若sin2x、sinx分别是sinθ与cosθ的等差中项和等比中项,则cos2x的值为:( ) A.1+338 B.1−338 C.1±338 D.1−24
问题描述:
若sin2x、sinx分别是sinθ与cosθ的等差中项和等比中项,则cos2x的值为:( )
A.
1+
33
8
B.
1−
33
8
C.
1±
33
8
D.
1−
2
4
答
依题意可知2sin2x=sinθ+cosθsin2x=sinθcosθ∵sin2θ+cos2θ=(sinθ+cosθ)2-2sinθcosθ=4sin22x-2sin2x=1∴4(1-cos22x)+cos2x-2=0,即4cos22x-cos2x-2=0,求得cos2x=1±338∵sin2x=sinθcosθ∴cos2x=1-2si...