Make a drawing: your drawing will look like an isosceles triangle such that the matching sides will each be 3 inches (the legs of the compass) and the remaining side (on the paper) will be 5 inches, for the diameter of your circle.

You can find the angle you need by calculating with half of your triangle.

Draw a line from the angle you want to the opposite side to make two identical right triangles. Each will have half of the angle you want, a 3 inch hypotenuse, and the base (opposite the angle you want) of 2.5 inches.

Right triangle rules say that sine (x) = opposite side length/hypotenuse length so:

sin(x) = 2.5/3

x = arcsin(2.5/3)

x = 56.44 degrees (approximately)

since x is 1/2 the angle you want, we find that the angle is: 2x = 2*56.44 = 112.88 degrees.