 Welcome to our review for Exam 2 for Math 1060 Trigonometry for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. If you're watching this video, it means you've probably already taken exam number one. And so specific policies, procedures, timing of the exam, I'm not really going to go into here because many of those policies and procedures will be identical to what you saw on exam number one, so refer to that. And also specific like time, place, and manner changes from semester to semester. So in that regard, please consult the syllabus or the information on Canvas, or if you do have any questions about those specific things, feel free to send me an email. No problem with that. For this review video, I want to focus primarily just on the topics, the contents of exam number two to help you in your studies, as that's probably why you're watching this video right here, of course. So for exam number two, I should mention that this will cover the topics from lecture nine all the way up to lecture 20, lecture nine to lecture 20, for which there's basically three main ideas covered on this exam. There's going to be the topic of graphing trigonometric functions. That's what we talked about in lectures nine through 13. Then the second topic will be about inverse trigonometry, the inverse trigonometric functions, arc sine, arc cosine, arc tangent, and the like. That was talked about very briefly in lectures 13 and 14, but still it's a very critical topic that you'll see lots of questions about inverse trigonometric functions in this exam. And then the remaining topics, specifically lectures 15 through 20, will be about trigonometric identities. This was the largest unit in our class, and so they're going to see a big portion of that on this exam. Also, one thing that you're going to see on this exam that you might have not seen on the previous exam is that a lot of the topics are going to be integrated together. That is, there will be questions that do involve topics from multiple sections. Just for a few examples, we've covered how to compute inverse trigonometric functions, things like sine of arc tangent of x, how do you compute something like that. But then later on in unit six, when we learned about trigonometric identities, we combined that so we could do things like sine of arc tangent of one half plus arc cosine of one third, things like that. So many topics will be integrated together and you'll see that and I'll point these things out throughout this video right here. The length of the exam will be the same as the previous exam. You'll see 15 questions. Nine of those questions will be in the multiple choice section, which they're worth five points each. Remember, in the multiple choice section, you will not be allowed any notes or calculator with the exception of a formula sheet that will be provided to you. And on that formula sheet, you will have many of your desired trigonometric identities, the fundamental ones and many of the families we've talked about. So that'll be of great help to you on that section. Questions 10 through will technically 16, but 10 through 15 will be the free response section for which most of those questions will be worth 10 points. Question 10 will be only worth five. Question 12 will be worth eight. And then also question 16, which is just turning your no card type question. That's worth two as usual. So let's get into the nitty gritty topics of this exam. So in the multiple choice section, we'll start with that. Let's look at question number one. Question number one, you'll be asked to compute without a calculator. Because remember, every question in the multiple choice section will have no calculator allowed. So you have to do this on your own. And question number one will ask you to compute the value of an inverse trigonometric function. So can you compute arc sine arc cosine arc tangent arc secant arc cosecant arc cotangent without a calculator. Now this this when you have an inverse trigonometric function, the number you put in is the ratio. The thing you get out will be the angle, which that angle could be expressed in radians as you see right here. Every option is in radians or that option could be in degrees. It will depend on the multiple choice questions. So you should be familiar with both. Now what ratios do you need to know? Well, this is going to coincide with our special angles. So thinking of our classic unit circle diagram right here, you need to know zero degrees, 30 degrees, 45 degrees, 60 degrees, and 90 degrees. You need to know sine cosine their ratios for those five special angles in the first quadrant. And as such, you should also be able to compute tangent cotangent secant cosecant for those five special angles. But that also encompasses the other angles on our unit circle diagram that reference these. So you should be able to do things like 120 degrees, 135 degrees in the second quadrant. She will do 300 degrees in the third quadrant, 270 degrees, etc. So you should know those angles from the first exam. And now the inverse trigonometric ratios are going to go the other way around. You've given the ratio, you need to find the angle. So be able to identify those for question number