Hi everyone, and welcome to our lecture on right triangle trigonometry. So let's, I guess, start from the beginning. Maybe it's a review, maybe it's brand new, I don't know, but let's make sure we have all the words written down. First up, we're going to be using right triangles. As a friendly reminder, a right triangle is one with a right angle, that's 90 degrees or Pi over 2 radians. What I was going to talk about, one angle that we're going to denote Theta, that's going to be the angle we're talking about, and the sides all have good names. The side that is next to the angle is going to be called the adjacent side. The side that, if you can imagine a little laser gun shooting across from the angle, that will give us the opposite side, and then the side that is opposite the right angle is called the hypotenuse. So our triangle has three sides; adjacent, opposite, and hypotenuse. The hypotenuse is always determined, that's always opposite the right angle, and then determining what angle you're talking about, everything is in reference to that angle. So we'll always be very clear what angle we're talking about, so we'll say the opposite side to the angle Theta, the adjacent side to Theta. So we have these nice triangles. This is all great and you can imagine there's lots of beautiful things in triangles in architecture, in nature, and the sort of thing. So again, put yourself back in ancient Greece. You want that which is beautiful. So what are Greeks really good at? Well, I guess besides sides, they want ratios of sides. So we're going to look at the ratios of the sides. So we'll do the first one together. So we want the opposite over the hypotenuse, we'll abbreviate that with opp and hyp, opposite over hypotenuse. You say, "Why ratios?" Well, we like things that are pretty. This is something if you look at architecture, you like things that are in different scales of three to two or one-to-one relationships. We want ratios of things, and if you grow or scale a picture, you want to preserve this ratio. So we want ratios, ratios are a fancy way to say a fraction if you're talking about two numbers. So I really want these fractions of these sides. This ratio, when you fix an angle in a right triangle, it's important that you start with a right triangle, has a name and this is called the sine of the [inaudible] So it's spelled S-I-N, but it's pronounced sine, not sin. In sine we use parentheses of the angle. So that ratio is called sine. Same thing, you can create another ratio, there's three sides here, so you get to pick and choose what you want. If you choose the adjacent over the hypotenuse, so the adjacent side over the hypotenuse, that is called the cosine of the angle. I know it's written C-O-S, we will say cos. Some people say that, but I think they're wrong. So if you want to say cosine of the angle, cosine, use your big words. Then last but not least, you can create another ratio called tangent. So we write T-A-N of Theta, and this is going to be opposite over adjacent. These three functions, this is trigonometry, we're getting into it. But they're periodic functions, they're going to repeat. So they fall into this category. Right now, before, when people see these words, they get terrified. They're like, "Well, I'm out. It was fun." Like cartoonish, like person jumping out the window when the car starting. We're going to take this nice and slow, and I'm going to try to avoid the scary stuff right now. If you're okay with the triangle, you should be okay with these things. We have a triangle, we fix an angle, and then we just create fractions, just make some fractions. If you're with two and you're cool with three, and you shouldn't be scared of two-thirds, this is the idea. So you want to have these ideas that they're just fractions, they are literally just fractions. Yes, they have fancy names, but we are a fancy folk, and so we will use fancy name, sine, cosine, and tangent. This of course though, however, whether you're done, may not love sine, cosine, and tangent, it has an amazing acronym, SOH CAH TOA, you've heard this before. Some people yell it. I used to have a teacher that screamed it obnoxiously SOH CAH TOA. SO CAH TOA is just there to help remind you the ratios. So we have, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, so SOH CAH, really stress the A there, and then TOA, tangent is opposite over adjacent, which is just there to help you out. But the key with this, remember the steps here. Again, treat this more like a recipe, I can't cook, but I can follow a recipe. Remember the steps here. Step 1 is you fix your angle Theta, fix Theta and then whatever that is, then you make a right triangle. So I'll say in a right triangle, I have to stop for a minute. There's a joke here that I'm not saying and I'm debating in my head if I should say it. I'm going to say it. What do you call a triangle that isn't right, or wrong triangle? There you go. I said it, it's on record. I'll probably regret that. So what did we do here? We fix an angle in a right triangle and then we form ratios. It's like a two-step process. We just form ratios and depending there will be a time and a place for each one, whatever it is, you're not scared and remember why not. Let's write it one more time so we can