 Here we go. We're going to be spending probably until around spring break about two plus months on trig. Most of you when I say trigonometry you think Sokotoa, sine, cosine, and tangent. That's how much trig that is. This is how much trig there is. This is how much we're gonna look at math well. What you think of as trig is really like saying drawing a circle is art. There's more to art than drawing one circle. Are you saying that trig is a form of art? Trig means three-angle measurement. Rotation angles can be measured in three ways. They can be measured in degrees or in radians. We're going to move to radians in a couple of days. We're gonna leave degrees behind. Radians, Trevor, are a much better way to measure angles for a number of mathematical reasons. There allows you a way to measure angles if you're trying to arithmetic in your head. How many degrees is this? Approximately 90. You know how many radians this is? Approximately 1.712. Not quite as convenient the number. Sorry. Such as life. But it has a whole bunch of other useful properties that we really really really really like. Okay. There is a third way to measure angles. They don't mention here. It's called gradians. GRAD. And if you have a scientific calculator, it probably had a DRG button or a degrees rad, rad button. Strangely, this is the one of the only things that your graphing calculators don't have and I haven't able to find it anywhere. A nerd that I am, I've looked. How many degrees in a circle? 360. Do you know how many gradians are in a circle? 400. They use it in engineering. Why is 400 nice? If there's 401 circle, how many gradians in this? 100, which means now you can describe an angle out of 100 gradians. Hey, that sounds like a percentage grade. When you're driving on the freeway and you see a sign in front of a hill or it says trucks gear down and there's a little hill symbol and a percentage, they've measured that hill in gradients. Now I had to look that up, Brett, because I'm only, I'm a nerd. I have a math degree. I did four years of university math. Never used gradians once. So, okay. I guess use them in engineering. That's the last I'm going to mention, gradians, but I had to mention it just for the completeness aspect. Angles can be measured in degrees where 360 degrees is one complete rotation and what we're going to try and do today, Eric, is we're going to try and link trig triangles, Sokotoa, to graphing because that's going to let us go in a general mode and that's going to let us expand what we could do with Sokotoa dramatically. A rotation angle is formed by having an initial arm that's fixed and rotating a terminal arm around a vertex. If you go in the positive direction, that's counterclockwise, so this way is positive. This way is negative. Positive angle comes from counterclockwise. A negative angle goes from clockwise. And Alex, if you have your initial arm on the x-axis, we say that it's in standard position. We're always going to start out in standard position. So example one says this. Draw the rotation angle in standard position and the angle they gave me is the rotation of 150 degrees. Trevor, if it's in standard position, I'm going to start right here. How many degrees have I gone? No, how many degrees have I just gone? Here. So far. 90. How far do I want to go? If I go all the way to this line, how far will I have gone? Too far. 150. About there. In fact, Emily, I actually know two angles. I know that this is 150. I also, with a bit of arithmetic, know that that angle right there is 30 degrees, which is really how I know to stop. B. Asks for a rotation angle. Do you have your book here? Did you find it? Yes. B asks for a rotation angle of negative 210. Mitsu, what's the angle that they want me to do? Negative means go this way. How far have I gone this way? Don't say 90. Fussy, fussy. Negative 90. How far now? Negative 180. If I go to here, I'm at negative 270. That's too far. In fact, how much further than negative 180 do I want to go to get to negative 210? Justin, how much further? In fact, I think I end up in exactly the same location. Eric, I end up in the same spot. You live? Okay. Eric, what's this last angle that they want me to draw? Positive or negative? Go this way. Eric, how far have I just gone? Look. Okay. Most of you know 360. Now, what I do, every time I go 90 degrees further, what I'm really doing is my nine times table. So you said 360, I think 36, 45, 450. How far? 54, 540? Too far. How far do you want me to go, Eric? About there. And you know what? As it turns out that the leftover angle is 30 degrees as well. In fact, all three of these have the same terminal arm. There's an infinite number of angles, Brett, that actually end up in this location. Because instead