one. Clearly question number one has to do with inverse trigonometric functions. So that was talked about in lecture 14. Now technically lecture 13 is also in our unit on that regard, lecture 13 focused primarily on the general idea of an inverse function. And then lecture 14, it focused exclusively on inverse trigonometric functions. So I would primarily point you to lecture 14 in that regard. Question number two is going to hit on the second main topic about graphing. So you'll be given a trigonometric function. So sine, cosine, tangent, cotangent, secant, cosecant, we learned how to graph each and every one of these transformations applied to the function change aspects about it. So this question specifically is asking about the amplitude, how do you identify the amplitude of a sinusoidal wave? What about a period change? What about shifting of sometimes shift up, shift down reflections? So really question number two is going to ask you to analyze the properties or the anatomy of a trigonometric graph. So I would tell you to primarily focus on for this lecture here, you're primarily going to focus on 10 and 11. Because this is when we talked about transformations to sinusoidal waves, so how you transform sine or cosine. Of course, you do need to know lecture nine, where many properties of the trigonometric functions were discussed, but these were without transformations whatsoever. And then also lecture 12 is relevant here as well. Because lecture 12, we focused on graphs of the non sinusoidal trigonometric graphs, so tangent, cosecant, etc. But like I said, primarily this question will focus on 10 and 11. But these other topics will be of interest to you as you're studying. Question number three will also be a question about graphing. In this question, you will be given the graph of a trigonometric function. And then you'll be asked to identify at least by the process of elimination, which of these formulas produced this graph. So there's really two ways you could approach this. Again, you could just do process of elimination. Well, that can't be right. That can't be right. That can't be right until you get to whatever you think the correct answer is. Or you come to this graph and analyze things like what's the amplitude? What's the period? Is there a shift? Is there a reflection? Does it look like a sine? Does it look like a cosine? Does it look like a tangent? So you want to recognize the graph and then find the formula you think is the correct answer, assuming that that one. I don't claim the one I just circled was the correct answer as well. So I tell you to consult the same sections, right? Focus primarily on 10 and 11. But of course, you do need to know lecture nine, which is the basic shapes of these graphs, especially sine and cosine. And then maybe you'll get a tangent or secant or something. So be familiar with lecture 12 as well. But again, primarily 10 and 11 is when we did things like this. Question number four, just the last question of the first page of the multiple choice section. In this question, you'll be asked to compute sine, cosine, tangent, secant, cosecant, or cotangent for an angle, which is a multiple of 15. Let's just say it that way. Because again, with that unit circle diagram we drew earlier, you should know multiples of 30. You should know multiples of 45 pretty well, right? So zero, 30, 45, 60, 90, and then continue on in that direction to 120, 135. You get the idea. Keep on going in that direction. You should be able to do those. But how do you do something like sine of 15 degrees? Well, 15 degrees, you'll notice, is half of 30 degrees. That's not what I meant to write. It's half of 30 degrees. So if we know sine of 30 degrees, we could use a half angle identity to find sine of 15 degrees, or even better, 15 degrees is 45 degrees minus 30 degrees, like so. So you use like an angle difference identity. Or another possibility is what about 75 degrees? 75 degrees is 45 degrees plus 30 degrees, for example. So could you compute secant of 75 degrees? Absolutely, because you could compute cosine of 75 degrees using the angle sum identities. And so that's how this is going to work. There's going to be these memorized angles. You need to know sine of 30, 45, all of these angles we talked about on question number one, you need to know all of those. Those need to be memorized, because you won't have notes that allow you to do those things. At the very least, remember how to reconstruct this unit circle diagram on the fly, right? You think of your hand, right? You have your five fingers. Oh boy, that's a great drawing there. Think of your fingers, right? And you have zero, you know, 30, 45, 60, 90, right? Those are the five special angles in the first quadrant. And remember the pattern, you know, for cosine, it's going to go one, root three over two, root two over two, you get the idea. Remember that pattern. You do need to know that for this exam. Otherwise, some of these questions will be basically impossible to do. Now, question number four, you didn't know those special angles and things I've referenced to them, but to do these multiples of 15, like 1575, et cetera, you need to use appropriate trigonometric