 Welcome to our review for the final exam for math 1060, trigonometry for students at Southern Utah University. As usual, I'm your professor, Dr. Andrew Missildine. In the final review, we're going to want to go through this pretty quickly because there's a lot of stuff one could talk about. The basic structure of the exams can be like the previous midterm exam, so I don't want to spend too much on that issue. There'll be two sections, a calculator section and a non-calculator section. The non-calculator section, there'll be multiple choice and the calculator section will be free response. You must show all of your work to get full credit. There will be 15 questions in the multiple choice section and 10 questions in the free response section. The time, place, and manner of the final exam changes from semester to semester, so I won't put that in this video. Instead, take a look at the core syllabus or Canvas or talk to me, your instructor, to get that information, right? This final exam is a comprehensive final exam, meaning that it potentially covers every topic we have studied this semester, including those topics that did not appear on exam three, like complex numbers, parametric equations, and polar coordinates and such. So we're going to talk about those topics in this exam and to, I'm going to point out what you should be aware of as you're studying for the exam. The short answer is you should really have to study everything they've learned because it's a comprehensive exam. In addition to this review video, this practice exam, the three previous exams that we've taken, their practice exams are good study tools for this test, with their accompanies syllabi, of course. And then I should also mention that the actual exams one, two, and three that you took in this class are also potential questions you could see again. They give you examples. So the three practice exams, for example, one, two, and three, the actual exams you took for one, two, and three, those six exams plus this practice exam are all tools you can use to prepare for the final exam. Some questions that you'll see in this practice exam didn't appear on any previous exams. I will specifically point those out when necessary so special emphasis can be placed upon those. So let's get into it. Some questions will have a lot of variability, some not so much. And I'll point those things out as we go through this exam. All of these tests will be, of course, randomly generated. So the things you see on this practice exam are only samples of what you could see on the final exam that you take and things you don't see on this practice exam. That doesn't mean it won't be on your final exam. Again, this is just an example, just a practice. And I want to point out to you the types of questions you should be expecting. So question number one, you will be given something very similar to this right here. That is, you'll be given the graph of some trigonometric function, maybe sine, cosine, tangent, secant, whatever. You'll be given the graph of the function and you'll be asked to identify what is the correct equation that gives you this graph. So we should be looking at things like period, amplitude, x-intercepts, reflections, the midline to help us out here. It is a multiple choice question. So at the very least, you could try to solve this by process of elimination. Removing answers you know are going to be incorrect. This question is similar to what we saw on exam two, number three. All right, the exact same type of question there. Question number two is going to be a question coming from exam number one. In this version of the question, you're asking about the midpoint. Here's two points, find the midpoint. Some variations we saw of this question were like find the distance between two points. We could also ask to find the equation of a circle, the general equation of a circle. These were all topics we saw on exam number one. This distance and midpoint question was asked in question number three. The question about circles themselves from question number four. So go back to exam one materials to help you study this type of question here. Question number three is going to be a topic from exam two. And it's going to have to do with inverse trigonometry. So like this one right here, we have cosine of sine inverse of x. How can you rewrite that without any trigonometry purely in algebraic form? That's something we did for question number seven on exam two. But also question number one about, can you compute something like tangent inverse of one? Can you do that without a calculator? That type of question you'll see. So question number three will be about inverse trigonometric functions. Question number two is going to be about a geometric question that doesn't really involve triangles whatsoever. Very elementary geometry stuff we did back in chapter one. Question number four has one of the most variety of all question types. But it's probably also going to be one of the easiest questions you can see on this final exam. Most of the variants you can see here will come from exam one. So this one wants you to convert from degrees to radians, but you could go from radians to degrees the other way around. Could you compute something like sine of say 150 degrees? Right? Could you compute something like cosine of 11 pi six? So using special angles and things that reference the special angles, either in degrees or in radians, could you compute those without your calculator? Or