 You also want to get up your calculator, preferably your graphing calculator, but A calculator, get up your calculator, do that right now, please. And what we're going to be looking for today is a pattern, a unique but very, very important pattern. We're also going to be waiting for my calculator to actually turn on, there we go. First thing, make sure you're in degrees, press the mode button and make very certain you're in degrees. Today is the last day we're going to be in degrees. Next class, we're going to enter the wonderful world, so Friday we're going to enter the wonderful world of radians. Who's in physics? For the rest of the year, you're going to have to get into the habit of physics degrees, math 12, radians. Do not write physics 12 tests in radians, you'll cry, you'll be able to find any of your answers. Call that the Jordan-Sakilomal, right? And don't write this down. What I'd like you to do is, on your calculator, as I'm waiting for this little journal program to open up, come on, this stupid antivirus program running an antivirus, come here, okay, let's do it this way, click, there we go. I would like you if you would be so kind, find the sign of 40 degrees and give it to me to four decimal places, Amanda, that's my way of saying go to your calculator, get it out and type it in. What'd you get to four decimal places? Point 6427, and could you also find the sign of 140 degrees, please, and give that to me to four decimal places? Is it also point 6427? Really? Could you also find for me the sign of 220 degrees and give that to me to four decimal places, please? Call me silly, do you get the same decimal, point 6427, but a negative in front of it? And how about the sign of 320 degrees? I'm going to bet that you get negative point 6427, do you? Really quickly, find the cosine of 40, I'm not going to write it out, just type it in. Nicole, what do you get to four decimal places, cosine of 40, point 766 to four decimal places? Okay, so point 7666, find the cosine of 140, I'll bet you get negative point 7666, yes? Find the cosine of 220, I bet you get negative point 7666, yes? And I bet you, Holly, if you find the cosine of 320, get positive point 7666, Amanda, do you think I have all these memorized, or do you think there's some kind of a shortcut pattern that I know that I'm just quickly doing almost effortlessly in my head? These four angles all have something in common, let's try another one. How about, are you going to make it, Andrew? You sure? Not looking good. Okay. How about, oh, the sign of 50 degrees, Andrew, can you give that to me, the four decimal places, please, point 7, 50 degrees, point 7666. Can you find for me the sign of 130 degrees, point 7666, really? How about the sign of 230 degrees, negative the same? And if you find the sign of 310 degrees, do you also get negative point 7666, that's supposed to be a graph, I know they're not straight lines, shut up. Where would 40 degrees be, if I started going, it would be right about there, this would be 40 degrees. What about 140 degrees, well, 140 degrees would be, how far have I just gone, right there, 90? How much further do I want to go? There's 140 degrees, how big is this angle here, how much further did you say I had to go, so how big is this angle here, 220 degrees would be, 220 degrees would be in here. Think about it, how big is this angle here, 40, are we spotting a pattern, and Trevor, what's all the way around, how many degrees? All the way around, how many degrees, 360, so 320 is right about there, can you tell me how big this angle right here is, 40 degrees. We call that a reference angle and we say that 40 degrees, 140 degrees, 220 degrees and 320 degrees, they have the same reference angle, if they have the same reference angle, their trig values are the same decimals. Some negative and some positive and Ashley, I'll teach you a shortcut to figure that out in your head too, but for what it's worth, they'll have the same decimal values. The tangent of 40 will be the same as the tangent of 320, but they might be positive or negative, but the decimal will be the same, here's 50, that's 50 degrees, 130 degrees is in here somewhere, you know what, if I think about it, I think that's 50 degrees because it's 180 minus 130 would give me 50 left over, 230 degrees is right there and you know what, that's 50 degrees, 310 is right there and you know what, that's 50 degrees. The reference angle, by definition, the reference angle is always measured to the closest x-axis, it's never measured to the y-axis, in other words, the reference angle is either this one here or this one here, think about holding your arms up and if you have the same reference angle, Kirsten, if you have the same reference angle, then I guarantee your sine, cosine and tangent values will be the same decimals, it wants us to walk through that, we just