Question

三角函数求值：

（1）sin6°sin42°sin66°sin78°；

（2）sin^2(20°）+cos^2(50°）+sin20°cos50°

Answer 1

(sin6°sin42°sin66°sin78°cos6)/cos6
=sin12 sin42°sin66°sin78/2cos6
sin78=cos12
=sin24 sin42°sin66/4cos6
sin66=cos24
=sin48 sin42/8cos6
sin42=cos48
=sin96/16cos6
=1/16

2 sinA sinB = cos(A-B) - cos(A+B)

sin^2(20°）+cos^2(50°）+sin20°cos50°
=sin^2(20°）+sin^2(40°）+sin20sin40
=(1-cos40)/2+(1-cos80)/2+[cos(20) - cos(60) ]/2
=1-(cos40+cos80-cos20+cos60)
=1-(2cos60cos20-cos20+cos60)
=1/2

Answer 2

(1)
=2sin6cos6cos12cos24cos48/2cos6
=sin12cos12cos24cos48/2cos6
=sin96/16cos6
=1/16
(2)
=(1-cos40)/2+(1-cos80)/2+(cos20-cos60)/2
=1-(cos40+cos80-cos20+cos60)
=1-(2cos60cos20-cos20+cos60)
=1/2