heck, could you just calculate the reference angle? If your angle is 150 degrees, can you compute its reference angle of 30 degrees? And so there's a lot of questions one gets from this. So on exam one, you have question number one, you did something like this question number five as well, where some questions we saw on exam one that you could do this, do things like this. I'm also putting it as a possible variation of this question from exam number two question number four. So as a reminder, on question number four from exam two, it's the same basic idea. Can you compute something like sine of 150 cosine of 11 pi six or what have you? But I actually didn't use one of the five standard angles. So not zero, 30, 60, 45 and 90, you should know those and those things that reference to it. On exam two, it was like, can you do sine of 15 degrees or cosine of 75 degrees? So using some of the trigonometric identities that we've learned, can we compute those? So for example, 15 degrees is 45, take away 30 degrees or you could use a half angle identity. 75 is 30 degrees plus 45 degrees. So using identities and special angles, can you compute this without a calculator? And I should of course mention that for the multiple choice section of the final exam, you will have the same formula sheet that you've used on the previous exams. Moving on to the next question number five. This question you see is asking a question about angular velocity. And so this was some of the stuff we learned about in chapter three of our series. And particularly this shows up on exam one, some of these applications of circles we've seen. So things like, can you compute the arc length? Arc length, things like s equals r theta. Can you find the sector area? That is the area of a pizza slice. So we got that the area is equal to one half r squared theta or in this case, can you do angular velocity where velocity v is equal to r omega? One thing that's very important with all of these formulas about angular velocity arc length are areas, the sector of an area. Your angle measure always has to be in radians in order to use this. The arc length formula, dependent upon radians and then these other ones derive from it and thus have that same requirement. You do have to use radian measure in all of these examples. So some conversion might be necessary. Some other units of conversion might be necessary as well. So you should know how to convert from degrees to radians. You should know how to convert from revolutions to radians. One revolution is one complete circle. That would be, of course, two pi radians. And also convert measurements of time and distance if necessary for a question like this. So we saw questions like this back on exam number one. I said, particularly question number six and number eight and the multiple choice section of exam one were similar to these type of things. And another question I should mention, number eight. So this is in line with question number six. Number eight was things like if you have the unit circle and here's a point, here's this, here's the x-coordinate, here's the y-coordinate. And then this gives us an angle of some kind, theta. Can you find cosine of theta? Can you find y, sine of theta? Well, that's just the x or y-coordinate. Could you find tangent? Well, that would be the ratio. So how does the unit circle give us trigonometry? What is circle trigonometry? That's what question number five is all about, that's circle trigonometry. Question number six is going to be a question about solving trigonometric equations. So we actually did this on exam three, in which case there is two questions in the multiple choice section, seven and eight. Question number seven was a basic equation of solving trigonometric, that is a basic trigonometric equation. Question number eight was when you change the periods, like this is a cosine of two theta. So how does that affect things? Be prepared to solve just a basic linear trigonometric equation where the period may or may not be changed. Question number seven, I'm going to kind of squeeze a lot of the graphical questions we've done on previous exams into this little botch, this little batch I should say, because it makes it easier to format the page. This is a question about finding the area of a triangle. So area we talked about on exam three, of course. In particular, when it came to finding the area that was question number five from exam number three. We learned a couple of different formulas like Huron's formula. If we have a side angle side situation, if we have an angle side angle situation, how do you find the area? You should know those area formulas in order to do that. But some other triangle diagrams we saw on previous exams, particularly exam number one, we had questions similar to number seven, number nine and number 10. Again, just as a reminder, questions nine and 10 from exam one, this was where we were given a triangle and asked to do some basic SO-KATOA trigonometry, like here's a three, four, five triangle, what's cosine of that triangle, what's sine of that angle, things like that. 30, 60, 90 triangles, 45, 45, 90 triangles, those type of calculations. Question number seven had to do with like the alternate interior angle theorem. So when we have parallel lines and we have a transversal, when can we guarantee that angles are congruent, vertical angles, supplementary angles, complementary angles, that type of business, course one and angles, all of that jazz. We