 So what we're going to do is we're going to memorize two special triangles for those of you who are watching online. We're doing lesson six. And I'm going to diverge Trevor from the textbook a fair chunk here because I think the textbook, its method I guess is faster because you can instantly recall everything. I'm going to say I can spend one second and cut my memorizing way down. The first triangle is this one here. How big is this angle Spencer? If that's 90, how big is this angle? And you know what? It's the same size. I'm going to put a pi by 4 there just to remind myself what it is in radians. It's 45 degrees pi by 4. And the triangle simply asks what if your x-coordinate is 1 and your y-coordinate is 1. So x is 1, y is 1. What's r? In fact r ends up being root 2. This is the 1, 1 root 2 triangle. I'm going to memorize it. Thankfully I think Dominique, those are pretty good numbers to memorize. It could have been a lot uglier. I mean it only easier to be 1, 1, 2. It's not 1, 1, 2. It's 1, 1 root 2. And you can remember the root 2 because it's quite digress. And you can remember the angles because they're the same size 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45. This is going to allow us to do sine and cosine and tangent as an exact value. Now here we're going to use opposite adjacent and hypotenuse. So if this is my angle right here, I shouldn't cover up the degrees sine, if this is my angle right here, sine is opposite over hypotenuse. As an exact value the sine of 45 is what over what? Nope. I go to degrees and I go sine of 45 and I go 1 divided by the square root of 2. I think those are the same. Yes? What's the cosine of this angle? Adjacent over hypotenuse, Ellen. Also conveniently 1 over root 2. Which makes sense because it's isosceles and the opposite and the adjacent are the same length. What about the tangent? 1 over 1 or just 1. By the way, do you also now know cosecant, secant, and cotangent? So if they give me an angle and it's a 45, pi by 4, 45 degree, I quickly sketch the 1, 1 root 2 triangle. I don't memorize that chart that I showed you earlier. Special triangle number 1, the 1, 1 root 2, 45, 45, pi by 4, pi by 4. Special triangle number 2, Tyler, how big is this angle? Just read it to me. How big is this angle up here have to be if this is 90, 30? How big is this angle? How big is this angle up here? I'm going to argue these two are actually the same triangle and although for some reason the textbook wants you to memorize them in two different orientations, I'm going to say, hey look, that's 60 degrees right there. I'm going to use this as my triangle. This is a triangle that has a hypotenuse, a radius of 2, and a short side of 1. How long does the base have to be if I use Pythagoras? Now I have to be careful. It's not going to be this squared plus this squared. It's going to be this squared minus this square root. So Sabina, what's 2 squared minus what's 1 squared, what's 4, take away 1. This is triangle number 2. Ryan, triangle number 2 is the 1, 2, root 3 triangle, which is also, I'm going to argue, pretty good numbers to memorize really. They're only usually 1, 2, 3, not 1, 2, 3, it's 1, 2, root 3, where the bottom angle is how big? 30. And the top angle is how big? Where the bottom angle is how big, pi by 6, radians, 1, 6 of 180, and the top angle is how big? Pi by 3, 1 third of 180. You see, now I can tell you as an exact value, what is the sign of 30? Here's 30, what's opposite over hypotenuse? What's the cosine of 30 adjacent over hypotenuse, root 3 over 2? And what's the tangent of 30 opposite over adjacent, 1 over root 3? Oh, and I can fill in this row up here. What's the sign of 60 opposite over hypotenuse? Tyler, opposite over hypotenuse. What's the cosine of 60 adjacent over hypotenuse? Yep. And what's the tangent of 60 opposite over adjacent? Yep. Am I going to write the over one? No. By the way, Tyler, you may notice the sign of this is the coast of this, and the coast of this is the sign of this, which kind of makes sense because when you change angles, what swaps, swaps is your opposite and your adjacent, but your hypotenuse is your hypotenuse. We have the 1, 1, root 2, pi by 4, pi by 4. And the 1, 2, root 3, pi by 6, pi by 3. So the first thing that I'm going to do is I'm going to cut your memorization down by 33.3% by a third. Justin, I'm going to say that this triangle and this triangle are identical, except one is flipped on its end. So I'm going to say, and I'm going to do one more thing. Frequently over the next few days, I'm going to end up quickly in my work drawing this along with the cast rule. These two triangles are going to become your special friends. In fact, for a lot of these questions, Trevor, you're going to cast rule first and you're going to draw a special triangle. If I have any piece of advice for you, it's when you draw the 1, 1, 2, 3, don't draw it like this. Sorry, let me try that again, Mr. Duke, with a little less. Don't draw it like this, where you can barely tell. Spencer, I always draw it really long and skinny and exaggerated because if I do that, it's obvious to me which one's 30 and which one's 60. The smaller one is 30. If you're sloppy or in a Rush Dominique and you draw it like that, you really can't tell. So I will always draw it big and long and that is, how many degrees? 