Hey, everyone. You may remember learning that some functions can be classified as being either even or odd based on the symmetry of their graph. Well, here we're going to specifically take a look at our trig functions and whether they're even or odd. Now, why that's useful might not be immediately apparent to you. But if we know that some function f(x) is even or odd, then we can easily find and simplify f(-x), which is something that we'll have to do more and more as you continue to work your trig problems. So here, I'm going to break down for you exactly which trig functions are even and which are odd and then use that in order to simplify some expressions. So let's go ahead and get started.

Now, let's start by taking a look at our cosine function and specifically the value of our cosine function at π2. Now, here on my graph, I see that f(π2) is equal to 0 and if I go to the other side of my graph and check out the value of my function for -π2, I see that f(-π2) is also equal to 0. So relating these two points to each other, I could say that f(-π2) is equal to f(π2) because they are the exact same value. But I can actually generalize this for the entire function. See f(-x) for this function is always going to be equal to f(x). So if I flip the sign of my input, like I saw here for -π2, the sign of my output remains the same. And that's because my cosine function is an even function.

Now, you might also hear even functions talked about in terms of their symmetry. And even functions are specifically symmetric on the y-axis, which just means that if I took my graph and folded it along that y-axis, all of my points would match up with each other on either side because it's symmetric about that line. Now we've seen that our cosine function is even. But let's take a look at our sine function over here.

Now, looking at our sine function again at a value of π2, I see that f(π2) is equal to positive 1. Then going to the other side of my graph, looking at f(-π2), I see that f(-π2) is actually equal to negative one. So relating these two points to each other, I could say that f(-π2) is equal to -f(π2) because my value for f(-π2) was negative 1, and for positive π2 was positive 1. Now again, we can generalize this for our entire function. So f(-x) here, we see, is always going to be equal to -f(x). So, whenever we flip the sign of our input value to -