did page 232, hugely important concept, the concept of a reference angle. In order to investigate pairs of angles with identical trig ratios, same decimals, we're going to introduce the concept of the reference angle and the reference angle is the acute angle formed between the terminal arm of the rotation angle and the x-axis, now Nathan, it's easier to draw, if you draw a graph and you put an angle there or there or there, the reference angle is going to be this one or this one or this one or this one, it's always measured to the x-axis, it's always measured to the x-axis, it's always measured to the closest x-axis, that's how it's defined. That's really the image I have in my head, when you saw me pausing for a half second, I was visualizing the graph, I was visualizing the angle and then I was dropping down to the closest x-axis and it was either going 180 minus or 180 plus or 0 minus or 360 minus, I was doing some basic math in my head. So there's the fancy schmancy definition, I'm not going to ask you that, I'm just going to ask you do you know what a reference angle is and find it. In each case it says sketch the rotation angle and then state the reference angle, so 120 degrees, well 120 degrees, Carson how far? How far have I gone right there? 90, how much further do I want to go? Now that's not the reference angle, 30 is not the reference angle, the reference angle is always closest x-axis, this is my reference angle, how big? 60, now I need, I don't want to write out reference angle is, so I have an abbreviation, this is my abbreviation, this is Amanda not an official one, so don't think everyone, you know if you're props next year in university you're going to use this, but I do, I call my reference angle capital R of theta, to me that stands for reference angle. Would you say it was, Carson? I agree, 243 degrees, B, ready Katie, here we go, how far? How far? Too far, so 243 somewhere in here and the reference angle is that one, how big? How can you calculate it? But there's no one rule, the rule is different in each quadrant, which is why I haven't given you a rule for calculating a reference angle, you got to think, how can you figure it out? How far? How far? 243, how can you figure it out? Can't you just do that? Yes? So how big? Katie, use your calculator if you have, but give me an answer, I'm going 63, yeah, 63 degrees, Roxanne 337, how far have I gone? How far? 180, 270, 360 would be too far, 337 would be right there, now the reference angle is this one, Roxanne how big? Because it's always measured to the closest x-axis, how can you calculate it? Can you tell me out loud? I think not quite, ready Roxanne? Look up, look up, look up, look up, look up, how far? No, no, look where I'm going, how far? Okay, how far is the angle they gave me? 337, what's left to get to 360? Because that's your reference angle, it's 360 minus 337, yes? In your head or on a calculator, what is it? I'm getting 23, yes? By the way, you want to be able to do most of these, or it's going to take you a long time. Ready Nicole? What angle did they give me? Okay, how far? 70, the reference angle is measured to the closest x-axis, what's the reference angle? Also, 70, in fact, the reason you never dealt with reference angles in earlier Soka Toa Trig was because all your angles are between 0 and 90, and by definition, if it's between 0 and 90, it's also its own reference angle, so you were using reference angles without realizing it. Okay, the only reason we have to introduce reference angles now is we're going beyond 90 degrees and we're going to combine this in a few minutes with x and y and r, and get some neat stuff. Jessica, what angle did they give me in E? Do you say negative? That means go this way. How far? Negative, how far? Negative 90, how far? Negative 180, negative 270 would be too far, so I'm right there. Reference angle is always measured to the closest x-axis. How big, and I'm going to tell you right now, reference angle is defined as always positive. How big is that angle there? 23? No, yes, no, yes, yeah, 20, yep, right, 180, negative 180, negative 203, how big is this space? The difference between them, 23 degrees. Yes, last one, ready Ashley? 