of going once around and stopping, I could have gone twice around and stop. Or three times around and stop. There's an infinite number of angles that will actually end up right here. Or I could go negative twice around, or negative three times around and stop, or negative four times around and stop. Angles that have the same terminal arm are called coterminal angles. These three angles are all coterminal. What's the correct answer? Well, they sort of all are. But there is sort of like a lowest terms answer. Don't write this down. Ellen, if I give you this, do I really want you to leave that as your final answer? How would I prefer you to write that as one hand? Same thing, okay? Well, what we have is something called the principal angle. It's the smallest positive rotation angle that's a long definition. Ryan, the principal angle is always between zero and 360. That's the lowest terms. In other words, these are all three the same angle is 150 between zero and 360. That's your lowest term. That's your principal angle. That's your principal angle. Okay. How can I find coterminal angles easy? Don't write this down. Just watch. Let's suppose that I went 140 degrees, making this angle right here 40 degrees. So my principal angle is 140 degrees because that's between zero and 360. How big is that angle? Can you find a mathematical way to get there without counting 90 by 90 by 90 by 90 by 90 by 90 by 90 by 90? Ellen, which is I think 500 if I do the math in my head correctly. Hey, how about add 360 twice? Or instead of adding 360, you know what I could have done? Minus 360 or minusing 360 twice or minus, you know what? Emily, I can add or subtract 360 as many times as I want. That's going to give us a mathematical expression for saying, here's all the angles. The mathematical expression is your principal angle and we're going to write this down in a second. Plus or minus multiples of 360. Don't write this down yet because that's way too much writing. How could I write multiples of 360? Well, instead of writing multiples of 360, I'm going to say plus or minus 360 times a number, times one, or times two, or times three, or times four, or times five, or times six, or times seven, or times eight. Except right now, if I just write that, you might say Spencer, oh, can I stick in a 1.2? Am I allowed to put in decimals? We need to somehow let everyone know, oh, plus or minus whole numbers. But I can also put a negative number here. So instead of going plus or minus, I could just go put a negative number there. I'll have to go plus or minus. Here's what we write. We say where m is an integer. Now, you guys have seen this all reals before. Now, we're saying this time it's not all reals. This time it's all integers. What are integers? Not positive or negative numbers, actually. Positive or negative non-decimal whole numbers. One, two, three, negative one, negative two, negative three are integers. Negative 3.6 is not an integer. So the principle, if I want to find every single angle theta, it's the principle angle plus multiples of 360 degrees where n is an integer. It's easier to do than to write. That's the fancy schmancy, me being pussy formula. It's really easier just to do it. Next page. Example one. Find the angle between negative 360 and positive 360. That's coterminal with 200. Okay. You know which angle they want? That one. How big? It's going to be the principle angle that they gave me. Am I going to add 360 or minus 360 to get that one? Minus 360, which is what? Which is what? I think, isn't it negative one four? No, no, it is negative 160. I stand corrected. Mr. Dewey, bad math. This angle right here is negative 160 degrees. It's coterminal to positive 200 degrees. Yeah, never the same spot. What about, oh, coterminal with negative 290 degrees. So that's negative 290. I want this one. How will I find the coterminal angle to negative 290 degrees? Specifically, the angle is between zero and 360. Yeah, so between negative 360 and positive 360. I didn't hear. I heard negative, which one? Either of those would give you a coterminal, but since they want my answer to be between negative and positive 360, I think I have to add 360 to this one, which gives me what? 70 degrees. Which of these two here is the principle angle? Which of these two here is the principle angle? Which of these two here is the principle angle? It's meant to be really easy. 200. Which of these two here is the principle angle? 