 Normally, I do this lesson in one shot and I made a deliberate choice because this seems to be an area where kids struggle. I'm changing, I always smoke for like first time in eight years. I'm trying to do this lesson over two days. What we're asking is, are there certain trig values that we can memorize because I've told you your first trig test, part one, is going to be no calculator whatsoever. Holy smokes. Well, yes there is. As it turns out, we can get away with memorizing a minimum amount. We can memorize these two triangles. The one, one, root two, pi by four, pi by four, 45, 45 degrees, but pi by four radians, pi by four radians. And we can get away with memorizing the one, two, root three. Now you're going to have to memorize that it's a one, a two, and a root three. You're going to have to memorize that the two is the hypotenuse. I hope your clue in the one goes on the short side and the root three goes on the long flat side. And I think I even told you whenever I draw it, I always draw it, stretched out just so it's very obvious to me that that's the root three and that's the one. Every year on the test, there's somebody who has not taken my advice, and Ellen, when they draw their triangle, they do this and then they go like this. And now they've got all the angles in the wrong place, the disaster. And in the one, two, root three triangle, this angle is 30 degrees or pi by six. This angle is 60 degrees or pi by three. Then Trevor, our second option, if you really wanted to, was we could memorize this. How many of you memorized this over the weekend? Good. I'm using a method that combines the best. I'm willing to write something for one second in order to avoid memorizing this. The second concept that I introduced to you, Alex, was the concept of the unit circle. What I said was, you can imagine a circle where r is one, and if r is just one, instead of sine being y over r, sine just becomes the y-coordinate. And what I asked you to imagine was me, the human unit circle where if my arm was one long, sine of the y-coordinate, sine of zero is zero because that's how high my, sorry, zero is over here for you guys. Yes? I got to reverse everything in my head. Sine of zero is zero. Sine of, what angle is this in radians? I'm here to hear. Pi by two, sine of pi by two is one. Back to here, Mr. Duk. Sine of pi by two is one. What angle is this in radians? Sine of pi turns out that's zero. Sine of three pi by two, negative one, back to zero again. We said, Carly was your x-coordinate on the unit circle where r is one. It's a nice shortcut for learning the certain angles. So if my arm is one long, cosine becomes just x. Cosine of zero is one. Cosine of pi by two is zero. Cosine of 180 or pi is negative one, and cosine of three pi by two or 200 degrees is zero. That gets you those ones, which gets us around, I think a few days ago when I told you to do the alarm bell of zeros and negative ones, I said, I'm going to show you a better way to do those questions when they're undefined or zeros, and the better way is to use the unit circle. In fact, there's even a better way, which I'm going to show you next class. Then we said we can combine all of that with the cast rule to do stuff like this, example four. Now it says, use the chart or the unit circle. Technically, I'm using a combination of both. So I'm just going to say, find the exact value. But I did say, now this is your trigger phrase almost always when you see the phrase exact value, that's on the test telling, hello, special triangles or unit circle. The only time exact value doesn't mean special triangles or unit circle is in a question like this, where they give you the trig ratio as a fraction, and then they ask you for the exact value of another trig ratio. We'll talk about this as we get closer to the test. So let's try this, example six. The secant of 225, what's the first thing I'm going to do? Well, the first thing I'm going to do is what I've been doing for most questions, I'm going to do a sketch. I'll do mine right here, maybe a little straighter. 225 degrees, okay, finally Brett, we're giving degrees a nod, but we're committed to radians now here, okay? 225 degrees, there's 180. I think 225 is, right there, C-A-S-T. First thing, can you tell me is the answer going to be negative or positive? What is the secant of 200, well we don't have a rule for secant. What does secant go with? Okay, cosine, positive or negative in that quadrant, negative, secant therefore, positive or negative in that quadrant. I would do this, now if I give you this question, it'll be a multiple choice question and you'll cross off all the positive answers, I hope so. I want my reference angle, now my reference angle is this one right here. How big is this angle? Yep, 45 degrees, have I memorized a triangle that has a 45 degrees in it? Yeah, which one? Okay, I'm going to draw carefully. The 1, 1, root 2, and this is the 45 degree angle. So is the