537, you know what, I'm going to take a shortcut. How far? 360, and remember I said it was your nine times table is the each axis, so for 50, 45, right to here would be 540, I want to go 537, so 537 is going to be like right there, and the reference angle is this big. Oh, what did I say? How far to there was 540, and we're at 537? How big? Yes, what? Three degrees. Not only if I give you the original angle can I say, hey, tell me the reference angle, I can go backwards, I can tell you the reference angle and the quadrant and say, hey, what was our original rotation angle? Oh, example two. Reference angle is 25 degrees, we're in quadrant two. First it says do a sketch, and even if it didn't say do a sketch, I would do a sketch. Quadrant two means I'm here. Holly, how big is the reference angle that they gave me? This right here is 25 degrees, so if that's 25 degrees, here's my rotation angle. I think 155, isn't it? I think you got to carry a one, don't you? Yes? Matias, second one, what reference angle would they give me? What reference angle would they give me? Quadrant four is here, and the reference angle is that one, 60 degrees. So the question is, what's the rotation angle? How far all the way around? And what's my reference angle? Yes, okay, figuring this out. I can't give you a generic way to sketch it each time, look at what quadrant you're in each time, and you're going to be sometimes subtracting from 360, sometimes subtracting from 180, you're adding to 180, do a bit of arithmetic. You had 300, yes? Good, good, good. Eight degrees and quadrant three, so here's quadrant three, mute. Eight degrees is the reference angle, and Nicole, that means this is eight degrees, and they want the rotation angle, which is this whole thing here. How big? 188, 180, and eight more. 180 and eight more would be 188, Mr. Good, there we go. Andrew, reference angle of 39, oh, quadrant one, quadrant one, and 39 degrees, so quadrant one, I'll put my angle there, says the reference angle is 39 degrees, and oh jeez, I want the rotation angle. Ah, it's its own reference angle, which is nice, and quadrant one, that will always happen. Ooh, what about a reference angle of 90 degrees, and we're in between quadrant three and four. Now if we're in between quadrant three and four, I think what we're talking about, there's quadrant three, there's quadrant, oh, we're talking about that line right there, are we not? Shannon, how big is the angle? 270? Yes? Yes? reference angles. If you're having trouble with this, make sure you do the homework, make sure you do the homework, fundamentally key. Now, I told you when I did this little trick here, that reference angles have the same decimal value, which is how I was able to do my little predictions, but you'll notice Nathaniel, I was also able to tell you sometimes whether it was negative or positive in my head without having to think about it too much, and there is a great trick for that as well. Okay? Nicole, sine is what over what? Y over r. Cosine is what over what? X over r, and tangent is what over what? Y over x. What we're going to look at is how those values change from quadrant to quadrant. Now, r, we said was the radius. Remember, r was when we rotated the arm around, sorry for you guys, it's this way. When we rotated the arm around the origin, the radius was the length of that arm. That means, Sabrina, by definition, the radius is positive. It's a length. X and Y are not necessarily positive. It depends on which quadrant you're in. I'll show you what I mean. It says this, in quadrant one, sine, which is Y over r, in this quadrant, Y is positive because you're above zero, and r is positive. You have a positive divided by a positive, which is a positive. Over here, sine is what over what? Sine is what over what? Trevor? Y over r. Now, in quadrant two, which is right here, is your Y coordinate positive or negative, positive, and r is positive. You also have a positive divided by a positive. What is a positive divided by a positive always, Trevor? Yes. Ah, but in quadrant three, Katie, sine is what over what? Now, in quadrant three, you're down here. Is your Y coordinate positive or negative if you're down here? Ah, and r is positive. In fact, in quadrant three, it's going to be a negative number divided by a positive number. And do you know what a negative divided by a positive always ends up being? Negative. Sine is negative in quadrant three. Look up. Sine is negative in quadrant three. Ah. Oh, positive in quadrant two and quadrant one. Ah. What about in quadrant four? Well, here, Y is negative, Y over r. I also have negative divided by positive. Sine is negative in quadrant four, which it is. What about cosine? You said cosine was X over r, Nicole. Let's write that in the first quadrant, X over r. And in the first quadrant, that's a positive number divided by a positive number. Cosine will be positive between zero and 90. But in quadrant two, over here, I'm to the left of the Y axis, Nathaniel. Am I not? If I'm over here, you know what my X coordinate is going to be? Negative. In fact, cosine is going to be negative divided by a positive. What's a negative divided by a positive? Hey, got your calculator? Find the cosine of, let's see, an angle between 90 and 100. Find the cosine of 110. I guarantee it's negative. Yes? So is the cosine of 170? So is the cosine of 153.2? Anything between 90 and 180, cosine is going to be negative. What about in quadrant three? Well, in quadrant three, I'm still to the left of the X