把tan(x)=(a/b)^(1/3)代入L=a/sin^2(x)+b/cos^2(x)

来源：学生作业帮编辑：拍题作业网作业帮分类：数学作业时间：2024/05/02 01:16:34

把tan(x)=(a/b)^(1/3)代入L=a/sin^2(x)+b/cos^2(x)
怎么得出 L = (A^(2/3) + B^(2/3))^(3/2)的?

tan(x=(a/b)/(1/3)
tan^2(x)=(a/b)^(2/3)
联立sin^2(x)/cos^2(x)=(a/b)^(2/3)
sin^2(x)+cos^2(x)=1
解得sin^2(x)=a^(2/3)/(a^(2/3)+b^(2/3))
cos^2(x)=b^(2/3)/(a^(2/3)+b^(2/3))
代入L得L=a*(a^(2/3)+b^(2/3))/a^(2/3)+b*(a^(2/3)+b^(2/3))/b^(2/3)
L=(a^(2/3)+b^(2/3))(a/a^(2/3)+b/b^(2/3))
=(a^(2/3)+b^(2/3))(a^(1/3)+b^(1/3))
=(a^(2/3)+b^(2/3))(a^(2/3)+b^(2/3))^(1/2)
=(a^(2/3)+b^(2/3))^(1+1/2)
=(a^(2/3)+b^(2/3))^(3/2)
再问：你怎么从联立sin^2(x)/cos^2(x)=(a/b)^(2/3) sin^2(x)+cos^2(x)=1 到解得sin^2(x)=a^(2/3)/(a^(2/3)+b^(2/3)) cos^2(x)=b^(2/3)/(a^(2/3)+b^(2/3)) 的？联立什么意思
再答：联立就是联合两个方程解。。。因为sin^2(x)+cos^2(x)=1 cos^2(x)=1-sin^2(x) 所以sin^2(x)/cos^2(x)=(a/b)^(2/3) sin^2(x)/1-sin^2(x)=(a/b)^(2/3) 解得sin^2(x)=a^(2/3)/(a^(2/3)+b^(2/3)) cos^2(x)同理
再问：。。。我猜到联立的意思了但是你怎么从 sin^2(x)/1-sin^2(x)=(a/b)^(2/3) 解得sin^2(x)=a^(2/3)/(a^(2/3)+b^(2/3)) 的
再答：我有点内伤。。。 sin^2(x)/1-sin^2(x)=(a/b)^(2/3) sin^2(x)/1-sin^2(x)=a^(2/3)/b^(2/3) 交叉相乘 b^(2/3)sin^2(x)=a^(2/3)(1-sin^2(x)) b^(2/3)sin^2(x)=a^(2/3)-a^(2/3)(sin^2(x)) 移项 b^(2/3)sin^2(x)+a^(2/3)(sin^2(x))=a^(2/3) (sin^2(x)(b^(2/3)+a^(2/3))=a^(2/3) sin^2(x)=a^(2/3)/(a^(2/3)+b^(2/3))

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