say it, SOH CAH TOA, sine is opposite over hypotenuse, cosine, adjacent over hypotenuse, tangent, opposite over adjacent. Now, the skeptic in you maybe saying, "Wait a minute, what if I started with a different triangle, the new? " So you drew some triangle, [inaudible] big, some right triangle, you put Theta off in the corner and you formed your ratios. But I want to draw a bigger triangle, a much bigger triangle. I'll try to make huge thing. Yes, sure. I'll have the same angle that we're talking about, but I really want things that are different. So just to make up some numbers here, let's say we have like 3, 4, and 5. That is a legitimate sides of a triangle. I'm going to make my other triangle twice as large. So let's say 6, 8, and 10, whatever, three times as large, four times large, who cares. The point is I want different size triangles. You see, does this all still work? Doesn't it matter what size triangle you start with? Well, the idea is that it turns out that it doesn't. Why? So let's just look at one of them together. So let's look at sine of our angle Theta, whatever that may be. This ratio sine from SOHCAHTOA opposite over hypotenuse, our ratio for this one would be 4 over 5, four-fifths, for our smaller triangle at 3, 4, 5 triangle. One that's twice as large or sine over here on the second example, still the same thing, the same rule apply, you don't change the ratio. Sine is opposite over hypotenuse, so we would have 8 over 10. The brilliance there is that, of course, that 8 over 10 reduces to good old four-fifths. So however you scale your triangle, and this is the video fractions, you get a fractions with different numbers, but they can represent the same thing. So 1.5 is two-fourths, four-fifths is the same as eight-tenths. So if you scale the triangle larger, as long as you preserve the angle that we're talking about, the ratios will be preserved. This is mildly amazing, and this property, if you remember this, this is called similar triangles. They turn out to have lots of things in common, that's why they get a name. The picture though, you normally don't see the several ones like I drew one left on the right, usually you see them in pose in the same triangle. I'll give you the picture that normally you see. See I've one big triangle with a right angle. You put Theta off in one of the angles, let's sought the right angle, and then you draw your second triangle, maybe it's a little smaller, whatever it is, but as long as it's a right triangle and it shares the same angle, it turns out that they will be similar. The beautiful thing about similar triangles is that ratios of their sides are equal. So this is why it does not matter the size of triangle that you want. This is the beautiful thing about it. You just want to be talking about the ratio of their side, sine of the angle, cosine of the angle, or tangent of the angle. It's all the same. It's nice. So ratios are defined for any right triangle, this is the nice thing. So it does not matter, pick your favorite one. It turns out in math that we do, I know one thing is we try to make life very difficult, but once in a while we try to make life easy. So since you can pick any right triangle, let's pick our default to be an easy one. So what does easy mean in this mathematical context? Well, we're going to live inside the unit circle, what easy means for us here is that we're going to be inside the unit circle. Now this is not obvious, this is a smooth move that math plays on here. It's like the study triangles, we're going to use another object. It's like I want to know everything about the triangle I have put in circle. You got to pause for a minute there and be like, "What?" If you ever cooking or something, there's some ingredient that just comes out of nowhere, like now use curry powder, "Well, I didn't see that coming." Certainly that. So you're going to take a triangle and you're going to dump it inside a circle. This will pay dividends later. It's on an obvious move, but is a clever one, a little subtle. Further reminder of the unit circle is the circle centered on the x, y plane right at the origin, and it has a unit circle of radius one. So its points, its radius is one. So if you want to label the points of its intercepts, you have negative one, one all around the circle. We want the angle that we study to have its vertex right at the origin as well and its initial side on the x-axis. So whatever triangle we're drawing, the terminal side will come right on the circle. So whatever triangle we are drawing, we're going to inscribe, we're going to embed it inside of the unit circle. So the key here is that its hypotenuse will be the radius. So what do we want to know? The hypotenuse to be the radius, we want our angle, our initial angle on the x-axis, and the beautiful thing about this is the unit circle radius is one. So we already are fixing our radius to be one. That's the value right there. If I can have any triangle I want, why don't I pick hypotenuse of one instead of say, I don't know, 14.5 or something like that, one is a nice number, we like one. That's the benefit of putting inside the unit circle. There's another benefit too, and that's if you noticed, the initial side, it starts from zero. Its length goes along the