30 degrees, Emily, 30 degrees is what in radians, 60 degrees is what in radians, Brianna, where kids sometimes get a little bit sloppy. Because this is 30 and this is 60, they want to put the pi by 6 with the 60 and the pi by 3 with the 30. No, no, no, no, no. Pi by 3 is a bigger angle than pi by 6. The other one, I'm a little sloppy here because the opposite and the adjacent are the same length, that's pi by 4 or that's pi by 4. So the first two triangles we're going to memorize, the 1, 2, root 3, where the 2 goes on the hypotenuse and the 1, 1, root 2 or the root 2 is on the hypotenuse, in fact there's going to be some kind of a 2 on each hypotenuse, whether it's a 2 or a root. Then it says, however, and we're not going to use those, that's if we want to rationalize the nominator and we're not going to use that. How does that help me find the exact value for the cosine of 150 degrees? Our trigger phrase, what's going to let us know that we're using special triangles is that phrase right there. Without a calculator, how can I find the cosine of 150? Easy. The first thing I'm going to do is sketch it. 150, right about there. Asar, can you tell me first of all, will the cosine of 150 be negative or positive? I know that my answer is going to be negative something and I cross out all the positive answers as it was multiple choice. Hey, can you folks tell me what's my reference angle? My reference angle is 30 degrees. Do I have a special triangle with the 30 degrees in it? Which one? Which angle is 30 degrees, the bottom one or the top one? This guy here and what is the cosine of this angle? The exact value of the cosine of 150 is negative, Castro, root 3 over 2 reference angle special triangle. I'll prove it. First of all, make sure I'm in degrees. Cosine of 150, that's the decimal my calculator gives me, negative square root of 3 divided by 2, that's the decimal my calculator gives me, they are the same. But you'll notice this lets me break away from calculator dependence. So turn to or go to example 1 here. Example 1a says find the sign, oh, first of all, Maria, can you read the instructions to me here? To find the what? There's my trigger phrase which says to me, don't be using a calculator, apparently you can do all this in your head and what do they want me to find in a sketch? Yes, Brett, we're going back just a little bit to our friend degrees, but it's only temporary. Brett, how far? Now we're in degrees, 180, 210 about there. First thing, Dominic, can you tell me my answer, will it be negative or positive? Yep, because we're in the tan positive quadrant and this is sine. So down here I'm going to go sine of 210 degrees equals negative, then I need my reference angle. How big is my reference angle, Brett? I agree. My reference angle is 30 degrees. Do I have a triangle with a 30 degrees in it? I do, then let's get which triangle, the 1, 2, root 3? Which of these angles is the 30, the bottom one or the top one? Tara, so that's the angle we're talking about. Which trig function? What's the sine of this? The exact value of the sine of 210 is negative from the cast rule. Sketch find the reference angle to figure out the triangle, 1 over 2 from the triangle and the reference angle. That's what they're walking you through here. The solution of rotation angle of 210 is the reference angle of 30 and quadrant 3, sine is negative and the sine of 210 is negative, sine of 30 and I sketch the triangle. They want you to memorize the chart, we've already said, you know what? The one second it takes me to sketch, I'll do that. Yes, not yet. Next unit. I'll show you the theory of how it's done and I might do one on your test. But also some of this is, this is still somewhat useful but most of it, you say, I got a calculator now anyway, whatever. C, Alex, what do they want me to find, the exact value of what? Let's sketch, C, A, S, T, what's my denominator, Alex? So I'll call this 3 pi by 3, 6 pi by 3 and I think it's going to look this. There's 3 pi by 3, 4 pi by 3 would be right about there. I think 5 pi by 3 is about there-ish. This idea will my cosine value be positive or negative. I'll put a plus sign just to remind myself that I did that step in my real work I wouldn't bother but in our notes I will. Hey, what's my reference angle? Ellen, pi by 3, do I have a triangle? It has a pi by 3 in it. Which one? The 1, 2, root 3? I'll sketch it right here. 1, 2, root 