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

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

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

\(R = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25.\) \(\tan \alpha = \frac{24}{7} \implies \alpha = \tan^{-1}\left(\frac{24}{7}\right) \approx 73.74^\circ.\)

(ii) To solve \(7 \cos \theta + 24 \sin \theta = 15\), we use the expression:

\(R \cos(\theta - \alpha) = 15.\)

Substitute \(R = 25\) and \(\alpha = 73.74^\circ\):

\(25 \cos(\theta - 73.74^\circ) = 15 \implies \cos(\theta - 73.74^\circ) = \frac{15}{25} = 0.6.\)

Find \(\theta - 73.74^\circ\):

\(\theta - 73.74^\circ = \cos^{-1}(0.6) \approx 53.13^\circ.\)

Thus, \(\theta = 53.13^\circ + 73.74^\circ = 126.87^\circ\).

Also, \(\theta - 73.74^\circ = 360^\circ - 53.13^\circ = 306.87^\circ\).

Thus, \(\theta = 306.87^\circ + 73.74^\circ = 20.61^\circ\).

Therefore, the solutions are \(\theta = 126.9^\circ\) and \(\theta = 20.6^\circ\).