70. Why? Because the principle angle is between zero and 360. It's the smallest positive one. It's the lowest terms equivalent. Then they ask us to write an expression which represents all of the angles. Okay. For the first one, all of the angles are going to be the principle angle plus or minus multiples of 360 degrees, where n is an integer. Now, Tyler, I was taught actually to write plus or minus, and I still will sometime, but a few years ago, they said, hey, look, if n is an integer, you're introducing a negative. You can also introduce a negative right here, which means the minus there is kind of redundant. And I said, okay, fair enough. So in the book of the book, they probably won't put the negative right there. I often will in my notes or in my hand. If I'm in a rush, you'll see it. I don't take marks on this. Either is valid. How could I write an expression for all of the angles for this second one? It's the principle angle plus multiples of 360, but the multiples of 360 have to be whole number, positive, or negative multiples, multiple. Example three, it says, for each of the following angles, determine the quadrant of the terminal arm and the principle angle. You guys got room up here, top left corner a little bit, throw a little tiny graph right there. This is quadrant one. This is quadrant two. This is quadrant three. This is quadrant four. Traditionally, we use Roman numerals to label the quadrants, and you do need to know, you need to memorize which quadrant is one. It starts with positive, and it goes in the positive angle direction all the way around. So having said that, for about 90, for the rest of this unit, oh, for about two months now, about 80% of the time, the first thing I'm going to do, my physics 12s, I know this is dull. What does dull stand for, physics 12s? I'm going to draw a little graph in this case. I guess I could say dog. They want the quadrant. Here's 265. I'm going to go like this. I'm going to start a SAR in standard position right there, and then ready a SAR. We're going to go for a walk. Here we go. Here we go. Here we go. How far? How far? If I went all the way to here, how far? 270. How far do they want me to go? Just shy of 270. In fact, I can tell you exactly how shy, five degrees shy, but they didn't ask me for that. They asked me, what quadrant are we in? What quadrant are we in? Quadrant three. What's the principal angle? Now that's the angle, the smallest positive angle. That's the angle between zero and 360. In this case, it's kind of a trick question. You know what the principal angle is? 265. It is the principal angle. Eric, you know what I'm going to do first? They're right. Maria, what angle did they give me? No, they didn't give me 111. That's hugely important. What angle did they give me? You said negative? That means I'm going this way. How far have I just gone? Now I haven't gone 90. Can't be doing that. How far have I just gone? Negative 90. If I go all the way to here, how far will I have gone? Too far, which means I'm in there somewhere. Care of what quadrant? Is negative 111 my principal angle? No. In fact, you know what is? That guy there is my principal angle as a picture. How can I calculate it without drawing a picture if I'm lazy or if I just want to calculate it? It's going to be negative 111. What? Plus 360, which is what? 249? Ready, Alex? What's the angle that they gave me? How far? How far? Now here's the trick. Use your nine times table. So 36, what comes after 36 and your nine times table? 450. What comes after 45 and your nine times table? 540. What comes after 54 and your nine times table? 630? Yes. 720. Now some of you might have been able to go twice around as a 720. Some of the basketball players might not have heard that term rate. That's fine. What comes after 720, Alex? 810? How about here? Yeah, 981 plus 9900. How about here? What comes after 900? 990 or 99, right? So nine times table. How about here? 1080. And what do they want me to draw? I meant 1080. That much further. There's my angle. Alex, what quadrant? Yep. What's the principal angle? Now there's two ways to find it. I'm pretty sure the principal angle is that I think it's five. But if I didn't have the graph in front of me, Ryan, I could take the angle that they gave me. I could minus 360. Am I done? Am I between zero and 360? Is this the smallest possible? Minus 360 again. Am I done? Is this the smallest positive angle between zero and 360? Minus 360 again, and now I reach math that I can do in my head, Spencer. What's 365? Take away 360 now that you're awake. Yeah. There's the principal angle. Another thing you learned with triangles was Pythagoras. Pythagoras is a squared plus b squared equals c squared. But what we want to do today, Tyler, is we're going to superimpose this triangle onto a graph. I'm going to tell you that this point right here is x comma y. It's on graph paper. And if it's x comma y, what that also means, Asar, is this distance right here, this distance right here, or this length right here is x. In other words, if you're 3 