axis. Cosine is still going to be a negative divided by a positive. It's going to be negative. If you find the cosine of, oh heck, 222, it'll be negative. I don't know what the decimal is, but I know it'll be negative. What about in quadrant four? Well, in quadrant four, although my Ys are negative, I am to the right of the Y axis. My Xs aren't negative. What are my Xs? They're not negative. What are my Xs? Positive, which means I have a positive number divided by and the radius is always positive. Cosine is positive. Find the cosine of, oh 300 right now. Guarantee it's going to be a positive number. Is it Cassandra? Did you try it? You did? I don't think you did. Now, just because of that, I'm tempted to start lying and see if you actually will catch me rather than nodding. We'll find out. Nicole, what did you say tangent was? Tangent over here is Y over X. Now, this is going to change. In the first quadrant, both X and Y are positive. It's positive divided by positive. It's positive. In quadrant two, my Y coordinate is positive because I am above the X axis. What can you tell me about my X coordinate? It's negative. You know what? The tangent between 180 and 90, it's going to be negative. Between 90 and 180, it's going to be a negative tangent. In quadrant three, my Y coordinate because I'm below the X axis is negative. But my X coordinate is also negative. What's a negative divided by a negative? The tangent between 180 and 270, thank you Trevor for testing it, is positive. Try the tangent of 220 for example. It's going to be positive. What about in quadrant four? Well, here I'll have negative Y value and a positive X value. It's going to be negative. You have to memorize this. Well, no. I'm going to give you an easy cheat. You will memorize this, but we don't memorize this. We'll memorize a shortcut. Which one is positive in quadrant two? I'm going to put a big capital letter S right there. S4, sign. Oh, and cosecant as well because cosecant goes with sign. Which one is positive in quadrant three? I'm going to put a big capital letter T there. Which one is positive in quadrant four? Which one is positive in quadrant one? Did you say all of them? We use the cast rule. That's what I was doing at the beginning of class when I was rattling off whether they were negative or positive. I was simply envisioning the word cast and it was pretty easy to do in my head. Or if you want to start in quadrant one, there's all sorts of acronyms. All add sugar to coffee. All students take calculus. Arnold Schwarzenegger takes cocaine. Well, whatever. Whatever acronym you want to come up with. Sign values are positive in quadrants one and two. Cosine is positive in quadrant one and four. Tangent is positive where? What quadrants? One and three. If you know where they're positive, can you figure out what quadrants they're negative in? I'm going to skip that. But I'll certainly ask that. This can be memorized by the cast rule or add sugar to coffee or whatever acronym you want to. The reciprocal trig functions follow the same pattern. So where is cotangent positive? The same place tangent is positive. Where is secant positive? What does secant go with? Same place cosine is positive. I don't really memorize those. I think about those for one second. I match up the pairs and I go back to the cast rule. I'm not going to come up with a second rule. I don't need it. This was hugely useful 50 years ago. Still useful now, but even more useful 50 years ago because 50 years ago you didn't have calculators to do trig values for you. You had a table and you had a table of trig values from zero to 90 degrees and that was it. You had a table that listed the sine, the cosine and the tangent of zero degrees of one degrees of two degrees of three degrees of four degrees of five degrees all the way up to 90. And if they said to you find the sine of 140, here's what you did. By the way, it says rewrite as the same trig function of a positive acute angle. What that's saying is rewrite it as a reference angle. We always would start out with a little sketch. Kirsten, what angle did they give me? 140, which I think Kirsten is about there. Kirsten, can you tell me will the sine be positive or negative? How do we know? So what does that mean? Will the sine be positive or negative? So what did the cast rule tell us? The cast rule told us what would be positive in that quadrant. So what does the S stand for? What does the S stand for? Sine. So ready? Here we go. Will the sine be positive or negative? I already know that this answer is going to be positive. What's the reference angle? Kirsten, the reference angle is that angle right there. How big? 50? 180, 140. Let's go with carry the one once again. Yes, Amanda? 