x-axis. Here we have a nice variable for that. What's a variable that measures your distance away on the x-axis? Well, that's just x. The height of the triangle, the opposite side, is the height off of the x-axis. Well, we also have a variable for that, that's just Y. So we have this nice triangle, let me just zoom out for a minute here, let me draw a little bigger on the screen, we have a nice triangle inside the unit circle where the base is X, the height is Y, and the radius is one, and these are just natural variables to use and again, picking my hypotenuse to be one is just nice number. This is all inscribed inside of the unit circle. So the point on the unit circle is X, Y and we got our X value and our Y value. This will turn out to be quite nice. Friendly reminder, the unit circle has an equation and it's X squared plus Y squared is one and you should see that equation in two different ways now, this is neat. So yes, it's an equation of a circle. So X squared plus Y squared is one, but where's it coming from? What is this thing? This is the Pythagorean theorem on this triangle. So now you flipped it a little bit, you said you want to study the circle, I want to study triangles. So these two things, circle's nice and rounded object but triangle's just a rigid thing with a right angle, they're really used to say lots of things about each other. So Pythagorean theorem will tell you that X squared plus Y squared is one squared, put it all together, you get X squared plus Y squared is one. So that's a nice relationship that just immediately pops out of this diagram. Now to the task at hand, within this context inside of the unit circle, let's define our sine of Theta, our cosine of Theta and then we'll also do tangent of Theta just to define it. Remember what these are, is just a practice. We have SOH CAH TOA, why not? Let's put that on the side, SOH CAH TOA. So we want opposite over hypotenuse for sine, cosine of Theta is adjacent over hypotenuse, tangent of Theta of course, is opposite over adjacent. These are how these things are defined, we're picking this super nice triangle inside the unit circle, hypotenuse is one, it's beautiful, couldn't get any better, I suggests it cannot. So what does it mean in this context? What is the opposite side to Theta? That's Y over the hypotenuse, it's Y over one, oh my goodness, Y over one, that's just Y, that's amazing. So now we took this difficult concept that everyone freaks out about like "Sine, what is that?" No, it's just the Y value, literally just the Y value. Tell me what the Y value is at this point. I can do that, maybe it gets harder. Adjacent, the adjacent side is just the X value at this point, the hypotenuse is one, again we like one, it's nice, and that's just X. Oh, my goodness, so this is amazing. So on the unit circle, even these are even worth putting off on the side here. Sine of Theta is your Y value and cosine of Theta is your X value on the unit circle, that's pretty neat. Tangent of Theta is opposite over adjacent, so we just put opposite over adjacent, this is y/x. That's nice, but it's a fraction, they all can't work out beautiful, but it's pretty good. We're going to focus mostly a sine and cosine as we move things along, but just remember that tangent is y/x as we go through, there's more out there than just sine and cosine, but I think understanding sine and cosine to start is a great foundation. So now let's try to find some values of things that we want and we'll start collecting these values in the table, and once we have a table, we never have to re-derive them, which is nice. So whenever I want to study some angles here and some values, I'm going to draw the unit circle, centered at the origin of radius one. I'm going to draw my little triangle inside of the first quadrant here, so it's inscribed of course, it's a right triangle. Hypotenuse is the radius, so that's one. I have my point on the circle X, Y, which is corresponding to my adjacent side and my opposite side, and don't forget, floating on the background is the Pythagorean theorem or the equation of a circle, however you want to view it, X squared plus Y squared is one. So let's get a couple of values from these and see what we can do and we'll collect them all on the table. So let's find zero, let's say what is sine of zero, now the beautiful thing about here is I can say zero degrees or zero radians. Again, put yourself on the mindset of radians. So what is sine? So now remember on the unit circle and why not? You can pick any triangle, the sine of some value is the Y coordinate of the point where the angle hits the unit circle. So what is an angle of zero degrees? Well, it's the world's worst triangle. So let's draw one more picture just to be specific here, so I have the unit circle centered at the origin, radius one and I want an angle with zero rotation. This one's like a dumb triangle, the worlds worse triangle. I have my initial ray is equal to my terminal ray. Can I call this a triangle? It's kind of lame. This is basically two lines segments that are on top of each other. This is like zero degrees or zero radians, does not matter. But the key thing that are more importantly than just arguing over it's the world's worst triangle, is realizing where this ray intersects