Question

高一三角恒等变换题

求sin10°sin30°sin50°sin70°的值（一串乘积）
5分求具体的解法

Answer 1

sin10*sin30*sin50*sin70
=cos10*sin10*sin30*sin50*sin70/cos10
=1/2*sin20*sin70*sin30*sin50/cos10
=1/2*sin20*cos20*sin30*sin50/cos10
=1/4sin40*sin50*sin30/cos10
=1/4sin40*cos40*sin30/cos10
=1/8sin80*sin30/cos10
=1/8cos10*sin30/cos10
=1/8sin30
=1/16

Answer 2

因为sin70°=cos20°，sin50°=cos40°
所以 sin10°sin30°sin50°sin70°=sin10°sin30°cos20°cos40°
=sin10°cos20°cos40°sin30°
=（cos10°sin10°cos20°cos40°sin30°）/cos10°
=（0.5*2cos10°sin10°cos20°cos40°sin30°）/cos10°
=（0.5*sin20°cos20°cos40°sin30°）/cos10°
=（0.5*0.5*2sin20°cos20°cos40°sin30°）/cos10°
=（0.25*0.5*2sin40°cos40°sin30°）/cos10°
=（0.25*0.5*sin80°sin30°）/cos10°
=（0.075cos10°sin30°）/cos10°
=0.075sin30°=1/16

Answer 3

原式=cos80.cos60.cos40.cos20.sin20/sin20
=0.5cos80.cos60.cos40.sin40/sin20
=0.5*0.5cos80.cos60.sin80/sin20
=0.5*0.5*0.5sin160.cos60/sin20
=0.5*0.5*0.5.cos60
=0.5*0.5*0.5*0.5
=1/16