The first term \(a = \cos \theta\) and the second term \(a + d = \cos \theta + \sin^2 \theta\) gives the common difference \(d = \sin^2 \theta\).

The sum of the first 13 terms of an arithmetic progression is given by:

\(S_{13} = \frac{13}{2} \left[ 2a + 12d \right] = 52\)

Substitute \(a = \cos \theta\) and \(d = \sin^2 \theta\):

\(\frac{13}{2} \left[ 2\cos \theta + 12\sin^2 \theta \right] = 52\)

Simplify and solve for \(\theta\):

\(2\cos \theta + 12(1 - \cos^2 \theta) = 8\)

\(2\cos \theta + 12 - 12\cos^2 \theta = 8\)

\(12\cos^2 \theta - 2\cos \theta - 4 = 0\)

Divide by 2:

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

Solving the quadratic equation for \(\cos \theta\):

\(\cos \theta = \frac{2}{3} \text{ or } \cos \theta = -\frac{1}{2}\)

For \(\cos \theta = \frac{2}{3}\), \(\theta \approx 0.841\).

For \(\cos \theta = -\frac{1}{2}\), \(\theta \approx 2.09\).

(i) The terms of the arithmetic progression are \(2 \sin x\), \(3 \cos x\), and \(\sin x + 2 \cos x\). The common difference \(d\) is given by:

\(3 \cos x - 2 \sin x = (\sin x + 2 \cos x) - 3 \cos x\)

Simplifying, we have:

\(3 \cos x - 2 \sin x = \sin x - \cos x\)

Rearranging gives:

\(4 \cos x = 3 \sin x\)

Thus, \(\tan x = \frac{3}{4}\).

(ii) Using \(\cos x = \frac{3}{5}\) and \(\sin x = \frac{4}{5}\), we find:

\(a = 2 \sin x = 2 \times \frac{4}{5} = 1.6\)

\(d = 3 \cos x - 2 \sin x = 0.2\)

The sum of the first 20 terms is given by:

\(S_{20} = \frac{20}{2} (2 \times 1.6 + (20 - 1) \times 0.2) = 70\)

The first term \(a_1 = 1\) and the second term \(a_2 = \cos^2 x\). The common difference \(d\) is given by:

\(d = a_2 - a_1 = \cos^2 x - 1\).

The sum of the first ten terms \(S_{10}\) of an arithmetic progression is given by:

\(S_{10} = \frac{10}{2} (2a_1 + 9d)\).

Substitute \(a_1 = 1\) and \(d = \cos^2 x - 1\):

\(S_{10} = 5 [2 \times 1 + 9(\cos^2 x - 1)]\).

Simplify the expression:

\(S_{10} = 5 [2 + 9\cos^2 x - 9]\).

\(S_{10} = 5 [-7 + 9\cos^2 x]\).

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

\(S_{10} = 5 [-7 + 9(1 - \sin^2 x)]\).

\(S_{10} = 5 [-7 + 9 - 9\sin^2 x]\).

\(S_{10} = 5 [2 - 9\sin^2 x]\).

\(S_{10} = 10 - 45\sin^2 x\).

Thus, \(a = 10\) and \(b = 45\).