had a question like that on exam number one, that was number seven, if I remember correctly. You could have one of those questions as a variant on number seven. Question number eight is going to be a question about complex numbers. We saw them sort of on exam three, sort of, right? Particularly question number four, what I mean by that is, unlike every other chapter, chapter three actually got broke up between exams. In fact, exam three did cover section 10.1 about complex numbers, but it actually only covered like the first half of 10.1, just the algebraic stuff. Can you add, subtract, multiply, divide complex numbers in their Cartesian form? You should be able to do that. Question eight could ask you to do something like that. But all of like the trigonometry we did with complex numbers, that didn't show up on exam three, but those questions did get covered in this class and thus could show up in the final exam. So for this version of question number eight, I wanted to give you a version of complex algebra that actually relies on trigonometry. So for example, these complex numbers are in polar form, they're in the trigonometric form, and you have to square them, right? Turns out the polar form actually makes it easier to do that. And so this is a good new question you can look at to help you prepare for and get anticipation on what type of things you could do. Can we multiply complex numbers in polar form? Can we divide complex numbers in polar form? Or in this case, can we take exponents of complex numbers in polar form? We should be prepared how to do that. In particular, this is stuff we did in section 10.2. Could we also just convert a complex number to polar form or back again? We will do that but not on this question. I'll actually show you where that shows up a little bit later. Section 10.3 will show up in the free response section, so we'll put a pin in that for right now. Question number nine is going to be a question involving trigonometric identities of some kind. So your formula sheet will be very helpful for this one. Trigonometric identities was the primary topic of exam number two. And so one version, so one type of question we saw on exam number two, number nine, like you see right here, we're given some type of trigonometric expression. We're given a trigonometric substitution and we're supposed to substitute this into the trigonometry and simplify it into just an algebraic thing. Since we have a cosine of 2 theta, a double angle identity seems appropriate here. Some other things we did on exam number two, you could have a product of some trig functions or some of trig functions and using the product to sum or sum to product identities, you can convert it into a product or sum depending on the direction you're told. That was like question number six. And then then also, can you compute trigonometric expressions using identities of some kind, like, oh, if sine theta is equal to one fifth and you're in the first quadrant, what's cosine of 2 theta? So using some identities, can we compute things like that? These are some questions that we saw on exam number two. You'll be asked one of those randomly for question number nine. So it's going to be a basic calculation that involves trigonometric identities of some kind. Question number 10 is going to be a new question and it's going to be about parametric equations. So this is what we did in section 11.1 in our series. That was not on any of the previous exams. So you will be asked a single multiple choice question about parametric equations. And so think about what we did in that homework assignments that if I remember correctly, that should be homework 35, I believe, about parametric equations. You'll be asked a multiple choice question very similar to what you saw in the homework. So be prepared for that on question number 10. Question number 11, this is what I forecast towards earlier. You're given a complex number in Cartesian form. Can you switch it to polar form? This is like we did in section 10.1, right? That did not show up on exam three, but it was something we talked about. So can you convert complex numbers to polar form or vice versa? Some other questions from exam three that you could be asked here instead, like questions number one and six from exam number three. Given a vector in geometric form, can we convert it to algebraic form? Or given a vector in algebraic form, can we convert it to geometric form? Can we find the components of the vector? This is essentially the same question because our whole coverage of complex numbers in chapter 10 was just about thinking of complex numbers as vectors in the plane. So this is like the vector 11. So this is the algebraic representation of the vector, finding its mod just an argument. It's just like finding its magnitude and direction. So the two questions are basically the same. So question 11 will ask you either about complex numbers, thinking of those vectors or just vectors themselves and convert between the two different representations we have. Question number 12 will be about polar coordinates. So you'll be given a point in either polar coordinates and asked to convert it to rectangular coordinates or you'll be given a rectangular point in Cartesian coordinates and asked to convert it to polar coordinates. So be able to convert back and forth between them. So this is a new question. We start talking about polar coordinates in section 11.2. But I should mention questions 11 and 12 are