a = 4 m

b = 3 m

c = 12 m

d = the space diagonal, d>0

x = the face diagonal, x>0

We need to find sine(<1).

sine(<1) = x/d

The space diagonal is calculated by the formula:

d^2 = a^2 + b^2 + c^2

d^2 = 3^2 + 4^2 + 12^2

........

calculation of the space diagonal

........

d = 13 m

The face diagonal is calculated using the Pythagorean Theorem:

x^2 = a^2 + b^2

x^2 = 3^2 + 4^2

..........

calculation of the face diagonal

..........

x = 5 m

Now we go back to calculating the sine of the angle that a diagonal of the box makes with its longest edge:

sine(<1) = 5/13

sine(<1) = 0.38461538461