(a) Start with the equation \(3 \cos \theta = 8 \tan \theta\).

Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), substitute to get:

\(3 \cos \theta = 8 \frac{\sin \theta}{\cos \theta}\)

Multiply through by \(\cos \theta\) to eliminate the fraction:

\(3 \cos^2 \theta = 8 \sin \theta\)

Use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\):

\(3(1 - \sin^2 \theta) = 8 \sin \theta\)

Expand and rearrange to form a quadratic equation:

\(3 - 3 \sin^2 \theta = 8 \sin \theta\)

\(3 \sin^2 \theta + 8 \sin \theta - 3 = 0\)

(b) Factor the quadratic equation:

\((3 \sin \theta - 1)(\sin \theta + 3) = 0\)

Solving \(3 \sin \theta - 1 = 0\) gives:

\(\sin \theta = \frac{1}{3}\)

Find \(\theta\) using \(\sin^{-1}\):

\(\theta = \sin^{-1}\left(\frac{1}{3}\right)\)

\(\theta \approx 19.5^\circ\)