basically the same question, although the language is a little bit different. All right. So be aware that we've done the same problem three times with complex numbers, with vectors, and with polar coordinates, this conversion. And this really is just so katoa, right triangle trigonometry. If you know that, you're gonna be okay on a lot of these questions. Question number 13 will be another question about vectors. It won't require you to convert between the two representations, but you could do that potentially if you wanted to. One question could ask you about work, like this one right here, for which work is the dot product between force and displacement. We should be able to do that either algebraically, like this one's presented or geometrically. Could we also compute dot products, dot products or linear combinations? Can we add, subtract, scale vectors? That's the type of stuff you'll be doing on question number 13, some computation with vectors. And so this is stuff we saw on exam number three, particularly questions two, three, and nine from the multiple choice sections or what you could be given again. And what I say could be, basic idea of this exam is actually written by a computer program that I wrote, which the way it works is I actually have pools of questions, different types of questions I like to give to students. And then I just tell the computer, randomly pick from these pools. That's how we do the midterm exams. For the final exam, since it's comprehensive, it's like, oh, pick randomly one of the question pools and then randomly pick a question out of it. So when I say things like two, three, and nine, I'm saying this question on the final exam will randomly select a question either from pool two, pool three, pool nine, from question, or from exam number three. That's what I'm trying to describe here. Question number 14 will be a question about graphs of trigonometric functions, although you won't be asked to graph anything. So this will give you like the equation of a trigonometric function and you'll be asked to identify what's its amplitude or what's its period. Was there a vertical shift by how much? Was it reflected? Those type of questions, where's the vertical asymptotes, where's the x intercepts, where's the midline? Those type of questions we can glean from the formula of a trigonometric function that's related to its graph. Question number 14 that you see on the screen right now is a variant of that. It actually kind of goes the other way around. You have to actually set up a model for simple harmonic motion, given its amplitude and its period, and go from there. So again, that's stuff we did in our unit about graphing trigonometric functions. So as you're preparing for question number 14, you should be going back to exam number two, right? Particularly questions two and five. Give you some other examples of what you might see for question number 14. And now the last question, the multiple choice section will be question number 15. This is yet another new question. It's a question about polar functions. So this comes from 11.2. In this situation, you'll be given a polar graph and asked to come up with the polar equation that gives this. So this is kind of like question number one at the beginning of this multiple choice section, which in that one, you're given a Cartesian trigonometric function and asked to identify, that is given the Cartesian graph of a trigonometric function. Question number 15, you'll be given the polar graph of a trigonometric function and you'll be asked to identify it. Just like question number one, the process of elimination might be sufficient that you can rule out specific functions because it's like, oh, this point, if I were to like, oh, you know, if theta equals zero, that would be two plus three should be five. Oh, that doesn't work out. You can use points to help you out on this one, just like you could on the previous one. The previous one was probably a lot easier because we spent a whole lot more time graphing Cartesian trigonometric functions, but trigonometric polar functions, we spent a little bit of time on. We talked about limousines and cardioids and roses and things like that or daisies, whatever you want to call that flower. So you should have some proficiency there, but I don't expect complete mastery on a question like number 15. So the process of elimination is an appropriate tool to use on question number 15. And that then brings us to the end of the multiple choice section. So continuing on to the free response, remember the free response section, you won't have your formula sheet anymore, but you will be allowed a calculator and you have your note card which you can use as well. So make sure you fill that note card with useful things, right? Question number 16 will be a question about applications, like story problems, right? Oh, assuming I can spell this word, applications. Story problems we saw on exam number one. So what were some of the things we saw? We saw some circle applications. So you had some story problems which basically boiled down to either finding arc length or area of a sector or in this example, you have to find the angular velocity with some bike that's spinning or something, or excuse me, this one, you have to find the linear velocity translated from the angular velocity, okay? We also did some story problems involving triangles. So like how tall is that tree, how tall is the building, angles of inclination, those type of