(i) Start with the equation:

\(3 \sin^2 x - 8 \cos x - 7 = 0\).

Using the identity \(\sin^2 x = 1 - \cos^2 x\), substitute:

\(3(1 - \cos^2 x) - 8 \cos x - 7 = 0\).

Simplify to:

\(3 - 3 \cos^2 x - 8 \cos x - 7 = 0\).

\(-3 \cos^2 x - 8 \cos x - 4 = 0\).

Multiply through by -1:

\(3 \cos^2 x + 8 \cos x + 4 = 0\).

Factorize:

\((3 \cos x + 2)(\cos x + 2) = 0\).

Thus, \(\cos x = -\frac{2}{3}\) or \(\cos x = -2\).

Since \(\cos x = -2\) is not possible for real \(x\), we have \(\cos x = -\frac{2}{3}\).

(ii) Substitute \(\cos(\theta + 70^\circ) = -\frac{2}{3}\) into the equation:

\(\theta + 70^\circ = \cos^{-1}(-\frac{2}{3})\).

Calculate \(\theta + 70^\circ\):

\(\theta + 70^\circ = 131.8^\circ\) or \(228.2^\circ\).

For \(0^\circ \leq \theta \leq 180^\circ\), solve:

\(\theta = 131.8^\circ - 70^\circ = 61.8^\circ\).

\(\theta = 228.2^\circ - 70^\circ = 158.2^\circ\).

Thus, \(\theta = 61.8^\circ\) or \(158.2^\circ\).