设f(a)=sin^Xα+cos^Xα,X∈{n|n=2k,k∈N+}.利用三角变换,估计f(α)在X=2,4,6的取值情况,对X取一般值时f(α)的取值范围作一个猜想.高一数学三角恒等变换中的题目,必修4中P144第5题,在线等答案.
2019-04-14
设f(a)=sin^Xα+cos^Xα,X∈{n|n=2k,k∈N+}.
利用三角变换,估计f(α)在X=2,4,6的取值情况,对X取一般值时f(α)的取值范围作一个猜想.
高一数学三角恒等变换中的题目,必修4中P144第5题,在线等答案.
优质解答
当α=2时,sin^2α+cos^2α=1
当α=2时,sin^4α+cos^4α=(sin^2α+cos^α)^2-2sin^2αcos^2α=1-1/2(sin2α)^2,所以1/2≤sin^4α+cos^4α≤1
sin^6α+cos^6α=(sin^2α+cos^2α)(sin^4α-sin^2αcos^2+cos^4α)
=(sin^2α+cos^2α)^2-3sin^2αcos^2α
=1-3/4(sin2α)^2
1/4≤sin^6α+cos^6α≤1
猜想当x=2k(k∈N+}时,1/2^（k-1)≤sin^Xα+cos^Xα≤1
当α=2时,sin^2α+cos^2α=1
当α=2时,sin^4α+cos^4α=(sin^2α+cos^α)^2-2sin^2αcos^2α=1-1/2(sin2α)^2,所以1/2≤sin^4α+cos^4α≤1
sin^6α+cos^6α=(sin^2α+cos^2α)(sin^4α-sin^2αcos^2+cos^4α)
=(sin^2α+cos^2α)^2-3sin^2αcos^2α
=1-3/4(sin2α)^2
1/4≤sin^6α+cos^6α≤1
猜想当x=2k(k∈N+}时,1/2^（k-1)≤sin^Xα+cos^Xα≤1