Periodicity: it's oh-so-polysyllabic, but oh-so-simple. Just think of a unit circle with a moving line. That's it. Since our unit circle has a radius of 1, the sine function can be represented by a line.
Or think of it this way: sine is always the opposite side over the hypotenuse, and the hypotenuse is always 1 in the unit circle (the hypotenuse is the radius). Anything over 1 is just itself (like 2/1 = 2), so when we're chilling inside the unit circle, the sine is just the length of the opposite side.
Let's see what happens when we slide our angles around a bit.The sine line function is highlighted in red below.
As our angle ɵ swings from 0 to 
(that's 0° to 90° for all you degree-lovers out there), sin ɵ swings from 0 to 1.
Next, rotate ɵ from to π.
This time, sin ɵ swings from 1 to 0.
Similarly, as we swing from π to 
, sin ɵ swings from 0 to -1. (From flipping to swinging—now it's getting interesting, if not dizzying.)
And as we swing from 
to 2π, sin ɵ swings from -1 to 0. Check it out:
We can see from our unit circle that the amplitude (maximum value) of y = sin x is 1, and the period (time it takes for one full cycle) of y = sin x is 2π. After that, the whole process repeats again, which means the sine function is periodic. It repeats itself. It repeats itself.
The cosine function is another periodic function.
We can follow it by using the unit circle again. Since cosine is always the adjacent side over the hypotenuse, and since the hypotenuse is always 1 inside the unit circle, the cosine of any angle in the unit circle is just the length of the adjacent side.
The cosine line function is highlighted in red.
As ɵ swings from 0 to 
, cos ɵ swings from 1 to 0.
As ɵ swings from 
to π, cos ɵ swings from 0 to -1.
Similarly, as ɵ swings from π to 
, cos ɵ swings from -1 to 0.
And as ɵ swings from 
to 2π, cos ɵ swings from 0 to 1.
We can see from our unit circle that the amplitude (maximum value) of y = cos x is 1, and the period (distance it takes for one full cycle) of y = cos x is 2π.
To sum things up, that means the sine or cosine of any angle will always have a value between -1 and 1, no matter how big or small the angle is.