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Tick tock, tick tock, up and down, back and forth. Trig functions truck along, keeping time to the beat better than any metronome. There's a lot of repetition going on, and we can use it to our advantage when solving problems. It's time to talk about the periodicity and symmetry of trig functions. We promise not to repeat ourselves too much.
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
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.
We tend to think about symmetry in terms of geometry more than anything else. That's understandable; it's easy to fold shapes in half in different ways, to see if they match up. Well, functions can have symmetry too, and trig functions are like the sumo symmetry champs.
There are two types of symmetry when we look at trig functions. Let's start by looking at y = cos x.
It has symmetry over the y-axis. We could fold the whole graph over x = 0 and everything would match up on the other side. Any function that is symmetric over the y-axis is an even function.
We don't have to look at a graph to show that a function is even. Stick a blindfold on us, we don't care. We can instead see if the function fits this equation:
f(-x) = f(x)
Take off the blindfold and take another look at the graph. Compare every -x value to x: they have the same value of y. That's an even function's symmetry, and that's exactly what the equation says. That, and "Feed me, Seymour," but we're not listening to that old song and dance.
We can check that cosine fits the equation by looking at the unit circle. To do this, we remember the memory device, ASTC. Some people say All Students Take Calculus, but we like to think about it as A Small Tangled Cat. There's so much yarn, how are you going to get out, cat? Oh, it also tells us which functions are positive in each quadrant:
A(ll) S(ine) T(angent) C(osine). Here are the signs for cosine (and secant) for the angles a and -a:
The absolute value of cos a and cos (-a) will be the same, because they have the same reference angle. Plus, Quadrants I and IV are both positive for cosine. That translates into cos a = cos (-a). That's totally what we wanted, and we got it.
If there exists something called "even" functions, is it really any surprise that there are odd functions as well? Pick your jaw up off the floor; it will get dirt on your chin. Odd functions have symmetric over the origin.
Turns out that y = sin x is an odd function. The shock and surprise continues.
If we take this graph and fold it over the y-axis, we can see that it doesn't match up, so it is not symmetric over the y-axis; it is not an even function. However, if we compare any x and -x, their y values are opposite: y and -y. This is symmetry over the origin, where we are flipping over both the x- and y-axes. It has a handy formula too:
f(-x) = -f(x)
For sine (and its reciprocal, cosecant), we have this breakdown of signs:
We can see that angle a is in the positive territory. Going in the opposite direction, -a is in negative territory. They've got all the territories covered. Knowing this, we can see that sin(-a) = -sin a, which is just like the f(-x) = -f(x) of an odd function.
We're on another tangent for tangent (and cotangent). It has a different distribution of signs than sine or cosine.
However, checking out the function values for a and -a, we see that works the same as sine. tan (-a) = -tan a, so it is an odd function. We always knew tangent was odd, but now we have proof.