things. So those types of questions we saw on our first unit on exam one, on exam one specifically, we had questions 15 and 13, 13 and 15, 13 was about triangle applications, 15 was about circle applications. Be prepared to answer one of those types of story problems. Question number 17 is a new question and it's exclusively covering section 10.3 about complex roots. How do you find the square root? How do you find the cube root? How do you find the fourth root of a complex number or how do you use that to solve a polynomial equation that is find all of its complex roots? The basic idea is you wanna switch this number into complex, the polar form, excuse me, and then in polar form finding the roots is easy and then you have to switch it back to Cartesian form at the end. This might be one of the most challenging questions on the test. It's worth the most points. Six points doesn't necessarily sound like a lot but there's 25 questions total. In the multiple choice section, I should have mentioned this earlier, all of the questions are worth three points each and it's all or nothing of course for each question. You only get credit for selecting the correct response. In the free response section, some questions are worth five points, some are worth six points, the harder ones of course are worth six points and partial credit is possible for partially correct work there. Question number 18 will be a question when you're asked to grab a trigonometric function. This is stuff we did in exam number two. You'll particularly remember on exam number two, we had questions like number 10, number, what was the other ones? 14 and 15 from that exam where we had to specifically graph trigonometric functions given the graph. So we could graph something with sign, cosine, tangent, secant, cotangent, c, cosy, can't be prepared to graph any of those. Remember question number 15 from exam two, which is actually where this question was drawn from for the practice exam. This one actually requires we apply trigonometric identities first. So we don't know how to graph something like sine squared, but using the right trigonometric identity, we actually can convert this into something much easier to graph, something we do know how to graph and then we would graph it here, number 18. Question number 19, there's two variants for this question and they both come from exam one. The first one, what you see on the screen right now, this was similar to question number 12. That is, given a diagram involving the unit circle and similar triangles, you're going to use circle trigonometry plus your knowledge of similar triangles to verify that some, one of these line segments is equal to a trigonometric ratio. Like this one right here is tangent. That's what you're asked to prove. You might be asked to prove that this one, if I remember correctly, I think that's secant or whatever. Again, I'm not positive at the top of my head. Pretty sure that's secant. I'm pretty sure this one's cotangent and this one is cosy, can't remember correctly. You can try to verify them on your own if you want to practice a question like number 19. How do you practice it? We'll try those other ones I just mentioned. So you could be given something like we saw on question number 12 from exam one. Also another question you could be given with something like question number 14. So this is one where you have like, you have like triangles and lines and you knew something about this angle, this angle, this angle. So that adds up to 180. Maybe you knew something about these two angles. Those are supplementary angles. And that is, there are some variables in play like variable X, variable Y. So using properties of angles like, oh, angle sum adds up to 180 degrees. Supplementary angles add up to be 180 degrees as well. Complimentary angles add up to be 90 degrees. Vertical angles are congruent. Alternative angles are congruent. Correspondent angles are congruent. That type of stuff we learned about in chapter one. Can you set up and solve a system of linear equations to find out the variables X and Y? That was a question we saw. That was number 14 on exam one. You'll be given one of those here for question number 19. It's worth five points. All right, moving our way along. Question number 20. Well, you'll be given an oblique triangle like you see in this diagram right here and you'll be asked to solve it. So using a combination of the law of cosines and the law of signs, solve for this triangle. Find the missing pieces. You'll be given three pieces, some combination of sides and angles and you have to find the remaining ones. Beware of the ambiguous case though because there could be multiple solutions. There could be two triangles. There might be none, right? And so if there's two triangles, you need to just give me both solutions for full credit. If there's no solutions, you need to provide evidence why it cannot be solved. If there's only one solution, you would have to then give me that solution but also explain why there's not a second solution. That's why this one's also worth six points. There's a lot that can go into solving these oblique triangles. These type of questions showed up on exam number three. If you want some more practice, I'll look at questions 11, 12 and 15 or look at their corresponding sections in our lecture series. On this final exam, you will have to prove some trigonometric identities. In fact, there'll be two identities you have to prove. There's going to be