 Welcome to our review for exam one for math 1060 trigonometry for students at Southern Utah University. As usual I'm your professor today Dr. Andrew Missildine. What I want to do is very briefly in this video talk about the types of questions you're gonna see on exam one. Now there's a lot that goes into preparing for an exam. One of the most important things is to know the time, place, and manner of set exam so that this video can be reused semester after semester. It will not include information about where, when, the exam will take place and some of those other more semesters since to semester specific policies. You are encouraged to look at the course syllabus, the information on Canvas or just talk to me, your instructor, to find out those specific ideas about time, place, and manner of the exam. Now there are some important things that will be the same for all exams throughout this whole semesters and I do want to establish some of those things right now. So with our exams this is a paper exam. You should write on the exam. There will be two types of questions that you do see on the exam and they'll come into two sections. There's the multiple choice section and there's the free response section. The multiple choice, the multiple choice section as you might assume, these will be multiple choice questions that you'll be given a question and you know four, five, six possible responses. You would then clearly indicate the one single answer that is correct. Without marking an answer or if your answer is unclear, you would get no credit for that. But if you select the correct response and no others, you would get full credit for that type of question. On a multiple choice question, on a multiple choice question, you don't get any partial credit. You get points for getting the question right or you don't get points if you don't get it right. So there's no partial credit available on those types of questions on the multiple choice section. This is in contrast, of course, to the free response section. In the free response section, you must show all of your work, not just the final answer. The final answer is worth maybe a little or just writing the right answer with no work could get you no credit whatsoever. So you have to show all of your work. And that's the main difference between the multiple choice questions and the free response in terms of grading. One other also important thing to mention here is that our exam will actually come in two pieces. In this video, I'm going to show you both pieces together. But when you take the test yourself, there'll be two portions that you can take separately, the multiple choice section and the free response section. This is the reason why on the multiple choice section, you will not be allowed any notes, you will not be allowed any calculator to use. These are questions that you should be able to answer just using basic rudimentary arithmetic. So nothing is going to be too complicated there. These can be on the easier side. It will require, of course, you have certain principles, formulas, memorized in preparation for that multiple choice section. That's one of the reasons why we have this video and the exam syllabus. So it's very clear what it is you need to be studying and memorizing of anything. There will be a formula sheet provided to you for the multiple choice section. This is mostly to provide you with trigonometric identities, which while that's not going to be super critical on this exam, that might be helpful on future exams. Of course, do not write on your formula sheet because these could be reused for other students as well. In the free response section, I should mention that things will be a little bit looser. So first of all, the thing you're probably going to like the most is you are allowed a calculator in the free response section, but it has to be a non graphing calculator, aka scientific calculator. You can't use one with a graphing utility or symbolic algebraic manipulation or anything like that. It's just a standard scientific calculator. You would want one definitely with the trigonometric function, sine, cosine, tangent, their inverses. It would be nice if you convert back and forth between degrees and radians. You should be able to also just use no basic arithmetic additions, attraction, multiplication, division, exponents, radicals, those type of things. So just a basic scientific calculator is what you'd be allowed on that. In terms of notes, you will be allowed a single three by five no card that no card should be three inches by five inches. So 15 square inches of area. You can write front and back of that no card. Of course, you can write as teeny tiny small as you want, but it does have to be handwritten. And this no card I should mention is actually part of the exam. Technically, there are 16 questions in the 16th question is your no card question. That is, you'll you'll get either two points or no points by turning in a valid no card or not turning that in. So you can put whatever notes you want on that no card. Those can be trigonometric identities, formulas, examples, whatever you think is appropriate. Draw Perry the platypus if you want to. I've seen people do that before. Whatever you need to help you do well in the exam, put that on your no card. It'll be stapled and termed in with your exam when it's submitted. So just be aware. Be aware of those things that is two points out of 100. You're going to want to want. You are going to want to do that no card there. And again, the no cards only available for the free response section, which we'll talk about both. And I'll make it very clear when we go from one section to the other. The other things I should mention. I guess the only other thing I really want to mention for this exam before we get into the more nitty gritties of questions is that there will be occasionally geometric diagrams that are