the circle. Right at the X intercept, what point is this? This is X is one and Y is zero, right at 1, 0. So sine of zero degrees, sine of zero radians, maybe you want to put it in there just to be sure is what? Is the Y value. So now in your head you can imagine this little bar on the x-axis and it's not rotating and you say, "Where does it cross the circle?" It crosses it right at 1, 0. What's the y value? I appreciate it, what is sine of zero? If I said to you, "What's the sine of zero?" It's like panic. If said to you, "Hey, I'm thinking of the point 1, 0, what's the y value?" You're like, "Oh, that's zero." No problem. That's absolutely correct. So sine of zero is in fact zero. We can also ask the same question for what is cosine of zero? This is how I want you to do this cosine of zero. So in your head, imagine the unit circle. I want you to imagine the center right at the origin, nice beautiful circle, radius 1. I want you to go out from the center, and go out with no rotation. So just come out on the x-axis and see that point, highlight it, turn it purple, turn it blue, give it a little strobbing there, I don't care. But right at the x-axis, right at 1, 0, I want cosine of that angle which is asking for, what is the x coordinate of that point? That's just one. So that's really nice. So we can start putting these things together, then we'll make a little table here. So degree, maybe we'll do degrees and radians, just so we can used to it. So zero degrees, same as zero radians. If I want sine of the value or cosine of the value, we just found that sine of zero is zero and cosine zero is 1. This is super important here. I don't want you to memorize these numbers. I don't want you to have flashcards. This drives me nuts. When I was in learning this for the first time, we used to have these speed quizzes, where you have to come up and there is rapid fire or there was a 30-second timer, and all you had to do is fill this in. What's the point? There's no point to that. See the circle, visualize it in your head, go out zero degrees. Imagine that point 1, 0 and then ask yourself, "Do I just want the x value or the y value?" Take your time, go slow, and visualize it. This is not call and response. This is understand, interpret, and apply. Really see this thing. If you can get this process, you get so many more values for free, and then you never have to memorize anything, just derive them every time. So for example, let's do cosine of 90 degrees. So we'll do 90 degrees. Again, you really should be thinking in terms of radians, so Pi over 2. So here's what I want you to do. I'm not going to draw, I'm just going to explain it. Close your eyes and picture this. Unit circle, right at the center of the origin, beautiful circle, radius 1. Rotate from the x axis. So maybe even take your left hand and point your fingers nice and straight across your body, and rotate that hand straight up 90 degrees Pi over 2 radians. Ask yourself, pinch your thumb and your pointer finger, pointer finger and pinch that point right at the top of the circle. You're right at the top of the circle, and ask yourself, "What point am I at on the unit circle? I'm x is zero and y is 1. What is cosine of Pi over 2? What's the cosine of 90 degrees? Cosine is the x value. So zero. Sine of 90 degrees? Sine of Pi over 2. Same thing. It's asking for the y value at that point. That's zero and that's 1. This is amazing. We're getting all these values and you can visualize this. I don't you memorize it, I want you to literally move your hand, do a little dance, whatever it takes to figure this out, but we have 90 degrees, and we have sine is 1, and cosine as zero. You can do this for any point. These are really nice to visualize on the coordinates. So maybe we'll do one more. What if I said cosine of 180 degrees, but also that's the same as Pi. So cosine of Pi. A lot of good math right there, cosine Pi. Scary? No. I don't know it. Do you know it? No. Who cares, let's figure it out. Imagine in your head, close your eyes, go to your calm place. Sit on the beach, your yoga pose. Imagine drawing on the sand, a circle with a stick. Everyone's looking at you because you're so smart. Maybe if you want draw the nice arc you want at top of the circle. You're moving along the unit circle and you stop after 180 degree rotation, starting from the point 1, 0. Where do you land on the unit circle? Again, key here is unit so you know exactly how far away from the origin you are. When you rotate around, you land right on the point negative 1, 0. So what is cosine of 180, cosine of Pi? That's the x value, just negative 1. Stay at that point, don't go too far. What is sine of 180 or sine of Pi? It's the y value and that's zero. So you can start filling in and collecting all this information, these pieces here, and you get this nice table of values. You can do it also for 3 Pi over 2. This is your little exercise here. So what is 3 Pi over 2? Sine of 3 Pi over 2 is negative 1. Cosine of 3 Pi over 2 is zero. Go through, imagine that, visualize that, and try to understand that. So let's do one more example. We're going to do another one that's a little challenging. I want to study the angle that is at 60 degrees. So this is one-third of 180, 180 divided by 3 is 60. So we