Let the angle of the smallest sector be \(a\) and the common difference be \(d\). The angles of the sectors are \(a, a+d, a+2d, a+3d, a+4d\).

Given that the largest angle is 4 times the smallest angle:

\(a + 4d = 4a\)

\(3a = 4d\)

The sum of the angles in the circle is 360°:

\(360 = S_5 = \frac{5}{2}(2a + 4d)\)

Substitute \(4d = 3a\) into the equation:

\(360 = \frac{5}{2}(2a + 3a) = 12.5a\)

\(a = 28.8^{\circ}\)

The largest angle is \(a + 4d = 4a = 115.2^{\circ}\).