8.166. $25 \sin ^{2} x+100 \cos x=89$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

8.166. $25 \sin ^{2} x+100 \cos x=89$.
## Solution.

From the condition

$$
25\left(1-\cos ^{2} x\right)+100 \cos x-89=0,25 \cos ^{2} x-100 \cos x+64=0
$$

Solving the equation as a quadratic in terms of $\cos x$, we get $\cos x=\frac{16}{5}, \varnothing$, or $\cos x=\frac{4}{5}, x= \pm \arccos \frac{4}{5}+2 \pi k, k \in Z$.

Answer: $\quad x= \pm \arccos \frac{4}{5}+2 \pi k, \quad k \in Z$.