Periodicity is a big tongue twister that means, "This thing is periodic." That seems like a pointless word. And we already know that trig functions are periodic. So, lesson over? That seems a little pointless, too.
Sorry, not quite. We've only scratched the surface. We know that periodic functions repeat themselves over and over, just like a circle goes around and around. Hey now, isn't that a little too on the nose? The unit circle has a circumference of 2π, which is also the period of sine and cosine. Any x value that is greater than 2π is going to be in the same position as x - 2π:
sin (x – 2π) = sin x
cos (x – 2π) = cos x
It works the other way, too:
sin (x + 2π) = sin x
cos (x + 2π) = cos x
Translation: moving 2π in any direction is going to bring us full circle (pun intended) to where we started. The reciprocals of sine and cosine, cosecant and secant, also repeat themselves with a period of 2π. We can't say we're surprised; those -secants were always copy-cat tag-alongs of the -sines.
Tangent and it's reciprocal, cotangent, also have periodicity, but they have a period of π instead:
tan(x + π) = tan x
tan(x - π) = tan x
Sample Problem
What angle between 0 and 2π is coterminal with an angle measuring 875°?
We like radians, we're really good friends with them, but sometimes we miss our old friends, the degrees. So we're going to start this problem out using them.
One full circle is 2π, or 360° around, and 875° is definitely bigger than that. Let's pare this down by a coterminal angle or two. Taking away one circle from 875° gets us:
875° – 360° = 515°
515° is still bigger than 360°; let's take another circle away.
515° – 360° = 155°
Now it's small enough to fit in our circle. Okay, goobye, degrees. See you later. Now is the time for radians.
Any trig function will have the same value for this angle as they would for the original massive angle we started with. This might be helpful as we use trig functions down the line. Just maybe.