identities. Now, while you do have to have those special angles memorized, the trigonometric identities will be available to you on a formula sheet. So this, these identities will include the fundamental identities we talked about. So the Pythagorean identity, the ratio, the reciprocal identities, the symmetry identities, the co-function identities, complementary identities. You need to know those ones, but they will be on the formula sheet. You'll have the angle sum and the angle difference identities. These will be on the formula sheet. You'll have the double angle identities that will be on the formula sheet. You'll have the half angle identities. Those will be on the formula sheet. So if any of those are appropriate for question number four or other questions, you are welcome to consult them. You will also have the product to sum identities. You'll have the sum, the product identities as well. Those ones are very difficult to memorize, but those, no worries. Those will be on your formula sheet. So combine those with, with the angle, special angles you've memorized and you could be able to do something like number four without a calculator whatsoever. So moving on to the next page in the multiple choice section, you got question number five. You'll be asked to identify, you know, basically some more properties of, to a bunch of graphs, particular lecture nine and lecture 12 will be very important here. So for example, you might be asked a question, when is sine equal to zero? That is, when are, where are the x-intercepts of sine, where are the x-intercepts of cosine, of tangent, of cotangent, where are the x-intercepts of secant or cosecant? Well, in that situation, they don't have any, so you'd select something like do not, they don't exist, DNA. Could you identify what the amplitude of a standard sine or cosine wave is? No transformations whatsoever, periods. Could you identify where the vertical asymptotes of tangent secant, cosecant or cotangent are? Could you identify where's a maximum for sine, where's a minimum of sine, where's a maximum for cosine, etc., those type of things. So you'd be asked to, to provide some basic information, some basic properties of the graph of tangent and sine, all these trigonometric functions. Question number six, this question, you'll be given either a product of trigonometric functions for which you'll turn it into a sum or difference, or you'll be given a sum or difference and be asked to turn it into a product of trigonometric functions. So as you might be able to guess, you will need to use the product to some identities or the sum to product identities. You do not need to memorize these identities because they will be provided to you on the formula sheet, but you do need to know how to use them to convert from products to sums or sums to products. And so this is something we talked about in great detail, of course, in lecture number 20. So that's definitely the one you want to see there. Question number seven, you'll be asked to compute the trigonometric ratio of an angle defined using an inverse trigonometric function. So you have tangent of cosine inverse of x right here. And this is going to be left as a variable, that's okay. In regard of your ratio, just treat the ratio as x over one. And so then you have to compute the cosine or tangent ratio of x over one, I would recommend drawing your right triangle diagrams right here, which you know you call the you call the inverse function theta or whatever. So you'll draw your triangle label sides and then compute the ratio from your triangle. That's how I'd recommend doing that. This was one of the main topics we talked about in lecture 14 about trigonometric inverses. So be prepared to do that for question number seven. Question number eight, you'll be asked to compute a trigonometric ratio using trigonometric identities where some information on trigonometric functions will be given. So like in this example, you are given the cosine ratio and you're given the angles for which a lived between. So okay, we see that a resides in the fourth quadrant. And then you'll be asked in this case to be asked to compute cosine of A over two. But we could be asked to find cosine of two A, maybe two ratios are given two angles are given A and B. So you might be asked to compute like what's cosine of A plus B. We saw several examples like this in the homework in the lecture series. So given some information about the trigonometric ratios about the angle, find other information using identities. That's going to be the main takeaway for question number eight. So this of course, you know, this this we could find things from lectures 15 all the way to 20. This particular question clearly is using the half angle identity, which was covered in lecture 19, I believe. But we did similar things with double angles in lecture 18. We did things with angle sums and angle differences in lecture 17. So be able to compute trigonometric ratios, knowing some information about the trigonometric ratios of the original angle and then modified using trigonometric identities. All right. And then question number nine, this is the last question, the multiple choice section, you'll be given