question number 21, which is on the easier side of things. I'm actually just pulling this question directly from exam to question number 12. Not the same question you've gotten necessarily. It'll be randomized, but you'll be given a fairly straightforward trigonometric identity just using the fundamental identities. There'll be another one that shows up a little bit later. It's question number 24. Will you be asked to prove a trigonometric identity that involves the more exotic identities? When you're proving a trigonometric identity, remember the things you need to do. First, you need to work from the left-hand side, working all the way to the right-hand side. If you want to start on the right-hand side, move left, that's okay, not a big deal. Never work on both sides simultaneously. It needs to always be connected by equal signs. Some expression is equal to some other expression. If you don't include equal signs, it's not actually a proof of a trigonometric identity. We need to connect expressions by equal signs. Every step, as you go from one side of the equal sign to the other, that step shouldn't be too big. It should be like, oh, I applied one fundamental identity, or I did one algebraic thing, like I simplified a fraction or factored it. Every step should be fairly justifiable. If you take too big of a step where it's like, well, is that step so big? Do you even know how you got there, or is that just a guess? If it's ambiguous, I can't give you credit for it. So make sure you have good communication with a trigonometric identity. A trigonometric identity is not a calculation. The final answer is not something you just put in a box, because the whole proof is the answer, right? And proper communication is necessary to get full credit on these identities, like this one or question number 24. Question number 22 will be one that basically requires you to draw a right triangle diagram of some kind. So you're going to draw a triangle. That doesn't exactly look like a right triangle, but you'll forgive me for that. Right triangle of some kind, theta. And so using some Sokotoa, you're going to get things like opposite, hypotenuse, adjacent. You can unravel information using this right triangle diagram. In this situation, that's very useful when you're dealing with these inverse trigonometric functions. Perhaps the right triangle diagram will actually be given to you. We've seen things like this, what have you, some combination of triangles or whatever. Be prepared to be working with some right triangle type questions, right triangle trigonometry, some more advanced questions, like inverse trigonometry, some diagrams of various things involved here. So the question you see on the screen, this actually came from exam number two. This was question 11. But also going back to exam number one, if you grab question 11 there, that's also another example that you should be prepared how to do. Question number 23 will be a story problem involving vectors. So this will be things like static equilibrium, headings, airspeed versus ground speed. Some type of, again, story problem involving vectors of some kind. You can solve the problem geometrically. That's how it'll be presented. You can do that doing some type of triangle chase of some kind. But you could also do it completely algebraically. If you want to convert all the vectors to algebraically, then you can avoid things like the law of sines and cosines and even what the heck is this angle right here. That's an option if you want to. But question number 23 will be a story problem involving vectors, just like we saw on exam number three. Particularly, this is question number 14. All right. Question number 24, I did already mention. This is a trigonometric proof. It'll be a little bit more advanced than question number 21. This one's worth six points instead of five points. And so thus bringing us to the last question on this review. Question number 25, which will be a question about solving a trigonometric equation. It won't be a linear equation. It'll be more advanced. It'll basically be quadratic of some kind, like you'll need to factor it or use the quadratic formula. Trigonometric identities might be appropriate. Notice I have a cosine here and a secant here. That's incompatible. So you might need to use a trigonometric identity to help you out here. We saw these types of equations on exam number three. Look at exam, look at questions 10 and 13 for some more examples or go back to the syllabus, the exam syllabi or their corresponding sections for some more practice. So that gets us through the the the practice final exam. I should mention that although we went through it very, very quickly we did talk about basically every single topic that we've talked about in this semester perhaps just a few emissions. So we really shouldn't be focusing on what do I not need to study because that's not going to do us a lot of good. We really do need to prepare to do everything. And so the things that you're really good at might not require as much practice but things you really struggle on those are things worth practicing some more. Of course if you can form study groups with your classmates go to the tutoring center look for other resources to help you prepare. I of course am available to help you as you study for this exam. Best of luck to you all and I hope it goes well for everyone. But in the meanwhile let me know if you have any questions and I'll be I'll be glad to answer them.