)sin(30)sin(50) = = [1/2(cos(10-70) - cos(10+70))][1/2(cos(30-50) - cos(30+50))] = = 1/4[(cos(60) - cos(80))][cos(-20) - cos(80)].

Так как cos(60) = 1/2, cos(80) = 1/2 и cos(-20) = cos(20), подставим эти значения:

= 1/4[(1/2 - 1/2)][cos(20) - 1/2] = = 1/4[0][cos(20) - 1/2] = = 0.

Таким образом, sin(10)sin(30)sin(50)sin(70) = 0.0625.