 Another important aspect when it comes to graphing of functions is the recognition of symmetry. Does our graph have some type of symmetry to it? Is it reflective with respect to the, is it symmetric with respect to the waxes? Is it symmetric with respect to the origin? Well, trigonometric functions are periodic, which is a type of symmetry. But in terms of like reflection across a line or a point, it turns out that trig functions have these type of symmetries as well. So we say that just as a reminder, an even function, that's a great, great grammar right there, an even function is a function with the property that f of negative x is equal to f of x. Now, geometrically what we're saying here is f of negative x is equivalent to reflecting a function across the waxes. And then f of x, of course, is the original function. So if f of negative x is equal to f of x, that tells us that the function is symmetric with respect to the y-axis. We call that an even function. And the reason for that is that the poster child of an even function is f of x equals x squared, or you can take any even integer for the power of x right here. Because notice if you take f of negative x, this is going to be negative x quantity squared, which this means negative x times negative x. You get a double negative, which actually then becomes a positive and you get back f of x. So putting that negative inside the function doesn't change anything, it's symmetric with respect to the x-axis. An odd function, again, if we correct that grammatical mistake there, sorry, an odd function is a function with the property that f of negative x is equal to negative f of x. So an odd function, there's two ways of thinking about it. A reflection across the y-axis is the same thing as a reflection across the x-axis. That's what the formula is telling you. But another way of thinking about it is a function is odd if it's symmetric with respect to the origin. Being symmetric with respect to the origin means that if you take a point and you go through the origin, you'll find that point on the other side. Or in other words, you could take a half spin. If you rotated the graph around the origin by half speed, that would give you the exact same picture again. Let me give you an example of an odd function. Let's take, for example, y equals x cubed. Its graph looks something like the following. If you were to take this graph and a half spin around the origin, you'll get the exact same graphing in. Algebraically, we can see that if g of x equals x cubed, then g of negative x, this is negative x cubed. If you take an odd power of a negative, it'll still be negative. It's like a triple negative there. So you get negative x cubed, which is g of x. So this is an example of an odd function. And again, just to complete our discussion here, for an even function like x squared, you get something like this. You see that the graph is symmetric with respect to the y-axis. So this is odd functions and even functions. You've possibly seen this before as you studied graphing functions in an algebra class. What does this have to do with trigonometry? Well, it turns out that the trigonometric functions are symmetric as well. Cosine is an example of an even function. And sine is an example of an odd function. What do I mean by that? Cosine is even. So the graph of cosine is going to be symmetric with respect to the y-axis. As a formula here, cosine of negative theta is just equal to cosine of theta. So if you think of the unit circle for a moment here, here's our x-axis, here's a y-axis. If you were to rotate your angle counterclockwise, like we generally do. If you did that same angle measurement but clockwise, it turns out here. So these are the same angle theta right here. The x-coordinate doesn't care about that. This would be x comma y. This one here would be x comma negative y. So if you do a clockwise rotation versus a counterclockwise rotation, the x-coordinate doesn't care. The x-coordinate will be the same, irrelevant whether you rotate clockwise or counterclockwise. And so this tells us that cosine of negative theta is equal to cosine of theta. So cosine is an even function. If we say the same thing for sine, right? If you rotate your angle theta clockwise or counterclockwise, the x-coordinate stays the same, but the y-coordinate will be different, right? If you go up, you're going to have a positive y-coordinate. If you go down, you'll have a negative y-coordinate. So we see that if you take sine of negative theta, so that's rotating clockwise, that actually sticks a negative sine out in front, so you get negative sine of theta. So sine is an example of an odd function. Sine of negative theta is equal to negative sine theta. And so these equations right here, I'm going to draw a box around them to emphasize them, these equations right here, we can add to our list of trigonometric identities. We can call these the symmetry identities. The symmetry identities tell us that cosine is even and sine is odd, or more specifically, cosine of negative theta is equal to cosine theta and sine of negative theta is equal to negative sine theta. Now, using these symmetric identities, we can say things about the symmetry of the other trigonometric functions. What about tangent, for example? If I take tangent of negative theta, this will be sine of negative theta over cosine of negative theta. Well, sine of negative theta, since it's odd, it'll stick out at negative sine. Sticking out your tongue is a very odd thing to do. Same thing with your negative sine there. It'll stick its negative sine out, that's an odd thing to do. Cosine, on the other hand, of negative theta will just be cosine of theta. So you get negative sine theta over cosine. Sine and cosine is just tangent, so you end up with a negative tangent. So this tells us that tangent is an odd function, because it sticks out that negative sign if you put into it. By similar reasoning, cotangent is an odd function, and I should mention my drawing is kind of covering it. But secant is even and cosecant is odd. Secant and cosecant are just the reciprocals of cosine and sine. And so if cosine eats the negative sine, so will secant. And if sine sticks out the negative sine, so will cosecant as well. So if we summarize what we've learned here so far, sine is an odd function. It spits out the negative sine. Cosine is even. Tangent is an odd function, so is cotangent. Tangent and cotangent are both odd. Secant and cosecant will mimic the reciprocal. Secant is going to be even, and cosecant is going to be odd, just like sine. So we have these symmetries here as well. And so this can be helpful when we try to calculate trigonometric ratios that involve negative angles, that is, if we have a clockwise rotation. How would you deal with something like cosine of negative pi sixth? Well, cosine's an even function, so cosine of negative pi sixth, if you ignore the negative sine, and you just get cosine of pi sixth, which, when we look at our unit circle diagram, pi sixth or 30 degree angles, a 30 degree angle, that coincides with root 3 over 2. So that's cosine of negative pi sixth. What about sine of negative pi sixth? Well, because sine is an odd function, that negative sine comes out, and so you get negative sine of pi sixth. Sine of 30 degrees or sine of pi sixth is one-half, and so you're going to end up with a negative one-half right here. And then let's just do one more here. If you want to do cosine of negative 2 pi thirds, well, in terms of reference angles, the identity, the symmetric identity says you can just erase the negative sine when it comes to cosine. We just have to do cosine of 2 pi thirds. Cosine of 2 pi thirds, that's in the second quadrant, 2 pi thirds. So in the second quadrant, cosine is actually negative, so the negative sign did come out, not because of the symmetry, but because of the quadrant. So you get negative cosine of pi thirds. Pi thirds is the reference angle there, and cosine of pi thirds is one-half, so you end up with negative one-half as well. So the symmetric identities allow us to compute sine, cosine, tangent, any of the trig functions when we have negative angles, clockwise angles, because sine is odd and cosine is even.