comma 4, you're 3 over. And 4 up, this distance right here is y. In other words, if they give me a point, what they're also giving me always from now on, Mitsu, is part of a triangle. And because we're rotating around in a circle, instead of calling this a hypotenuse, we're going to call this r. I went to go see a pirate movie the other day, but I couldn't get in because it was rated r. What does that mean? It means Pythagoras. This means, Eric, that Pythagoras can be written as x squared plus y squared equals r squared. This is hugely important. Highlight it. You need to memorize this. You need to memorize this. You need to memorize this. You need to memorize this. You need to memorize this. In fact, Trevor, this is so important that in a few minutes, I'm going to say, repeat after me and you're all going to say x squared plus y equals r squared. We're going to do that right now. We're going to chant it together. Are you ready? Repeat after me. x squared plus y squared equals r squared. Eric had no idea what was going on, so we're all going to have to do it again. Are you ready? Repeat after me. x squared. Justin never moved his lips at all, so he's going to solo. Justin, repeat after me. That's very profound. You don't know it yet, but it is. Not only is x squared plus y squared equal to r squared, get the x by itself. Now my physics 12s can probably do this pretty easily because you've been rearranging formulas with square roots and squares like crazy this year. Justin, how would I get the x by itself? No, I would not square root. I would minus the y squared. I would get r squared minus y squared equals x squared. Then what would I do? Square root. The square root does not cancel out those squares. Get the y by itself, Justin. I don't have these two memorized. You can if you want to, Tyler. I just rederived them. I know this and I can get any letter by itself. By the way, this gives me r squared. How do I get the r by itself technically? Square root. x squared plus y squared equals r squared and x equals the square root of r squared minus y squared. y equals the square root of r squared minus x squared. Terribly profound. Next page. I picked the physics 12 student because we've been doing trig in physics 12 like crazy. We have what are called the primary trig ratios. Now there's something called the reciprocal trig ratios. You're going to learn three new trig ratios today. But the primary ones are the ones that you did in grade 10. In terms of opposite adjacent and hypotenuse, sine is what over what? My physics 12s are like, oh, for Pete's sakes in my sleep, opposite over hypotenuse. Yes. And cosine is what over what? Adjacent over hypotenuse, which we write as a over h and tangent is what over what? Opposite over adjacent. And for 100 years or so, probably more than that, English speaking students have used this acronym right here, SOCA TOA, to help them remember that sine s is opposite over hypotenuse. See the opposite over hypotenuse in there, Ryan? And cosine is adjacent over hypotenuse and tangent is opposite over j. Eric, there are three new trig ratios. By the way, it's sine, not sin. Sin is when you swear in my class or are late. That's a sin. Sine is the trig function. Cosine and tangent. Although often we abbreviate tangent as tan and cosine as cos. We don't abbreviate sine as sin. Sine is sine. There's something called the reciprocal trig ratios. The reciprocal trig ratios are cosecant, secant, and cotangent. We'll pause for a second. So let's keep going. Trevor, the reciprocal trig ratios are cosecant, secant, and cotangent. And they're actually easy to remember. It sounds confusing. It's not. Cosecant is the reciprocal of sine. It's one over sine. So if sine is opposite over hypotenuse, what's cosecant? Hypotenuse over opposite. If secant, abbreviated sec, is one over cos, what is secant in terms of opposite adjacent and hypotenuse? If secant is one over cos, what's secant in terms of opposite adjacent and hypotenuse? Kara. Yeah. Cotangent is one over tangent. Okay. What's cotangent in terms of opposite adjacent and hypotenuse? Adjacent over opposite. I don't memorize these. I memorize, okay, really and truly, put the report cards away. There's nothing on there significant anyways. Thank you. So how do you know which goes with which? First of all, you need to memorize that cotangent goes with tangent. Is that too difficult to memorize? I'm going to suggest that it's fairly easy to remember that cotangent goes with tangent. For the other two, what everyone remembers is cos never go together. Cosine does not go with cosecant. And sine goes with cosecant. Cos never go together. So I don't memorize what secant, cosecant and cotangent are in terms of hypotenuse. I just know if you