Notice that the teacher first has them recall the period for cos(x) before discussing cos(5x). This will give the opportunity to see how the 5 affects the period.
He also points out how much larger the arclength is for cos(5x). The student realizes it is 5 times as large, therefore it will take 1/5th of the time to make a revolution. This is a nice way of incorporating arclength into the discussion.
