(i) Start with the equation \(\frac{4 \cos \theta}{\tan \theta} + 15 = 0\).

Replace \(\tan \theta\) with \(\frac{\sin \theta}{\cos \theta}\), giving \(\frac{4 \cos \theta}{\frac{\sin \theta}{\cos \theta}} + 15 = 0\).

This simplifies to \(\frac{4 \cos^2 \theta}{\sin \theta} + 15 = 0\).

Multiply through by \(\sin \theta\) to clear the fraction: \(4 \cos^2 \theta + 15 \sin \theta = 0\).

Use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\) to get \(4(1 - \sin^2 \theta) + 15 \sin \theta = 0\).

Expand and rearrange: \(4 - 4 \sin^2 \theta + 15 \sin \theta = 0\).

Rearrange to \(4 \sin^2 \theta - 15 \sin \theta - 4 = 0\), as required.

(ii) Solve \(4 \sin^2 \theta - 15 \sin \theta - 4 = 0\).

Let \(s = \sin \theta\). The equation becomes \(4s^2 - 15s - 4 = 0\).

Use the quadratic formula: \(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -15\), \(c = -4\).

Calculate the discriminant: \(b^2 - 4ac = (-15)^2 - 4 \times 4 \times (-4) = 225 + 64 = 289\).

\(s = \frac{15 \pm \sqrt{289}}{8} = \frac{15 \pm 17}{8}\).

This gives \(s = 4\) or \(s = -\frac{1}{4}\).

Since \(\sin \theta\) must be between -1 and 1, \(s = 4\) is not valid.

Thus, \(\sin \theta = -\frac{1}{4}\).

Find \(\theta\) using \(\sin^{-1}(-\frac{1}{4})\), which gives \(\theta = 194.5^\circ\) or \(\theta = 345.5^\circ\) within the range \(0^\circ \leq \theta \leq 360^\circ\).