Answer: (a) \(a=5,\ b=\dfrac34,\ c=-4\); (b) \(p=1\) or \(p=5\).

We start with the main method. Use the measurements and relationships shown in the diagram, then translate them into algebraic or trigonometric equations. Use amplitude, period and vertical shift to identify the key points of the graph.

From the graph, the maximum value is \(1\), and the minimum value is \(-9\).

The amplitude is

\(a=\frac{1-(-9)}2=5.\)

The midline is

\(c=\frac{1+(-9)}2=-4.\)

The period shown by the graph is \(480^\circ\).

For \(y=a\cos bx+c\), the period is \(\frac{360^\circ}{b}\).

So

\(\frac{360}{b}=480,\)

giving

\(b=\frac34.\)

For part (b), the curve

\(y=3-2\sin6\theta\)

has minimum value \(3-2=1\) and maximum value \(3+2=5\).

A horizontal tangent \(y=p\) occurs at these extreme values.

Therefore

\(p=1\quad\text{or}\quad p=5.\)

This completes the solution and gives the required result.