top one by the way, but I usually use the bottom one if I have a choice. Dominique, what did secant go with, which trig function? Cosine would be adjacent over hypotenuse, secant therefore is hypotenuse over adjacent. It's root 2 over 1, am I going to write the over 1, no. By the way, I'm not giving you a degrees question on your test. I like b, I like b, I like b. I'm going to give you radians and you know what, I'll probably give you one of the primary trig functions and then one of the reciprocal trig functions, although maybe I'll just do both in one because it's like this, it's a nice way to test. Cotangent, and you'll notice I know the angle, what's the angle, pi pi by 3. So you know what I'm going to do first, sketch the angle. Now I'm into radians, so I'm going to call this 3 pi by 3 and 6 pi by 3. Ready Alex, how far? And 4 pi by 3, right about there, 5 pi by 3, about there, C A S T. Let's see what does cotangent go with because it's my reciprocal. Okay tangent, positive or negative in this quadrant. So cotangent therefore positive or negative. So I would happily go, hey I know the answer is negative, yay. Brett, I would also like the reference angle. Now the reference angle is this one, how big, pi by, see if you get the common denominator trick, I'm hoping you're recognizing, you can almost intuitively figure it out. Yeah, pi by 3, have I memorized a triangle with a pi by 3 in it? Yep, which one? I'll give you a hint, the other one, sorry. I'm going to draw it like this, 1, 2, root 3, that's a 90 degree angle. Which angle, Brett, is the pi by 3, the bottom one or the top one, right there? Tangent is opposite over adjacent, so cotangent will be adjacent over opposite, help me out. Now the back of the book, if they give you this, will rationalize the denominator, they will write this as root 3 over 3, you guys don't need to. So when you're checking your answers, you may need to do that whole change it to a decimal on your calculator, and then change their answer to that small on your calculator, and see if you're right trick. Now this says sine squared, something plus sine squared, remember this really means the sine of 3 pi by 4 times the sine of 3 pi by 4 plus the cos of 3 pi by 4 times the cos of 3 pi by 4. You really need to find the sine and the cos of 3 pi by 4. Thankfully, Ellen, it's the same angle for both, so you know how I'm going to start, sketch. My denominator is 4, so I'll call this 4 pi by 4. I guess this is 8 pi by 4, but I'm not going to bother labeling it, Vlad, because I don't think I'm going to need it. 3 pi by 4, Vlad, I'm almost positive, is, right there, 2 pi by 4 is straight above, right, C, A, S, G, sine is going to be positive, cos is going to be negative. Although is that going to make a difference? I don't think so, because when I square the negative, I'm going to lose it, but I'll put that there. Stanley, what's my reference angle right here? I think you're right, Vlad or with authority, pi by 4, isn't it, because this is 2 pi by 4 right here. My reference angle is pi by 4. I memorized a triangle with a pi by 4 in it, which one? Let's write that over here, 1, 1, root 2, where this angle here was pi by 4. Stanley, what is the sine of this angle? This is going to be 1 over root 2 all squared minus, and what's the cosine of this angle? I've got that all squared. Turns out, because this is a 45 degree or pi by 4 angle, the sine and the cosine are the same. It's not an athosceles triangle, so the opposite and the adjacent are the same length, and the hot news is going to be the same. Hey, let's keep going, Stanley, this is fun. What's 1 squared over, what's root 2 squared, just plain old 2? What's a negative squared 1, you know what, I think this ends up being 1 half plus 1 half, which I'm guaranteeing you can do in your head. What is 1 half plus 1 half? All of these are questions where I know the angle, where they gave me the angle. In a few minutes, we're going to ask, what if instead of giving me the angle, what if they give me the answer and they want to know what angles could have been here going backwards, and that's when we'll use the cast and reference angles as well. That's on example 5. Log and Trigs together? Wow. That's like, Prince William getting together with Kate Middleton, look at marriage made in heaven, look at that. Oh boy, I can hardly wait, I got to wipe a little drool off my chin, this is so awesome. Find the, Sabina, what's the word after the, oh, there's my little trigger phrase which tells me special triangles, unit circle, and also X, Y, and R. We're going to do bed mass, which means we're going to do it inside the brackets first. By the way, what's my base? How much do you want to bet I get a 2 or a root 2 or something in here, because a 3 is really going to screw this question up without a calculator. What's my base over here? How much do you want to bet I get like a 2 or a root 2, because a 3 is really, a 3 doesn't play