you'll have to compute a trigonometric substitution using some type of identity, right. So you're given the substitution x equals four sine of theta. So remember this leads itself to some type of trigonometric right triangle, excuse me, right triangle diagram with respect to theta with that diagram, you can then say things about trigonometric ratios like sine cosine tangent, et cetera. But then you have to simplify or you have to write the trigonometric expression in this case, sine of two theta over four in an algebraic sense. So how do you rewrite this thing with no variable theta whatsoever? Well, because you have things like double angles, right? Double two theta is not the same angle as theta. So you'd have to use some type of trigonometric identity, the double angle identity seems appropriate here to rewrite this in terms of just the algebraic expressions, you have to rewrite this expression just in terms of theta. And then once you have in theta, you can use the triangle to write in terms of just this algebraic expression whatsoever here. And so again, we did several of these examples throughout throughout unit six about trigonometric identities. So consult those ones as well, lectures 15 through 20 30 not that's not right 20. We did these type of trigonometric substitutions back on exam one. But now we're using trigonometric identities to help us compute these things. And that brings us to the end of the multiple choice section, the no calculator section. So do remember that in this section, you're not allowed to calculate no no no no notes except for the foremost you I mentioned. There's also no partial credit just based upon the answer you select. So make sure you're very careful and check your work there. There will be some space provided, some blank space provided on the page where you can write, but of course, no work is necessary, you won't be graded on that whatsoever. So moving to the free response section, the first one section is shorter and that it only contains six questions, although these questions are a little bit more involved, they're worth more points in general. And you can get partial credit based upon the work you show and you were expected to show all of your work on these questions as is appropriate. So let's talk about the specifics of this. Question number 10, you'll be asked to graph a trigonometric function. This one will not be signed or cosigned, there'll be a question later on that asks you to graph sign and cosine, because we spent a lot more time on graphing sign and cosine than we did the other four. So question number 10, it's only worth five points. So it's actually on par with a multiple choice question. Really though, I put this in the free response because graphing becomes a lot harder. I shouldn't say harder graphing becomes a lot different when it's a multiple choice versus you actually draw the graph yourself, which is what you're expected to do. So on this one question or 10, you'll be asked to graph just a single cycle, one complete cycle of a tangent, a cotangent, a secant or a cosecant graph. There will be some very minimal transformations going into play here. So like this one is just tangent of 2x. What does that two do to the graph? Tangent, cotangent, secant and cosecant do have vertical asymptotes. So do make sure to include any of those asymptotes on there. Label the x-intercept. If it exists, if there's a midline that's been modified, if there's some shifting of some kind, you might want to add that midline to your graph. And there you go. There's really not a lot of work you can show. I mean, there are some things like you could list. What is two doing here? Is it a period change? Is it an amplitude change? Is it a shift? You could list some of the transformations over here. That's some good partial credit. And then draw your things on the grid lines provided. While some of these things are listed for you, some of these things might not, you can of course change the labels if you think that's very appropriate for you as you graph this thing. This one's worth only five points. And this will be a question directly taken from lecture 12 about graphing the trigonometric functions other than sine and cosine. We'll get to that one in a second. Question number 11, I mentioned earlier at the beginning of this video. In fact, you'll be asked to compute a trigonometric expression that involves inverse trigonometric functions, but also will require identities of some kind. So on this question, you have these two inverse geometric functions, each each and every one of them represents an angle like angle alpha angle beta, but then we've added them together. So this thing that's like equivalent to sine of alpha plus beta, you might want to use an angle sum or angle difference identity to help you out here. We did some other examples similar to this involving double angles or half angles. So use your, use your knowledge of inverse trigonometric evaluation combined with your trigonometric identity. So this is really combining unit five about inverse trigonometric functions again, that's primarily lecture 14, but it's going to be combined with our topics of from unit six about trigonometric