40. The sine of 140 is the same as the positive sine of 40. Try it on your calculator. Go sine 140 right now equals and go sine of 40. They are exactly the same and the sines match positive and negative b. Tangent of 323 degrees. Sketch, C, A, S, T. 323 degrees. Let's see. Shannon, how far? Shannon, how far? No, how far have I gone right now? 90. How far? Keep going. 180. How far? How far? 362. 323 in here. So first of all, can you tell me is my answer going to be negative or positive? Got that. Next, I want the reference angle. How big is the reference angle? Yes, that's the whole package now. Maybe I was going too slow. The tangent of 323 is the same as negative the tangent of 37. Check it. Try it. Yes, Andrew, you too. Yes? This is how we did it in the old days. Now, Mr. Duk, if our calculators will do this for us, what do we need? Reference angles are hugely useful for other reasons as well. This is only one of the areas that they were useful. Cosine of 235 degrees. You know what I'm going to do first? Ashley, I'm going to sketch. Oh, and I'm going to write the cast rule. C-A-S-T. By the way, you're going to be drawing this now probably on about 80% of the questions for the rest of this unit. So get good at it and quick at it. Ready, Ashley? Here we go. Here we go. Ashley, how far? How far? How far right to there? 270. Too far? 235. First thing, Ashley, can you tell me in this quadrant, will cosine be negative or positive? Did you say negative? Because if you did, you're right. If you didn't, say you did anyways and no one will know. The cosine of 235 degrees is the same as the negative cosine of, oh, oh, reference angle, this guy here. I think 235 minus 180. Huh? 235 minus 180? 55? Yes, I meant, Gary, use a calculator if you have to. You did? It's 180, right? How much past 180 did we go? So what was it, 55? Cosine of 235 degrees is the same as negative cosine of 55 degrees. This is what I was doing here, by the way. A reference angle of 50 degrees, I went 180 minus 50. 130, and I'm in this quadrant, sine is positive. I went 180 plus 50. Hey, that's 220. Oh, and in this quadrant, sine is negative. Sorry, 230. Hey, and this quadrant, sine is negative. I went 360 minus 50. Ah, 310. And this quadrant, sine is negative. That's all I was doing. And for what it's worth, you can do these in your head in theory. D. Ooh, negative. Okay, no problem, no problem, no problem. Amanda, you know what I'm going to do first here? For D? Is this sketch? Yep. In physics, my kids would say dull, right? Draw a little picture. Oh, and the cast rule. Now it's negative 105. Negative means go in the other direction. So here we go, here we go. Ready, Amanda? Ready, here we go. Here we go. How far? No, gotta be fussy. No, no, no, gotta be fussy. Negative 90. Yes, yes, yes, yes, yes, yes. Okay. Now, ready? Ready, ready, ready, ready? How far all, if I went all the way to there? Too far, because they want me to go to negative 105, which I'm going to argue is right about there. Now the reference angle is this one. Oh, first of all, can you tell me, is the sine going to be negative or positive in this quadrant? Negative. And it's going to be the same as the sine of, knock me out, give me the answer, tell me you got it. What do you got? Yes. Now you can sit up straight and crowded. Secant of 358 degrees. I'm not going to panic yet. You don't know who first? Carson? Darn right. And right in the cast rule. Carson, how many degrees did they give me? You know what? That's going to be right there, right? Because what's all the way around? Oh, so can you tell me what the reference angle is? Okay, so that's two degrees. Now, this is going to be the same as, secant goes with which trig function? Secant goes with cos. Is cosine negative or positive in this quadrant? Positive. Is secant negative or positive in this quadrant? Positive. So positive, secant of two degrees. How would I find the secant of two degrees? One divided by the cosine of two degrees, because secant is the reciprocal of cosine. That's, by the way, that's why again, you don't have secant, cosecant, and cotangent buttons. You just go one divided by the trig function. Last one. Cassandra, what am I going to do first? Darn right. Now, maybe you're getting good enough. I think Shannon was, where you can start to say, hey, 107, there. Just past 90. How big is this angle? 73 is correct. Is it, right? Yeah, that is. Yeah. Oh, and you know what I forgot to put? My bad. C, A, S, T. Now, Cassandra, cot, that's for cotangent. What does cotangent go with? Which trig function? Okay, tangent, positive or negative in this quadrant. Ah, so it's going to be negative cotangent of 73 degrees. Yes? Carry the one. Woo-hoo. Skip number one. Skip number two. Three all. You want to reach the point where this becomes boring in a waste of your time. You have to be that good. So I'm going to be giving you a fair chunk, but these, if you get the hang of this, they go quick. So three all, four all, six all, seven. Yeah, skip eight.