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________．

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 $$\sin\theta+\cos\theta=\begin{cases} \pm 1,& 2\mid n,\\ 1,& 2 \nmid n.\end{cases}$$