(i) To express \(24 \sin \theta - 7 \cos \theta\) in the form \(R \sin(\theta - \alpha)\), we use the identity:

\(\begin{aligned} a \sin \theta + b \cos \theta &\equiv R \sin(\theta + \alpha), \\ R &= \sqrt{a^2 + b^2}, \\ \tan \alpha &= \frac{b}{a}. \end{aligned}\)

Here, \(a = 24\) and \(b = -7\). Thus,

\(R = \sqrt{24^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25.\)

\(\tan \alpha = \frac{-7}{24}.\)

\(\alpha = \tan^{-1}\left(\frac{-7}{24}\right)\), which gives \(\alpha \approx 16.26^\circ\) (to 2 decimal places).

(ii) To find the smallest positive value of \(\theta\) satisfying \(24 \sin \theta - 7 \cos \theta = 17\), we use:

\(R \sin(\theta - \alpha) = 17.\)

\(25 \sin(\theta - 16.26^\circ) = 17.\)

\(\sin(\theta - 16.26^\circ) = \frac{17}{25}.\)

\(\theta - 16.26^\circ = \sin^{-1}\left(\frac{17}{25}\right)\).

\(\sin^{-1}\left(\frac{17}{25}\right) \approx 42.84^\circ\).

\(\theta = 42.84^\circ + 16.26^\circ = 59.1^\circ\).