1. The set of values for the real number $a$ such that the equation $\cos 2 x+\sin x=a$ has real solutions is $\qquad$.

将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 

Note: The last sentence is a repetition of the instruction and should not be part of the translation. Here is the correct translation:

1. The set of values for the real number $a$ such that the equation $\cos 2 x+\sin x=a$ has real solutions is $\qquad$.
$$
=1 .\left[-2, \frac{9}{8}\right] \text {. }
$$

Since $a=-2\left(\sin x-\frac{1}{4}\right)^{2}+\frac{9}{8}, -1 \leqslant \sin x \leqslant$ 1, to make the equation have real solutions, the maximum value of $a$ can be $\frac{9}{8}$ (which can be achieved when $\sin x=\frac{1}{4}$), and the minimum value of $a$ is obtained when $\sin x=$ -1, which is -2. Therefore, the set of values for $a$ is $\left[-2, \frac{9}{8}\right]$.