【化简求值sin^4θ+sin^4(π/3-θ)+sin^4(π/3+θ)】
1人问答
问题描述：
化简求值sin^4θ+sin^4(π/3-θ)+sin^4(π/3+θ)
江晶回答：
sin^4θ=[1/2(1-cos2θ)^2]^2=1/4{1-2cos2θ+(cos2θ)^2}
=1/4{1-2cos2θ+(1+cos4θ)/2}=3/8-1/4cos2θ+1/8cos4θ
∴sin^4θ+sin^4(π/3-θ)+sin^4(π/3+θ)
=sin^4θ+sin^4(θ-π/3)+sin^4(θ+π/3)
=3/8-1/4cos2θ+1/8cos4θ+3/8-1/4cos(2θ-2π/3)+1/8cos(4θ-4π/3)+3/8-1/4cos(2θ+2π/3)+1/8cos(4θ+4π/3)
=9/8-1/4{cos2θ+cos(2θ-2π/3)+cos(2θ+2π/3)}+1/8{cos4θ+cos(4θ-4π/3)+cos(4θ+4π/3)}
=9/8-1/4{cos2θ+cos2θcos2π/3+sincos2θsin2π/3+cos2θcos2π/3-sincos2θsin2π/3}
+1/8{cos4θ+cos4θcos4π/3+sincos4θsin4π/3+cos4θcos4π/3-sincos4θsin4π/3}
=9/8-1/4{cos2θ+cos2θcos2π/3+cos2θcos2π/3}+1/8{cos4θ+cos4θcos4π/3+cos4θcos4π/3}
=9/8-1/4{cos2θ-1/2cos2θ-1/2cos2θ}+1/8{cos4θ-1/2cos4θ-1/2cos4θ}
=9/8-0+0
=9/8
数学推荐
数学推荐
最新更新
精品分类
优秀数学推荐
热门数学