well with 4, but 2 plays well with 4s and 2s, let's see. First thing we're going to do is we're going to do our humble sketch. Here's 4 pi by 4, here's 8 pi by 4, 7 pi by 4, Spencer, I'm pretty sure is right there. This would be 6 pi by 4 down here, yes. C A S T, Spencer, which trig function did they give me? Cosine, positive or negative in this quadrant. By the way, I knew that was going to happen because can I take the log of a negative? See, I almost could have skipped that step if I was really trying to cut corners. I wouldn't, but I was like, it's got to be positive, otherwise this question seemed pretty simple. No answer. Spencer, what's my reference angle? I totally agree with you. Have I memorized a triangle with a pi by 4 in a, yes I have, this is the 1, 1, root 2. Now you probably have that drawn up here. I would encourage you in your notes and in your homework, draw it every time because from personal experience, Trevor, every year there's a kid that gets to the test, the non-calculator test and you know what they do? They draw the triangle wrong and all I can think is they probably only did it once on each page in their homework and assumed that was enough to remember it. In fact, by the way, when you get the test, you know what you'll do, the very first thing at the top of the page, probably draw both special triangles and then some kids also write out sine is y over r, cos is y, and then they even draw the cast rule and you've got almost like a little cheat sheet for the rest of the test. I would certainly consider doing that. I think that's clever. You said the 1, 1, root 2, where this angle here is pi by 4? Okay. What is the cosine of this? Cosine is adjacent over hypotenuse, come on, you're in physics. This should be walking the park trick, even first thing in the morning when your eyes are fluttering. It's x over r in terms of x and y and r, but you have a triangle here. I just want the solcatoa one. In other words, what over what? Cosine of this is what over what? Nope, that'd be 10. One over, there is no 2. Strike 3, oh no, at least you bumped it on 3, okay. One over root 2. So here's what this really is. This is the log base 2 of, now I always have to pause here to see who can tell me what the answer is. We're going to simplify this if you can't see what the answer is. But the best of you should be able to right now tell me 2 to what power equals 1 over root 2. Can you see the, I know logs was a while ago, Brett, oh thank you for restoring my faith in you after that whole radian's degrees fiasco last week. Thank you. Yes, negative 1 half. And by the way, if you've got there, great. Let's prove it. First of all, this is the same as the log base 2 of 1 over 2 to the 0.5, or to the 1 half. You know what? I'll put a 1 half there. Cos square root Maria was the same as 1 half power. Dominique, where is that 2 to the 1 half on the top or on the bottom of a fraction elevator to an elevator reintroduce the negative, yes. And now Maria, I can move the negative 1 half to the front. And I get negative 1 half log base 2 of 2. And what is the log base 2 of 2, Maria? What's the log of anything of itself? You get negative 1 half. Well played for remembering that, by the way. Also a nice review of logarithms. Are we going to beat these over 10? I would feel comfy putting one of these as a tricky multiple choice. Not going to be 10 of these. But my little math nerd heart right now is going pitter pat like you wouldn't believe. B, a log and a cosy can. OK, sketch. 510 degrees, well, Eric, how far have I just gone? 360, and then it's the 9 times degrees, 9 times, that's 450. That would be 540, so 510 is in there somewhere. Where this is 540 degrees, C, A, S, T. Justin, cosy can't positive or negative in this quadrant. By the way, I knew that because it can't take a log of a negative. And Justin, while we're at it, what's my reference? We'll get a 30 degrees. Have I memorized a triangle with a 30 degrees in it? I have, which one? Draw the little skinnier method of exaggerating. One goes there, two goes there, root three goes there. What's 30 degrees? The bottom angle or the top angle? OK, so what's the cosy can't of that? Let's see. Cosy can't goes with what trig function? Sine would be opposite over hypotenuse, which would be 1 over 2, 2 over 1, or just plain old. In fact, this whole thing works out to the log base 4 of 2. OK, this one I am going to see if you can do in one step. 4 to what power is 2? Ellen, square root, 4 to the 1 half. OK, the nerd within me sort of wonders what mode am I in? Oh, I'm in degrees from physics. The log of how the heck would I do cosy? Oh, 1 divided by the sine of 510. That's base 10. How do I change it to base 4? Divide by log 4. I've always kind of wondered how the heck it is. My calculator is doing it, though, because it's doing some kind of numerical approximation. Anyways, it can't do the way we did. It doesn't know the exponent rules or the cast