Hey everyone. So up to this point, we've spent a lot of time dealing with the sine and cosine functions, and we've discussed their graphs and different ways that the graphs can be transformed, like being stretched or shifted in some kind of way. Well, in this video, we're going to learn that we can graph more trigonometric functions such as the secant and cosecant. And this might sound a little bit scary at first, but don't sweat it because it turns out we can actually use the graphs that we learned about for sine and cosine to graph the cosecant and secant. So without further ado, let's get right into this.

Now recall that the cosecant and secant are reciprocals, they're reciprocal identities for the sine and cosine. So cosecant is 1sin and secant is 1cos, and you can use these relationships to figure out what their graphs look like. So, I'm just going to jump right into the cosecant graph. And if I go ahead and take the reciprocal for all these outputs that I see, the reciprocal of 1 is just 1. The reciprocal of negative one is negative one because 1 divided by negative one would be negative one, and then the reciprocal of 1 is just 1 there. So that means at an x-value of π2, we'll have 1. For 3π2, we're going to have negative 1. And then for 5π2, we're going to have 1 again.

Now I'm also going to need to take the reciprocal of these values, but notice that all of these values are 0. And if I take the reciprocal of 0, that's just going to be 10. But this is actually a problem, because recall that in math, it's a fundamental rule that we cannot divide by 0. So what does this mean? Well, that means that every place we see 0 for our sine value, our cosecant is going to be undefined. And for undefined values, what we're basically saying in this instance is that they're approaching infinity. And since infinity is not a number, we can't define it. So what we can do is we can take asymptotes and draw them where we see the sine function reach 0. So, we're going to have an asymptote at an x-value of 0. We'll have another asymptote at an x-value of π, and we'll have another asymptote at an x-value of 2π.

Now from here, I need to figure out how the rest of the graph is going to behave. And the way that I can do this is by simply observing how the sine function behaves. Notice how the sine function gets smaller and smaller as we go to the left and as we go to the right. And what happens if you take the reciprocal of a number that gets smaller and smaller? Well, the whole fraction is going to get bigger and bigger. And because of this, our cosecant function is going to blow up as we go to the left and to the right. So we in essence end up with this kind of smiley face thing happening right here, and it turns out that this logic stays true for all the other points. So notice how we end up with these smiley and frowny faces where we have the peaks and valleys respectively.

Now let's also take a look and see how the secant function behaves. Well, I can use the same logic by taking the reciprocal of all these values for the cosine. The reciprocal of 1 is 1, and then we have the reciprocal of negative one and the reciprocal of 1. So going to our graph, our points are going to be at 1 at the peak there, at negative one for the valley at an x-value of π, and then at another peak, we're going to have 1 as well. Now because secant is a reciprocal of the cosine, all the places that we see 0 for the cosine are going to be undefined for the secant. So when doing this we can actually draw asymptotes at all these undefined values. So at an x-value of π2, we're going to have an asymptote. At 3π2, we're going to have another asymptote. And then at 5π2, we're going to have another asymptote. Now just like we saw with the cosecant, this graph is a reciprocal of the cosine. So just like with the cosecant graph, we're going to see these kind of smiley faces and frowny faces where the peaks and valleys are. So notice how these two graphs end up looking very similar. The really key difference between these two graphs is where the asymptotes are located. Because notice for the cosecant, we ended up with asymptotes at 0, π, 2π, basically just integer multiples of π. And for the secant, we ended up with asymptotes at π2, 3π2, 5π2, or basically just odd multiples of π2.

Now something that's really nice about the cosecant and secant graphs is it turns out that you can use all the same transformation rules that you use for the sine and cosine. So all the stretches and shifts for the cosecant and secant function that we learned about are going to hold true. And to see this, let's actually take a look at an example.

In this example, we are asked to graph the function y = 1sin(2x). You may be a little bit curious where we need to start with this example or how we could even do this, but what I personally like to do when dealing with these types of problems is to first figure out what I know and then build off of that. Now one of the things I know about the cosecant is I know the cosecant is a reciprocal of the sine. And so what I'm actually going to do is first find the graph of y = sin(2x). Now this is not the graph that we're looking for, but this will give me an idea as to what the cosecant is going to look like since I know their relationship. So recall that for sine of 2x, well, we first need to figure out what the period is. And the period for a sine or cosine graph is 2πb. Now the b value that we have in front is 2. So we're going to have 2π2, where the twos will cancel and give us π. So that means that we're going to have a sine graph that has its period at π. So what that means is that our sine graph will start at the origin, and we're going to reach a peak right about there, then we__*/