can write that as Pi over 3 radians. If you want to visualize the angle we're talking about, draw the x-y axis, draw a nice unit circle on it, and then I want you to imagine some angle going out on the x-axis. That's our initial side, and then one-third of Pi. This is the first part of the p sign if you want to break this thing up into the p sign here. So we have a nice one-third of our half rotate 180 degrees. So Theta is Pi over three. Now this is going to form a triangle of course, you can form a right triangle, but it's not obvious what x and y are. It's not obvious what those are going to be. So I want to blow this up out of the picture and I want to study this triangle. This triangle has some secrets in it. Now we're going to be able to explore it here. So I have my x value, I have my y value on triangle. I know that I'm staring with 60 degrees because I think it'll be maybe a little more obvious. Don't forget the unit circle guarantees that our hypotenuse is one. Again, we can pick any triangle, so why not the one that has hypotenuse one. This turns out to be nice. So lets try to fill in the blanks a little bit here. What am I after? Remember I'm after this x and my y. I want to know what is sine of 60 degrees? Perhaps more formally what is sine of Pi over three. I want to know what is cosine of 60 degrees? What is cosine of 60 degrees? I want to know what cosine of 60 degrees is as well as cosine Pi over three, same thing. So this is what I'm after, what's the goal? Remember, the sine is just your y value, and my cosine will be the x value. So stare at this triangle, and our goal is to find these things. I claim that we can do it. The first observation to make is that I have a triangle with two angles given, a 60 degree and a 90 degree. So I can always find that given two angles, you can always find the third. The sum of the angles in a triangle is always 180. So fill in the third one, and you get 30 degrees. Now this may trigger some flashbacks here, but this is a 30-60-90 triangle. This guy has a name. The name is not very descriptive in terms of why it's nice, but it's just 30-60-90. I actually don't like this name at all because it completely hides the fact of what it is. Yes, it describes the angle, but so what? The point of this thing is, and this is the beautiful moment here, is that you're staring at half of something that's absolutely gorgeous. Think of the most beautiful thing you've seen and cut it in half, take scissors and just cut it right in half, top to bottom. So what's going on? Draw the reflection, the other half of this thing, and maybe you'll see it. Now this picture, I drew a pretty bad picture here, it's not drawn to scale. I guess I should've drawn this a little taller, picture that drawn to scale. If you reflect this thing and draw the symmetry about left to right, you have 30 degree angle on top again, and you have a 60 degree angle on the other side, what is the big triangle? So you have 60 degrees, 60 degrees, and then the top angle is 60. That's the name for that. That's when all angles are equal, which imply by the way that all the sides are equal. This is better known as an equilateral triangle. So equal angular or equilateral triangle. So a 30-60-90 triangle is famous or it's known because it's half of one of the most beautiful geometric objects out there. An equilateral triangle is gorgeous. It's all sides equal, all angles equal. It's a really nice symmetric shape and you can get a lot of information from it just by knowing the very little things. So you're staring at half of it. This tells you immediately what x is. Pause the video for a second and see if you can figure out what x is. Are you ready? Again, my picture is going to be a little misleading here, but one is the hypotenuse because we start with the unit circle. Remember, all sides are equal. So the entire base, again, not quite drawn to scale, is also equal to one. So x is just half of that, and that immediately tells you what x is, that's one-half. Remember, our cosine of this angle is x. That just tells you from just knowing what an equilateral triangle is from the picture is one-half. You can draw this real quick. You can always just draw this, and I'm going slow as we go through it. But to draw this on the side if you forget it takes two seconds. So cosine of 60 degrees, cosine of Pi over three is one-half, that falls out nicely. Now, I'm going to redraw the picture over here just to see this. You have this left side of equilateral triangle, you know your hypotenuse is one, you know your adjacent side is a half, and you want to find y. We have a right triangle where two sides are known. How do you find the third side? Pythagorean theorem will do it. So what does that say? Says one-half squared plus y squared is one squared. One-half squared is a fourth plus y squared is 1. So this y squared is 1 minus a fourth, which is three-fourths. So y, take the square root of both sides, is root three-fourths. Some algebra Fourier for the reminder, square roots breakup over fractions. So this is really root three over root four. Root three, there's not much you can do with that, but root four of course is two. So you get that from the other side by Pythagorean theorem is root three over two. So we just found sine