rule. I'd like to see the software sometime. Turn the page. Using exact values to solve simple trig equations. I want you to notice the difference. Joel, here, I don't know the angle. They gave me the triangle portion. Here, I knew the angle. I needed to find the triangle portion. See the subtle difference? This is what I'm going to ask you, among other things, on the written section of your test. In other words, I'm telling you more than I like this question. I'm married to this question, except I hate one thing about this. What do I hate about A? And why do I like B better? OK, let's scribble out the degrees. And let's put a 2 pi right there. We're done with them. Now, we have solved trig equations before. But Ellen, we used our calculator. We're going to have to replace the calculator portion now, shift sine, shift cos, shift tan, with, hey, what triangle? So you ready? What trig function did they give me, Ellen? What trig function did they give me? Cotangent. I'm going to go like this, C-A-S-T. You'll notice on the previous ones, Maria, where I knew the angle, I drew the angle first and then wrote in the cast rule. Now, I'm starting out with a blank canvas. The cast rule, I need to figure out what quadrants we could have been in. Now, it's cotangent. I don't have a rule for cotangent. What function goes with cotangent, Ellen? Tangent. And they told me tangent was negative or positive. So where is tangent negative? No, no. Here, here. That's step one. By the way, this is a fraction. No, it's not. Yes, it is. It's root 3 over what? So I'm really saying, now I need to find the reference angle. How will I find that? From my special triangle. Which special triangle? Do I have a triangle that has a root 3 and a 1 in it? Let's draw it. 1, 2, root 3. Which trig function, Ellen? Cotangent. Cotangent is adjacent over opposite. Which of these angles has an adjacent over opposite of root 3, the bottom one or the top one? Which of these, when you go adjacent over opposite, do you get root 3? The bottom one or the top one? Bottom. What angle is that in radians? That's my referencing. This is pi by 6. This is pi by 6. Pi by what? Justin, I'm going to call this right here 6 pi by 6. And I'm going to call this right here 12 pi by 6. Pi and 2 pi. But why not give myself a common denominator and cut myself some slack? The first angle is that big. Emily, how big is that first angle? Well, let's see if that's 6 pi by 6. And my reference angle is pi by 6. You're right, I think. Say it nice and loud. Did you say 5 pi by 6? Yes. My second angle is that big. Brianna, how big is that one have to be? Got to be 11 pi by 6. Got to be. Similar to the process when we knew the angle, except this time, we're using the Castro to figure out the quadrants. We're using the answer, the fraction answer that they gave us to figure out which triangle we're going to draw. And from there, we can figure out what reference angle using good old Soto. And once you know the reference angle, it's some basic subtraction or addition. Caitlin, can you see I don't know the angle? So I'm going to start out, hey, with the cast rule. You'll also notice, fib a little bit to you, when we started doing secant and cosecant and cotangent with our calculator, I asked, how do we solve secant and cosecant and cotangent? We don't, I always said, we flip it. Actually, the nice thing about this is I really can do the flipping when I go adjacent over opposite instead of opposite over. I actually flip the trig ratio. I don't actually rewrite this as, well, this is secant. I could, I guess. Secant goes with which trig function, Caitlin? Cosine, was it negative or positive? Here or here? And what did they tell me the answer was as a fraction, Caitlin? Negative, ignore the negative, sorry. Two over what? Two over what? It's meant to be, next you do. Yes! Said that again, ready? As a fraction, what did they give me the answer? Two over what? Do I have a triangle with a two and a one in it? You see why we had to add that one in there this time? Which triangle has a two and a one in it? Yeah, careful, not the one, one, root two, that has a root two and a one in it. The one, two, root three, let's sketch. Caitlin, secant goes with which trig function? Cosine is adjacent over hypotenuse, so secant, therefore, is hypotenuse over adjacent. Yes? So here's what you wanna ask yourself. Which of these two angles has a hypotenuse over adjacent? Top one or the bottom one? This one? I can't hear you, I'm sorry. Yes, let's see, hypotenuse over adjacent would be two over root three. Is that what we have here? So which of these angles has a hypotenuse over adjacent? Yes, hypotenuse, the top one, the top one. Which is how big? And that's my reference angle. And Caitlin, how big is right here, straight across? Three pi by three, can you see now my two angles? I hope, I hope, I hope, I hope, I hope. Here's the first one, boom. Theta one, how big, kiddo? I totally agree. Theta