表达式 $8 \sin \theta - 3 \cos \theta$ 的定义域为 $0^\circ \le \theta \le 180^\circ$。
通过使用 $R \sin(\theta - \alpha)$（其中 $R > 0$ 且 $0^\circ < \alpha < 90^\circ$），

1. Express $8 \sin \theta - 3 \cos \theta$ in the form $R \sin(\theta - \alpha)$

Let $8 \sin \theta - 3 \cos \theta = R \sin(\theta - \alpha) = R(\sin \theta \cos \alpha - \cos \theta \sin \alpha)$.
Comparing coefficients:
$R \cos \alpha = 8$
$R \sin \alpha = 3$
$R = \sqrt{8^2 + 3^2} = \sqrt{73}$
$\tan \alpha = \frac{3}{8} \implies \alpha = \tan^{-1}\left(\frac{3}{8}\right) \approx 20.556^\circ$
So, $8 \sin \theta - 3 \cos \theta = \sqrt{73} \sin(\theta - 20.556^\circ)$.

2. Set up the equation using the R-formula expression

$\sqrt{73} \sin(\theta - 20.556^\circ) = 5$
$\sin(\theta - 20.556^\circ) = \frac{5}{\sqrt{73}}$

3. Find the basic angle and solve for $\theta$ in the given range

Basic angle $= \sin^{-1}\left(\frac{5}{\sqrt{73}}\right) \approx 35.831^\circ$
Since $0^\circ \le \theta \le 180^\circ$, the range for $(\theta - 20.556^\circ)$ is $-20.556^\circ \le \theta - 20.556^\circ \le 159.444^\circ$.
$\theta - 20.556^\circ = 35.831^\circ$ or $180^\circ - 35.831^\circ = 144.169^\circ$
$\theta = 35.831^\circ + 20.556^\circ = 56.387^\circ \approx 56.4^\circ$
$\theta = 144.169^\circ + 20.556^\circ = 164.725^\circ \approx 164.7^\circ$

1. Express the denominator using the R-formula result from part (i)

From part (i), $8 \sin \theta - 3 \cos \theta = \sqrt{73} \sin(\theta - 20.556^\circ)$.

The expression is $\frac{1}{(\sqrt{73} \sin(\theta - 20.556^\circ))^2} = \frac{1}{73 \sin^2(\theta - 20.556^\circ)}$.

2. Determine the smallest value of the expression

For the expression to be as small as possible, the denominator must be as large as possible.
The maximum value of $\sin^2(\theta - 20.556^\circ)$ is $1$.
Smallest value = $\frac{1}{73(1)} = \frac{1}{73}$.

3. Find the corresponding value of theta

This minimum occurs when $\sin^2(\theta - 20.556^\circ) = 1$.
Since $0^\circ \le \theta \le 180^\circ$, we have $-20.556^\circ \le \theta - 20.556^\circ \le 159.444^\circ$.
$\theta - 20.556^\circ = 90^\circ$
$\theta = 110.556^\circ \approx 110.6^\circ$ (to 1 d.p.)