化简：(cosA)^2+[cos(2/3pai+A)]^2+[cos(2/3pai-A)]^2

来源：学生作业帮助网编辑：作业帮时间：2024/05/02 06:19:18

化简：(cosA)^2+[cos(2/3pai+A)]^2+[cos(2/3pai-A)]^2
化简：(cosA)^2+[cos(2/3pai+A)]^2+[cos(2/3pai-A)]^2

化简：(cosA)^2+[cos(2/3pai+A)]^2+[cos(2/3pai-A)]^2
(cosA)^2+[cos(2π/3+A)]^2+[cos(2π/3-A)]^2
=(cosA)^2+[1+cos(4π/3+2A)]/2+[1+cos(4π/3-2A)]/2
=(cosA)^2+1/2+1/2*cos(4π/3+2A)+1/2+1/2*cos(4π/3-2A)
=(cosA)^2+1/2*cos(4π/3+2A)+1/2*cos(4π/3-2A)+1
=(cosA)^2+1/2*cos(π+π/3+2A)+1/2*cos(π+π/3-2A)+1
=(cosA)^2-1/2*cos(π/3+2A)-1/2*cos(π/3-2A)+1
=(cosA)^2-1/2*(cosπ/3cos2A-sinπ/3sin2A)-1/2*(cosπ/3cos2A+sinπ/3sin2A)+1
=(cosA)^2-1/2*[(cosπ/3cos2A-sinπ/3sin2A)+(cosπ/3cos2A+sinπ/3sin2A)]+1
=(cosA)^2-1/2*(cosπ/3cos2A-sinπ/3sin2A+cosπ/3cos2A+sinπ/3sin2A)+1
=(cosA)^2-cosπ/3cos2A+1
=(cosA)^2-1/2*cos2A+1
=(1+cos2A)/2-1/2*cos2A+1
=1/2+1/2*cos2A-1/2*cos2A+1
=1/2+1
=3/2

(cosA)^2+[cos(2/3pai+A)]^2+[cos(2/3pai-A)]^2
=(1+cos2A)/2 +[1+cos(4π/3+2A)]/2+ [1+cos(4π/3-2A)]/2
=3/2+1/2[cos2A+cos(4π/3+2A)+ cos(4π/3-2A)]
=3/2+1/2[ cos2A+2cos4π/3cos2A ]
=3/2+1/2[ cos2A+2(-1/2)cos2A]
=3/2

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