Question

Which of the following represents the measure of all angles coterminal with 418° angle?

Answer 1

The correct answer is:

58+360n, for any integer n.

Explanation:

412° is more than 360°, which is a full circle.  To find the smallest angle that is coterminal with 412°, we subtract 360° from it:
412-360 = 58°

This means that 58° is the smallest coterminal angle with 412°.

Every whole circle added on to this will also be coterminal with it.  Again, a whole circle is 360°; this means that every 360° increment that is added to 58° will be coterminal with it.  

To represent multiples of 360, we write 360n.  Adding this to our original angle, we have 58+360n.

This works for positive and negative numbers; for example, if you use n = -1:

58+360(-1) = 58-360 = -302

Graphing this angle, we would go counterclockwise from the positive x-axis; we would land at 58°, so it would still be coterminal with 412°.

Answer 2

418° is the same angle as 360° + 58° .

And then, if you add any whole number of 360s onto it,
you get the same angle again.

So it's    58+360n  with n=any whole number 
                                 (n=0 doesn't work, so it's not 'integer'.  '0' is an integer.)