(a) Start with the equation:

\(5 \cos \theta - \sin \theta \tan \theta + 1 = 0\)

Multiply through by \(\cos \theta\) to eliminate \(\tan \theta\):

\(5 \cos^2 \theta - \sin \theta \frac{\sin \theta}{\cos \theta} \cos \theta + \cos \theta = 0\)

Simplify:

\(5 \cos^2 \theta - \sin^2 \theta + \cos \theta = 0\)

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

\(5 \cos^2 \theta - (1 - \cos^2 \theta) + \cos \theta = 0\)

Simplify further:

\(6 \cos^2 \theta + \cos \theta - 1 = 0\)

Thus, \(a = 6\), \(b = 1\), \(c = -1\).

(b) Solve the quadratic equation:

\((3 \cos \theta - 1)(2 \cos \theta + 1) = 0\)

Set each factor to zero:

\(3 \cos \theta - 1 = 0\) gives \(\cos \theta = \frac{1}{3}\)

\(2 \cos \theta + 1 = 0\) gives \(\cos \theta = -\frac{1}{2}\)

Find \(\theta\) for \(\cos \theta = \frac{1}{3}\):

\(\theta \approx 1.23, 5.05\)

Find \(\theta\) for \(\cos \theta = -\frac{1}{2}\):

\(\theta = \frac{2\pi}{3}, \frac{4\pi}{3}\)

Convert to approximate values:

\(\theta \approx 2.09, 4.19\)

Thus, \(\theta = \{1.23, \frac{2\pi}{3}, 2.09, 4.19, 5.05\}\).