3. Ellen, what did you say the reference angle was? Which of those two angles is pi by 3, the bottom one or the top one? The top one is because 60 degrees is pi by 3. Alex, which trig function did they give me? Cosine of that triangle is 1 over what based on the angle that I'm dealing with? Cosine of pi by 3 is, sorry, cosine of pi pi by 3 is positive half. B, dumb. C, clearly can you read C to me? What do you want me to find? That's degrees. You know what? How about negative 3 pi by 4? That'll be a little more interesting. They want me to find the tangent of negative 3 pi by 4. I think I'll sketch this. Negative means going the other direction. I'm going to go this way. Here would be negative 4 pi by 4. Here would be negative 2 pi by 4. I think negative 3 pi by 4 is right about there. You make it? Is that okay, Joel? C, A, S, T. First question, Joel. My final answer is the tangent of negative 3 pi by 4 going to be positive or negative? How do you know? Yeah. I'll put a plus sign in my notes in your homework if you don't have to. Just to remind myself that I did that step. Now, I would like the reference angle. Eric, how big is the reference angle? Yes. Thanks for coming through in the crunch. Do I have a triangle with a pi by 4 in it? Which one? Not the 1, 2, root 3. Let's sketch that really quick. 1, 1, root 2. Which angle was pi by 4? The bottom one or the top one? Both. Usually, I stick it there because that's what you're used to seeing from grade 9. Which trig function did they give me? What's the tangent of that? What's the tangent of this? Put your calculators and radians, if you're not already, and try going tangent of negative 3 pi over 4. When you hit Enter, you should get a 1. That's how we're going to deal with. Any multiples of 30, 60, 45, pi by 6, pi by 3, pi by 4. I also, though, Ryan said to you, we're going to learn to deal with. You may remember a few lessons back when we started solving trig equations. We said there's two answers in the cast rule. I said to you, Dominic, we've got a problem though. If a trig equation works out to negative 1, positive 1, or 0, well, there's an alarm bell going off because the cast rule no longer worked with that. What that really meant was you were here, or here, or here, or here, and that means you're not really in a quadrant. So we're going to add a second trick of the trade. Turn the page, 268. We're going to talk about a special circle called the unit circle. I'd like you to imagine my arm is the terminal arm of an angle and it's slowly rotating its way around. So this would be pi, or 180, pi by 2. 3 pi by 2. How long is my arm? Ah, here's what we're going to do, Justin. We're going to pretend to make it as mathematically as simple as possible that my arm is one meter long, or one centimeter, or one unit long. That's why we call it the unit circle. The unit circle is where your radius is 1. That has a very profound impact because, Brett, sine is what over what in terms of x and y and r? What if r is 1? Sine becomes y over 1, or just plain old. On the unit circle, how high you are is your sine value. What's cosine? Well, it's x over r, but Alex, if r is 1, if r is 1, then cosine just becomes your x value. Over here, as good a circle as you can, mine is not going to be stellar, and make a little note that this is the unit circle by saying, hey, radius of that circle is one unit. And the points we're interested in, Trevor, are here, here, here, and here. So let's start out right here. What angle am I at in radians or in degrees? Zero. Let's do cosine first. What's the cosine of zero? Cosine is your x-coordinate. If my arm is 1 long, what's my x-coordinate right now? If my arm is 1 long, what's my x-coordinate right now? 1. Got you got this? Find the cos of zero. It's 1 whether you're in radians or degrees. You're not. What about right here? What angle is this in radians? Pi by 2. What's my x-coordinate right there? Zero. Try to make sure you're in radians. Try cosine of pi by 2. Better get zero. Do you? What will the cosine of pi be? What's my x-coordinate right here? Careful, it's not 1. You know what? Try it. What about the cosine of 3 pi by 2? Zero. What about for sine? Now sine is how high you are. It's your y-coordinate. What's the sine of zero? How high are you right there? Zero. What's the sine of pi by 2? Sine of pi by 2. Sine of pi by 2. Sine of pi by 2. Sine of pi by 2. Sine of pi by 2. Sine of 90 is one. What's the sine of either 180 or pi? Zero. What's the sine of 270 or 3 pi by 2? Not one. Negative 1. The unit's circle can bail you out of those. Oh! By the way, in a small aside, Note, if sine on the unit circle is y, and cosine on the unit circle is y over x, and tangent is y over x, tangent is also sine of the cosine. The original calculators, I don't believe they had a tangent button, you didn't need it. If you wanted to find the tangent of