of 60 degrees. Add this to the table. Sine of Pi over three and cosine of Pi over three. Now we did a lot of work to get this. You may say, "Well, that's not that obvious." Well, I went slow to go through it, but once you have it, eventually as you use it a lot, I'm sure you can recall it, repetition leads to memorization. Sometimes if I haven't used these in a while and it comes up, I'll go off on the side and just calculates real quick. I understand where it comes from. I think this is the big thing is realized that a 30, 60, 90, like why is it famous? Why do they have these things? Well you're staring at half an equilateral triangle. An equilateral triangle gets a lot of people excited just due to symmetry, how nice it is. So that's what we want to have. Other thing you'd get for this for free by the way, if you change your perspective and stare at like sine of 30 degrees, which is sine of pi over 6 and cosine of 30 degrees, you know everything there is to know about this triangle, we get these values for free as well. Just change your perspective of what this is. Remember on a unit circle, sine, SOHCAHTOA, is opposite over hypotenuse. In this case, here's the x value, so the values just switch. Then if you're 30 degree, its adjacent, SOHCAHTOA, adjacent over hypotenuse, the y-value, and we said that was root 3 over 2. So all of this work you can see is paying dividends already. I get, sine and cosine of 30 degrees when I wasn't even looking for it. It's also in there and they just switch, which is really nice. One more example. I want to study the angle theta, which is 45 degrees, 45 degrees is half of 90. So it's half of pi over two, so pi over four radians. On the unit circle, if you want to think about it in quadrant 1, just draw some ray that splits quadrant 1 in half. So you have 45 degrees between that ray and the x-axis. Then draw the unit circle doing its thing. I want to study where the radius of this triangle that's formed is one. So we're right in quadrant 1 in this circle. So if I want to perhaps scrap this triangle and make a little bigger, what we have going on here. Again, I'm going to put it in degrees just because I think we're more familiar with this. As I tried to explain it, I don't want you guys to have to convert. But you're staring at a right triangle with hypotenuse 1 and at 45 degrees. This is all embedded inside the unit circle. So we're going to play the same trick. So what is this thing? So I have two angles. I claim this one is nice too. I have two angles of a triangle, I want to get a third one. So of course, the other side is 45 degrees as well because they have to add up to 180 so you can subtract and get 45 degrees. Now you start seeing the other nice triangle that people care about. So the angles in this one is a 45-degree, 45-degree, 90, 45-45-90, maybe you've heard that before. Again, the world's worst name, like why do we care that these are the angles? I really wish they stressed the geometry of this thing. But once again, I claim that this is a very nice object, just cut in half. Instead of reflecting left to right this time, we going to reflect over the hypotenuse. We get a square. Why do we love squares? Well, all angles are equal, 90, 90, 90, around the horn and then all your sides are equal as well. I don't know that the sides are. Remember the hypotenuse is one. But I'd like to know what the sides are. I really would love to know what x and y are. But of course, because I have a square labeled as y, but x equal to y, the sides are all equal. So I'm after what is sine of 45 degrees, and that's the same as sine of pi over 4. The beautiful thing about this here is, I'm after cosine of 45 degrees, but because it's a square, the adjacent side is equal to the opposite side. So they're going to be equal. So cosine of pi over 4. They're all going to be equal. This is really nice. So I have this right triangle. Let me just draw it again, nice and big, so we can see it. I have this right triangle. I know that I'm half a square, it's what a 45-45-90 means. So that's the key realization, you're half a square. Once you realize that, off to the races you go. So you have a right angle in here, you have your 45 degrees or pi over 4, and your hypotenuse is one. What was the trick last time when we wanted to find a missing side? We use Pythagorean theorem. Let's do that one more time here. So now I have x squared plus x squared is 1, put it together you get 2x squared is 1, or x squared is a half. Take a square root, you get x equals the square root of a half, which is 1 over root 2. Normally, we rationalize the denominator. So let's multiply top and bottom by root 2 over 2. You get root 2 over 2. That's our answer, that's our number. So sine of 45 degrees, sine of pi over 4 is root 2 over 2. You can say 1 over root 2 is not wrong, it's just more common to see it as root 2 over 2. Because x is equal to y and the square of the sides are equal, your cosine of these values are as well. So start collecting a table together, put it all together, things you have, this is going to be a reference. Knowing these foundational angles, sine and cosine, are going to be great building blocks to go off and find other angles that we can get as well. Great job on this one. I'll see you next time.