identities. So this is like 15 through 20, which trigonometric identity will be appropriate? Well, it could be any of them honestly, but now you don't have the formula sheet on this portion of the test, but you all, you are allowed your no card and you can put whatever on that, whatever you can put whatever on that no card that you want, anything whatsoever. So if you want to just vomit all of the trig identities that were on that formula sheet onto your no card, do so, I would just suggest you write small and be prepared to do things like this. So what I would want to see is I definitely want to know what identities you're going to do. You should represent these things as angles, you know, like alpha and beta would be appropriate here and then draw the triangles, right? There's a triangle for alpha. If two angles are in play, there's a triangle for beta. You could do something like that as well. And then in the end, you should have a number. In the end, this should be a number. That's the final value you should be looking for. Question number 12 and question 13 will be asking you to prove trigonometric identities. We did this a lot in the homework. It's not just about using identities for computation. It's also about proving trigonometric identities. And you'll see these, of course, in the free response section. Question number 12 is only worth eight points. Question number 10 is worth 10. Excuse me. Question number 13 is worth 10. And that's because on park 13 will be a harder question than number 12. And I'll explain why. When it comes to question number 12, you will only need the fundamental identities to prove this thing, right? Only the fundamental identities will be necessary, none of the more exotic ones. So again, the Pythagorean identities, symmetry, complementary ratio, reciprocals will be what you see here. You see, of course, with the angles, all of the angles in play are theta. So you probably wouldn't need the symmetry. Symmetry comes in the play of a negative angle. You probably don't need the complementary ones either because that's if you take like 90 degrees minus theta. So probably you can get away with just the Pythagorean identities, maybe I don't see any squares. So I don't suspect any type of Pythagorean identities necessary, maybe just reciprocals and ratios, what have you. But remember the tips we've learned about proving trigonometric identities. Some of the important ones I'll mention right here, start with the left hand side, or you can start with the right hand side. If you think it's more complicated, pick the more complicated side, and then doing, you know, equality after equality, after equality, end up with the right hand side. That's what I should expect to see. If I see like a stack, you know, you're just you're doing, you're working on the two sides simultaneously, that's a big no-no that would forfeit most of the points on this question, potentially all of them. So don't do that. Remember, you can use algebraic things like foil, distributed, distribute things, clear denominators, multiply by conjugate, algebraic stuff is useful here. If you get stuck, you can switch the things, switch everything to signs and go signs. Admittedly that one's already there. But go back to the principles, the tips that we gave in lectures 15 and 16 here in our lecture series. There's a lot of tips given about proving trigonometric identities. That's what you're going to see, and that's what's going to be expected on question number 12. Question number 13 is going to be the same basic idea where you're asked to prove a trigonometric identity. But in addition to the guidelines and principles that you need to demonstrate from question number 12, you'll also need some of the more advanced identities like you look here, there's this cosine squared of theta over two, that tells me that maybe a half angle identity would be appropriate. There's also a sine squared over here, maybe Pythagorean identity or half angle identity could be useful there, maybe you have to go from theta halves to theta. So I would think the half angled identity would be appropriate. So can you prove trigonometric identities that use the more advanced trigonometric identities, like angle and angle sum and difference identities, product to sum, sum to product, half angles, double angles, all of those angles we saw there. So really, I will list again lectures 15 and 16, because you do know the principles, the guidelines of proving trigonometric identities, but you also need to know those identities, those more advanced identities we introduced in lectures 17 through 20. And again, these things could be listed, of course, on your note card. That's okay, but you'll have to use them to help you out here. And that's why this one's worth a little bit more points, because there's a little bit more going on on question number 13. So that's going to be the second last page, you're going to be asked to prove true trigonometric identities. Then the last page of the exam will be about graphing question 14 and question 15 will ask you to graph a sinusoidal wave, that is to graph either sine or cosine. So like on question 14, we're 10 points, you see this one right here, f of