Answer: \(c=9\) or \(c=-1\).

The key point is to differentiate first, then substitute the required value or solve the resulting equation.

A horizontal line \(y=c\) is tangent to the curve \(y=5\sin\frac{x}{3}+4\) when it touches the curve at a maximum or minimum value.

Since

\(-1\leq \sin\frac{x}{3}\leq 1,\)

the maximum value of the curve is

\(5(1)+4=9,\)

and the minimum value is

\(5(-1)+4=-1.\)

Therefore the possible values are

\(c=9\quad\text{or}\quad c=-1.\)