6. When $\theta$ takes values in the interval $\left(0, \frac{\pi}{2}\right)$, the minimum value of $\frac{8}{\cos \theta}+\frac{1}{\sin \theta}$ is ( ).
A. $5 \sqrt{5}$
B. $\sqrt{102}$
C. $\sqrt{113}$
L. 10
6. A
$$
\begin{aligned}
\frac{8}{\cos \theta}+\frac{1}{\sin \theta} & =\sqrt{\frac{64}{\cos \theta}+\frac{1}{\sin \theta}+\frac{16}{\sin \theta \cos \theta}} \\
& =\sqrt{\frac{64\left(\sin ^{2} \theta+\cos ^{2} \theta\right)}{\cos ^{2} \theta}+\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin ^{2} \theta}+\frac{16\left(\sin ^{2} \theta+\cos ^{2} \theta\right)}{\sin \theta \cos \theta}} \\
& =\sqrt{64 \tan ^{2} \theta+64+1+\cot ^{2} \theta +16 \tan \theta+16 \cot \theta} \\
& =\sqrt{65+\left(64 \tan ^{2} \theta+16 \cot \theta\right)+\left(\cot ^{2} \theta+16 \tan \theta\right)} \\
& =\sqrt{65+\left(64 \tan ^{2} \theta+8 \cot \theta+8 \cot \theta\right)+\left(\cot ^{2} \theta+8 \tan \theta+8 \tan \theta\right)} \\
& \geqslant \sqrt{65+3 \sqrt[3]{64 \times 8 \times 8}+3 \sqrt[3]{8 \times 8}}=5 \sqrt{5}
\end{aligned}
$$

When and only when \(64 \tan ^{2} \theta=8 \cot \theta\) and \(\cot ^{2} \theta=8 \tan \theta\), i.e., \(\tan \theta=\frac{1}{2}\), \(\frac{8}{\cos \theta}+\frac{1}{\sin \theta}\) can take the value \(5 \sqrt{5}\).