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.
This Looks Odd To Us
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.
Oh, By the Way…
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.