Express \(4 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
Hence find the smallest positive value of \(\theta\) satisfying the equation \(4 \sin \theta + 4 \cos \theta = 5\).
Solve the equation
$$4 \cot 2 x = 5 + \tan x$$
for \(0 < x < \pi\), showing all necessary working and giving the answers correct to 2 decimal places.
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