(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:

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