x is equal to three plus two sine of one half x minus pi halves right there. So you see, it's basically the whole enchilada. There is potentially a period change, a amplitude change, maybe we can potentially have a horizontal shift, a vertical shift of reflection could all be involved in these things right here. And so you'll be asked to graph this function. Now, you only have to graph, let's see what does it say, you really only have to graph one complete cycle, that's sufficient for these periodic functions. If you get one side, you can get it all. There's no labels on the graph on the grid lines whatsoever. So label them as appropriate. And one of the tricks about graphing a trigonometric function, honestly, is that, you know, if you put the labels at the very end, it's like always the exact same picture would have you. You do need to identify the transformations applied to sine for this one or cosine, if it was. So you'd be like, what does this three do? What does this two do? What does this one have to? And so we'd say things like, oh, the amplitude has now been changed to whatever, the period's been changed to whatever, there's been a shift of this upward, there's been a shift to the right by this. So list specifically what those transformations are. I should of course mention that if you look at the solutions to the practice exam, you can see exactly the type of things you need to show. And then of course, you need to graph the function over here. You need to include at least three points. So I know what's going on here. But when it comes to graphing these sinusoidal waves, really there are always five points you want to graph, right? So like for sine, you have the original x-intercept, you have a maximum, you have the next x-intercept, you have a minimum, then you have an x-intercept again. And so that's your typical sinusoidal cycle there. So you do that for cosine, of course, you start out a max, then you get an x-intercept, then you get the min, then you get an x-intercept, then you get the max again. So really there's like five points you really should be including. I'll put it as a minimum of three, but if you really want to get full credit here, you should be shooting for five to make sure your graph looks pretty good. All right. And then the last question, question number 15, which admittedly there's question number 16, but that's just a reminder to tell you to turn in your no-card when you turn your exam. That's just a pass-fail thing. That should be pretty easy. But question number 15 will ask you to prove, excuse me, ask you to graph a trigonometric function. This one will seemingly look harder than question number 14. But that's because we need to first simplify this thing and then graph it. So using trigonometric identities, again, like the Pythagorean identity, double-angle identity, etc., so using all of those identities, for which again the identities we talked about in lectures 15 through 20, using some type of identities, which you probably have these on your, either have a memorized or have them on your no-card, using some type of identities, you can simplify the thing. And then it'll simplify to be something like you saw above. It'll look like, you know, worst-case scenario, y equals k plus a sine or cosine of, say, b, all right, this way, x minus h over b, something like this. Again, sine or cosine, worst-case scenario, you get something like this. But then you can graph this using the techniques you saw, the same techniques used in number 14. Now, the thing is this one, once you've simplified it, won't be as complicated as you saw on question number 14. So you won't see like every possible transformation in play here, graph it here when you're done, and then graph it from zero to two pi. So that might be multiple cycles, that could be half of a cycle. The thing is we're going to graph it from zero to two pi, so it doesn't exactly predict what type of potential period changes could be happening. Maybe there are some, maybe not. So in terms of graphing, this will be an easier graphing question, but you do have to use identities to simplify it first. So you do need to know the principles of graphing. It will be a sine or cosine, of course. So you're going to be graph, you know, we learned how to graph those in lectures nine through 11. It'll be an easier graphing question, but you do have to use the identities to simplify it. Question number 14 will be the most challenging graphing question possible here, and it will be sine or cosine. So again, be able to graph something as complicated at the end of lecture 11. So lecture 11 is really where it's at. We build up to these more complicated graphs at the end of number 11. So that's what you should be able to do. All right. So that finishes exam number two. That finishes the free response section as well. And so hopefully this gives you a good idea of the topics that are going to be expected for this exam. Please consult the other review materials that you have available to you for this course. And of course, if you have any questions whatsoever, feel free to ask me. I'm here to help you as much as I'm able to do. So best of luck on this exam and I'll see you next time.