每日一题[743]你中有我

设$\alpha\in\mathbf R$,若$\left\{ x \,\big|\, \left|\sin x\right|^{\alpha}+\left|\cos x\right|^{\alpha}=1\right\}\subseteq\left\{x \mid \sin^4 x+\cos ^4x=1\right\}$,则$\alpha$的取值范围是________.


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分析与解 $(-\infty,2)\cup (2,+\infty)$.

情形一 当$\alpha>2$时,根据指数函数的性质,有$$\left|\sin x\right|^{\alpha}\leqslant \left|\sin x\right|^2,\left|\cos x\right|^{\alpha}\leqslant \left|\cos x\right|^2,$$等号当且仅当$\left|\sin x\right|,\left|\cos x\right|\in \{0,1\}$时取得.此时有$$\left\{ x \,\big|\, \left|\sin x\right|^{\alpha}+\left|\cos x\right|^{\alpha}=1\right\}=\left\{x \mid \sin^4 x+\cos ^4x=1\right\}=\left\{ x\mid x=\dfrac{k\pi}2,k\in\mathbf Z\right\}.$$
情形二 当$\alpha=2$时,显然有
$$\left\{ x \,\big|\, \left|\sin x\right|^{\alpha}+\left|\cos x\right|^{\alpha}=1\right\}=\mathbf R.$$
情形三 当$0<\alpha<2$时,根据指数函数的性质,有$$\left|\sin x\right|^{\alpha}\geqslant \left|\sin x\right|^2,\left|\cos x\right|^{\alpha}\geqslant \left|\cos x\right|^2,$$等号当且仅当$\left|\sin x\right|,\left|\cos x\right|\in \{0,1\}$时取得.此时有$$\left\{ x \,\big|\, \left|\sin x\right|^{\alpha}+\left|\cos x\right|^{\alpha}=1\right\}=\left\{x \mid \sin^4 x+\cos ^4x=1\right\}=\left\{ x\mid x=\dfrac{k\pi}2,k\in\mathbf Z\right\}.$$
情形四 当$\alpha\leqslant 0$时,显然有
$$\left\{ x \,\big|\, \left|\sin x\right|^{\alpha}+\left|\cos x\right|^{\alpha}=1\right\}=\varnothing.$$
综上所述,$\alpha$的取值范围是$(-\infty,2)\cup (2,+\infty)$.


练习 已知正整数$n\geqslant 3$,且$\sin^n\theta+\cos^n\theta=1$,则$\sin\theta+\cos\theta=$________.

答案 $\begin{cases} \pm 1,& 2\mid n,\\ 1,& 2 \nmid n.\end{cases} $

提示 由于$$\sin^n\theta\leqslant \sin^2\theta,\cos^n\theta\leqslant \cos^2\theta,$$于是$$\sin^n\theta+\cos^n\theta\leqslant 1.$$考虑到当$|x|\leqslant 1$且$n\geqslant 3$时,$x^n\leqslant x^2$的取等条件是$$x=\begin{cases} -1,1,0,& 2\mid n,\\ 1,0,&2 \nmid n,\end{cases} $$ 又$\sin^2\theta+\cos^2\theta=1$,于是$$\sin\theta+\cos\theta=\begin{cases} \pm 1,& 2\mid n,\\ 1,& 2 \nmid n.\end{cases}$$

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