something, you found the sine, you found the cosine, you divide them. That's not that important yet. It will be in about three weeks, but I always mention, oh, and cotangent will be cos over sine because it's the reciprocal of tangent, means you're recovering your mouth during your yarn. Let's try some of this. Example two, page 244, 269, sorry, not page 244, 269. Example two, now it says use the unit circle, okay, I can scribble that out. This is Mr. Dew Expected, which combines the best of both worlds with a bit of extra work, but only about one second worth of work and drops your memorization down to nearly zero. Two triangles, the one, one, root two, the one, two, root three, and spans are the concept of the unit circle that if you're here, or here, or here, or here, you better imagine your arm as one long and then ask how high or how far left right you are. Mitsui, what's the angle that they gave me? Let's sketch over here. 240 is, there, past 180, but not up to 270. What's gonna be positive, tangent, and negative, negative, negative, negative, right? What's my reference angle? Let's see, there's 180, the angle they gave me was 240. How big is my reference angle? Do I have a triangle with a 60 degrees in it, Alex? Which one? I think you said too many roots. It's one, the one, two, root three. Looks like this, one, two, root three, where that guy is 90 always. Which one is 60 degrees, Alex, the bottom one, or the top one? Which one matches your reference angle, in other words, okay? First trig function they want, Alex is sine, sine of that triangle is what over what? And I already put the negative in front, so I don't need to worry about that. Oh, and then they want a cosecant, which is the reciprocal. That's gonna be negative two over root three. Ryan, what's the cosine of 240 degrees? I don't know. What's the cosine of this reference angle here, and two over one? Tyler, what's the tangent? Sorry? And one over, I guess some of my i here is driving me crazy, I keep blanking. We're all teach like this in the empire. That's what we're gonna do. That's how I can find an exact value. Example three. Trevor, once again it says use the unit circle. I'm gonna say, once I cross that out, Tara, can you read the instructions to me? Find what? Did you say exact value? That's my trigger phrase, special triangles and unit circle. No calculator, they want this left as a fraction. Probably with some square roots in it. Sketch. Here's four pi by four. Meow. There's three pi by four. Maria, can you tell me, is the cosine gonna be negative or positive for my final answer? Hey, I saved myself some work, I'll bet you. Do you see what my reference angle is? What's my reference angle, folks? There's three pi by four. There's four pi by four. I heard it. Do I have a triangle with a pi by four in it? Yeah, which one? One, one, root two. Which, where am I gonna draw that? I guess I can squeeze that in over here. Except that looks terrible. One, one, root two. Where that, both angles in fact are pi by four is 45 degrees. What is the, which trig function? Cosine of that. So the cosine of three pi by four, negative one over root two. Try B on your own. See if you get that. Tyler, is that okay? C. Cosicant of three pi. Hmm. Well, let's draw it. How far? Pi. How far? Three pi. Trevor, are we on one of the arms? Then now, Castro, no, represent, no. Unit circle. Okay. Cosicant is what over what in terms of X and Y and R. Sorry. R over Y. And on the unit circle, what did we define our radius to be? So we're gonna use this thing where R is one. If we are right there, what's the Y coordinate? Or the Y coordinate, how high you are? How high are you right there? Hmm. What's one divided by zero? Sorry, what? You're in radians? I'm a little curious. Try going one divided by sine three pi. Do you have an error of some type? Yes? There's another more famous one, by the way. Put your calculators into degrees for a moment. Oh, I am in degrees. And find the tangent of 90. What's the tangent of 90? What's the tangent value up there? What did you get? Erich, why? Tangent is one over what in terms of the graph? Y over X, what's my X coordinate right here? Zero, dividing by zero again. I'm curious though, what's the tangent of 89 degrees? What's the tangent of 89.9? Erich, sorry? What's the tangent of 89.49? 89.9999. Justin, what did you get? What's happening to your answers as you're getting closer and closer to 90 degrees? Calculating students, the limit is approaching infinity. Math, 12 students. Oh, we have an asymptote right there. We're going to be graphing these shortly, a couple of days from now. For now, I'm going to pause here and just assign you the following for homework. You can do 1, 2, 3, 4. 1, 2, 3, 4. Now, sometimes it says use the unit circle. Use whatever method that you want. The method that I've showed you is the combination of both.