say to me, hey, what's cosecant? I go cosecant is sine, sine is opposite over hypotenuse. Hypotenuse is over opposite. It takes one second and I can do that in my head every time. And that's actually the truth. Usually when I say to you guys, I haven't bothered memorizing this, I'm lying because in 15 years I have memorized it after teaching all this. This I haven't bothered. Not worth my time. Not even worth making an effort. So you can remember it. Each pair, come on, come on. Each pair has only one co in it. And now, for the most profound moment in the next two months, we can also rewrite these in terms of x and y and r. What we can do is take this triangle right here and put this part of the triangle at zero, zero on a graph so that this is still opposite and this is still hypotenuse and so hypotenuse, Mr. Dewick. Good gosh. Adjacent and this still is hypotenuse, but instead of saying opposite, Asar, I can say what? Because I asked you to what? Oh yeah, that joke is back with a vengeance this unit. Why? And instead of saying a, what can I say? Adjacent is actually, and instead of saying hypotenuse, what can I say? These are your new definitions for trick. Sign is still opposite over hypotenuse, but in terms of a graph, sign is what over what? In terms of x and y and r, sign is what over what? Cosign is what over what? Tangent is what over what? You're not getting it. Ellen, see this right here, this sentence beginning with the word the? In just a second, I'm going to ask you to read it in a nice loud voice. Are you ready? Read it out. Under enlightening should go right now. It's profound. There should be a deep rumble in the distance. You have no idea just what this has opened up for us. The fact that from now on, sign is no longer or not only opposite over hypotenuse, it's y over r. Cosign is x over r. And tangent, there should be a breeze blowing through this classroom right now. Your hair should be ruffled and often the distance you should hear violins or something like that. It's profound. Ellen, could you read this to me? I'd like you all to cross out the word should and instead write, what does all caps mean in an email? So much so that for the next oh, three or four weeks, I'm going to start every class by saying, Mitsu, sign is what over what in terms of the graph. Alex, cosine is what over what in terms of the graph. Ellen, tangent is what over what in terms of the graph. Too slow. It has to be at your fingertips. Now you'll notice, Trevor, have I filled these ones in? I don't have them memorized. You know what I do have memorized? I know that cosecant is the reciprocal of sign. So if sign is y over r, Trevor, what's cosecant? Yeah, I don't waste my brain power having that at my fingertips. And if secant is the reciprocal of cosine, Justin, what's secant? And if cotangent is the reciprocal of tangent, Maria, what's cotangent? I re-derive those every time in my head. In fact, whenever we're doing a secant or a cosecant or a cotangent question, you'll see me pause for a second. What I've just done is I've associated it with its primary trig ratio. Ask what's the primary and flip it. Now, there is one easy way to remember one of these. Eric, can you read to me this one here? No, the whole thing. Physics 12s are going to hear something familiar. Sine is y over r, but it's not y over x. You have to memorize these. In fact, I'll say if you don't have these memorized, you'll flunk the test and you'll be dropping out a math flood. Now, having said that, Mitsu, almost all of you, it'll happen naturally because you're going to be writing this probably five or six hundred times in the next couple of months. But what I am saying is, if in about three weeks you're hesitant on this, you in trouble. You better sit down and write it out 50 times each. Then you'll have it. Why is this so profound? Why should thunder and lightning flash next page? Example five says the point 15 comma 8 lies on the terminal arm of theta as shown. Calculate the value of r and hence determine the exact values of the primary and reciprocal trig ratios. What this question is telling me is that we can find sine theta, cos theta, tan theta. Which reciprocal trig function went with tangent? Let's write it right opposite like that, cotangent theta because that helps me stay organized. Which one went with cosine, not the code? Which one was not the code? Secant, sec. Which one went with sine, the code? Cosicant. We're going to fill in all six of those. What have they given me here? An x and a y. What letter is right there? What's the first thing they want me to calculate? R. Did I have you guys a few minutes ago chant an equation that had x and y and r in it, oh pray tell? How are x and y and r related? Someone say it out loud, sure. Someone besides Kara, well done by the way, I'm impressed. See you're in teacher mode now, you're just like x squared plus y squared equals r squared. In this question, what's x15 squared plus, in this question, what's y? 8 squared equals r, oh Justin, how do I get rid of the r squared? Okay, I'm going to put a big square root over here. This does not cancel out those two squares. You get r equals the square root of, what is 15 squared plus 8 squared? 