two, how big? Four pi by three. By the way, let's prove it. Secant, you said it was one, one divide, oh, I gotta go into radians, Mr. Dick. Secant is one divided by the cosine. One of our answers is four pi by three. This should work out to negative two, does it? Woo! Of course, you can't check on your test with no calculator, but I think we need to do one more. It's no room, Mr. Dick, yeah, there is. Can you scroll, turn the few pages, get to the answers, because below the answers, I believe you guys got a half page-ish. Solve, giving your answer as an exact value. Cosy cant squared theta equals two. Zero less than or equal to theta, less than two pi. Go once around the circle and stop. Don't write this down, but just watch. Is that the same question? How would I get the cosy cant squared by itself? Five by two, or is that the same question? How would I get the cosy cant squared by itself? So in your homework, you may be asked to do some fairly basic equation solving, getting the trig function by itself, by either dividing or multiplying or crossing. That's fair game. But I don't have the quick trig function by itself. I have cosy cant squared by itself. How do you get rid of a squared? Okay, first thing I'm gonna do is recognize that this is actually cosy cant of theta equals the square root of two. Oh, no, wait a minute. When I square root both sides, what must I remember? This question. Do I know the angle here? No, no, no, no, no. So this isn't quite as easy as the other ones. I'm gonna have to use the cast rule to figure out what quadrants I'm in. Now let's see. C, A, S, T. Sorry, what is cosy cant go with? Which trig function? Sine. This says sine is positive or negative. Sine is positive here and here. Sine is negative here. Oh, there's gonna be four answers. And Caitlin, what is this answer as a fraction? Root two over what? What we're really asking is, do I have a triangle with a root two and a one in it? Do I? Why, yes I do. That would be the one, one, root two triangle. Now cosy cant, a star goes with which trig function? Sine is opposite over hypotenuse, so cosy cant's gonna be hypotenuse over opposite. Which of these angles has the hypotenuse over opposite of root two? The bottom one or the top one? It's actually a trick question. Turns out, you know what? They both do. I'll use this one here. It's cos. How big is that angle, Asar? Oh, you don't need to ask that question. I told you by giving you a domain. Nope, nope, nope. That's the other triangle. Five by four, okay? By the way, pi by three goes in the triangle with the three in it. Is six also multiple of three? That goes in the triangle with the three in it. Don't wait to remember, but you know. That's my reference angle, Vlad. What that means is that this angle, this angle, this angle, and this angle are all pi by four. Pi by what? Let's be clever. I'm gonna call this four pi by four and I'm gonna call this eight pi by four. Why not cut myself a bit of slack? And now I go, Asar, there's my first angle right there. I'll call that theta one. How big is theta one? Right here, just one pi by, right? The reference angle. Remember, each of these little squiggles is pi by four and that's just a squiggle by itself. Now the second one goes this far, which means Asar, it's gonna be four pi by four, take away a squiggle, three pi by, yeah, now we're onto it. Good, good, good. Let's keep going. This is kinda literally fun. How big is this one? My pi by four is absolutely, absolutely, absolutely. And how big is the very last one? Seven pi by fours. That's solving trig equations as exact values without a calculator. I have to give you a multiple of pi by four or pi by six or pi by three, or I have to be right here. So in addition to the homework from last day, so right now I think I gotta sign one, two, three, four, all, you can try nine, eight. You can try number 10, except you can cross out E and wonder where D went to for that matter. 11 all is good, 12 is good, 13 is good. And then I also have for you this sheet right here. Let me hand it out. So I have this sheet, it says solve for, without using your calculator. There's two questions, grand total, the answers are attached. This may look familiar, this was actually from a couple of days ago when you were doing the homework in your book and I was saying to you, why is the answer in the back of the book an exact value but we're only doing decimals is before we knew how to do these special triangles. I just recopied this. And some of them will get fairly repetitive because there's only so many multiples of 30 degrees, 60 degrees and whatever I can ask. But you need practice on this, okay? So the answers are attached. You got holy smokes like a half hour more than, hey probably no homework if you hustle. Kick butt. Take home quiz next class. Officially about three or four more lessons until we're at the test. So I'm thinking about two and a half, three weeks from now ish.