Answer: \(a=3\), \(b=\dfrac23\), \(c=-4\).

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 \(-7\).

The amplitude is half the distance between the maximum and minimum values:

\(a=\frac{-1-(-7)}{2}=3.\)

The midline is halfway between \(-1\) and \(-7\):

\(c=\frac{-1+(-7)}{2}=-4.\)

The graph has one full period of \(540^\circ\).

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

So

\(\frac{360}{b}=540.\)

Hence

\(b=\frac{360}{540}=\frac23.\)

Therefore

\(a=3,\quad b=\frac23,\quad c=-4.\)

This completes the solution and gives the required result.