15 squared is 225, 8 squared is 64, 225 and 64, 289? What is the square root of 289? Does it work out evenly? Oh, so you know what r is? 17. Sine is what over what in terms of x and y and r. If you have to turn back and look, go ahead, but sine is what over, I've heard two different answers, I need the right one. y over r, specifically here then, what's it going to be? What over what? And off in the distance, there was a rumble of thunder. Oh, and now I can do cosecant right away as well, 17 over 8, yes? That's why I made sure to write them in their pairs, it makes the second part way easier. Cosine is what over what in terms of x and y and r. So specifically for this one, what's it going to be? And secant will be 17 over 15. Tangent is what over what in terms of x and y and r. y over x, specifically for this question, what will it be then? Do you realize what we just did? We got all six trig ratios and we don't even know the angle. And we didn't need our calculators to do it. Didn't hit a sine button. Didn't hit a cosine button. Didn't tangent button. We calculated it. Fairly easily, in fact, I would argue if I had given you smaller numbers here, this is completely fair game as a non-calc question. As a matter of fact, folks, this next test is going to be all non-calc. Think about what we've just done though, Mitzi. This is why linking graphing and trig, this is why thunder and lightning should roll. We were able to do trig, sine, cosine, tan, no calculator, not even finding the angle. I like example six. I like example six. I like example six. I like example six. So example six says this. Angle A terminates in the first quadrant. You know what? I should draw that. There's angle A with the sine of A equaling one-third. Sketch a diagram, already did, and then find the other five trig ratios for angle A. So we know the sine of A equals one-third. We want to find cosine of A, tangent of A, which reciprocal goes with sine, cosecant? I guess this one I can already do, three over one. What goes with cosine, secant? And what goes with tangent, cotangent? What did they tell me the sine was here, SR? Sorry, you can't say it louder. Sine is also what over what in terms of the graph, in terms of x and y and r. I think what they've really done is they've told us one over three equals y over r. So based on that, what's y have to be? What's r have to be? Three? What's x have to be? Oh, I don't know. It would be wonderful if there was some kind of an equation that related x and y and r, that maybe we get chin. Is there? What? Spencer get the x by itself and then r squared minus y squared to the other side. Square root. What was r squared? I think that's fair game as a non-question. Put your calculator away, Spencer. Minus. What was y squared? I think asking you to go nine minus one in your head is completely fair game. What is nine? Take away one. We want you to leave it like that. I don't want decimals. I want exact value as well. I don't quite want you to leave it like that. We're actually going to ask you to write it as two root two. Put your pencils down, look up. You're going to be asked, Joel, to simplify square roots. For example, if I tell you the answer is root 20, 20 is what times what? Now two times 10 is not what I'm thinking. 20 is what perfect squares times something. 20 is what times what? 20 is root four, root five. What is the square root of four? It's math 10. Hey, what about root 40? What times what? I know it's five times eight. I'm not going to waste my time because none of those are perfect squares. It's what times what? So root 40 is root four, root 10. What's the square root of four? Two, root 10. What if they give you something like this? Root 75. What times what? Now I'm going to, Ryan said three times 25. I'm going to say it the other way around always. Perfect square first because I'd like to write it this way, Ryan. Root 25, root three. Doesn't that look nicer because what is the square root of 25? See, I get my five root three popping out of here. You guys remember doing this in math 10? Now I expect my math 10 kids to be able to do this in their heads. I push my kids a little bit. Some of your teachers might have had to do like the factory tree and all that crap. There comes a point when I say, for Pete's sakes, find the perfect square that goes into it. Take the square root and you're done. Root 80 is what times what? Eight times 10 doesn't mean no good. Those aren't perfect squares. Now a lot of kids will say root four, root 20. Which is fine, Emily. What's the square root of four? Here's my question. Is 20 as low as it can get? Because 20 is what times what? This is going to be two, root four, root five. What's the square root of four? This is going to be a two and a root five. Hey, what do you think I do with that two and that two? Now some of you who are better at your time tables than Emily, sorry, not 16, Mr. Dewick. Some of you who are better at your time tables than Emily might have noticed 80 is 16 times five, which gets you there in one step, but you don't need to get there in one step. You're going to hop Scotch your way there just fine. That's a really quick review of simplifying square roots, but that's where the two root two came from. Let's root four, root two, take the square root of two. Okay, here we go. Cosine is what over what in terms of x and y at r. x over r, more specific for here since x equals two root two, two root two over three. And this is going to be three over two root two. Think about what we're doing here. We're finding exact values of trig functions and we still don't even know the angle. We don't care. By linking, graphing and trig, all we need to know is where the terminal arm goes through. This is how they generated a bunch of the old trig tables, by the way, in case you ever wanted. What they do like 200 years ago before calculators, how do they get trig tables? Tangent bret is one over what in terms of x and y and r. Good recovery. I wasn't sure because you were out of the room. I was going to see you. Oh, so here specifically, what's it going to be? Three is r, one over root two, one over two root two and two root two over one. In fact, you know what? I feel embarrassed that I wrote the over one up here too. It's like washing my hands afterwards. Thunder, lightning, bumbling. Also, we're going to do some standard what you guys have all thought of as trig. Using Soka Toa to find ratios, angles and sides. Somewhere along the way, you're probably going to get a triangle with a missing side. Here's my angle. Lad, never mind, you're not physics 12, you're physics 11. Emily, opposite of Jason, you're hypotenuse. How about the m? Thank you, m for the m. Mr. D, which trig function? Yep. What always goes next to the trig function, the angle, which in this case is 37, equals whatever what Emily, and here just once I will put that over one, even though it makes me feel like I've washed my hands, because I can solve this how much cross multiply. In fact, I think you're going to get m equals 83 divided by cosine of 37. Try this, all of you on your calculators, because this is also the time for you to find out if your graphing calculators are in the light mode. Your graphing calculators by default, if you've done a reset or a battery swap, they tend to go back to radians. So you want to just see if you get 103.9, if you don't, you're in radians. Anybody not get 103.9? So press your mode button right here, and then make sure degrees is highlighted, not radians. To highlight it, just press enter on top of it. And you will notice, as I said, it has radians, it has degrees, it does not have radians. I'm quite surprised by that actually. Not only can you use trig to find sides, you can use trig to find angles. So it says, in example eight, use a diagram to determine sine A and the measure of angle A to the nearest degree. Okay, sine is what over what in terms of opposite adjacent hypotenuse? Opposite over hypotenuse, if that's angle A. I think it's opposite 29 over hypotenuse. Sine A is that. How do I find the angle? Now my physics 12 shush, you've been doing this all year long. How do I find an angle? If I know that the sine of A is 29 over 34, well I do. Y'all remember? K physics 12, what do you do? Inverse, second function, shift. In fact, I like the way the graphic calculators display this because I go second function, sine, and that little negative one, which was a stupid symbol, but it is a symbol for inverse. It appears there. It's saying the inverse sine of 29 over 34, the angle is 58.5 degrees. Oh, it says to the nearest degree, angle A equals 56 degrees. 56? Tara was giving me a bit of a frowny face. How about 59 degrees? Yes, thunder, lightning. What's your homework? Folks, all of it. 1 to 11. You need to practice this. Now, I am not going to give you new homework on Friday, so ideally you'll be able to get this all done before your Christmas break. I'll touch on a few things on Friday, but we'll also have some math nerdy fun, I think. So homework, 1 through 11.