drawn. Do not assume that these diagrams are drawn to scale. That's not necessary. That is, you shouldn't necessarily assume they're drawn to scale. So only use the information that's provided from what the diagram says and don't infer anything beyond that. And I guess one of the things that she mentioned in the multiple choice section, of course, all the answers will be exact because, of course, it's multiple choice. You'll select the correct answer in the free response section. You will have a calculator. So it might be there might be times that it's appropriate to round your answer. If you do so, do so, of course, three decimal places. But your best bet is to write all answers exact as much as possible. You don't actually need a calculator in the free response section. There'll be very few topics that actually require the use of a calculator. There aren't any on this exam either. And so if you put all of your, if you didn't have a calculator, you could put all of your answers in exact form and that would be perfectly appropriate. But if you do want to round, round your answers to three decimal places. So let's look at the specific types of questions that you're going to see on this exam. One of the things I should mention is with regard to our lecture series, exam one covers lectures one through eight. Just as a reminder, lecture one introduced us to the Pythagorean equation, a review of triangles, distance, coordinate geometry and circles. Lecture two, continue on with this, talking about angles and triangles. Lecture three was about similar triangles, alternate to your angle theorem, among other things. That was chapter one. Chapter two, we started learning about right triangle trigonometry, solving story problems with triangles, reference angles, computing trigonometric ratios without a calculator. For some of those special angles, those were all in lectures four, five and six. And then lecture seven and eight for chapter three was circle trigonometry. We learned about radiant measure. We also learned about degree measure back in chapter one. We learned about how circles are involved in trigonometry and how they relate to triangles, sector, area, arc length, angular velocity. These are all topics that will be found somewhere on this exam. So what you see in front of you is the practice exam for this exam, exam one. You can find this, of course, on Canvas and solutions are attached to that. There are also solution videos you can find on Canvas if you want to see some more details on how you solve specific problems. What I'm going to do in this review video is not actually solve problems, but just tell you what type of question types you should expect. Because after all, this practice exam is only a sample of the types of questions you should see. There are question types that appeared in lectures one through eight that may not be appearing on this practice exam. They could be on the exam you take. So just because something's missing, doesn't mean it won't be on the test. And also just because something's here on the practice test doesn't mean it will necessarily be on the test. So this is just a sample to help you prepare. And I want to talk about specific question types. So in the multiple choice section, all questions will be worth five points. No calculator or notes will be allowed on this except for the formula sheet. Question number one, you'll be asked to compute a trigonometric ratio without a calculator. So you might be asked to compute sine, cosine, tangent, cotangent, secant or cosecant of an angle. And that angle will be one of our special angles. It will be something like maybe 30 degrees. So remember in the first quadrant, of course, you have these special angles here. Whoops, that doesn't look like a circle. So we have zero, of course, 30 degrees, 45 degrees, 60 degrees and 90 degrees. So in the first quadrant, you should, you will need to have memorized, of course, sine and cosine evaluated at these five angles. And because of sine and cosine, you can also compute tangent, cotangent, secant and cosecant given that those four functions can be written as fractions or reciprocals of sine and cosine in some capacity. So you should be able to compute sine, cosine, tangent, cotangent, cosecant and secant using the angle 0, 30, 45, 60 and 90 degrees. You can also compute a trigonometric ratio for any reference angle. That is to say, if you take an angle whose reference angle is 90, 60, 45, 30 or 0 degrees, that you should also be able to compute without a calculator. So if you got something like, oh, let's take tangent of 315 degrees, could you do something like that? Of course, because you could compute the reference angle of 315 degrees. You can use the fact that 315 degrees is in the second quadrant to determine whether it's positive or negative. And then you can compute the trigonometric ratio in that regard. You should be able to do this in degrees, but you should also be able to do this in radians as well. So could you do something like what's sine of pi over 2? And if you have to convert pi over 2 into 90 degrees, that's also very appropriate. I don't care. They'll be able to compute the six trig ratios for the five special angles in degrees or radians or any angle that references to the five special angles. That sounds like a lot, but this is very important, very fundamental stuff. Basically, if you were to take the unit circle diagram that we introduced, of course, earlier in this lecture series, lecture seven specifically, can you use that diagram to do a calculation without your calculator? That's the type of stuff we're asking about here. And so to be able to do this, if you need some more practice, please reference lecture six and lecture seven. Lecture six, of course, introduced to us the special angles 0, 30, 45, 60, and 90. Also reference angles were introduced in lecture six. And then lecture seven introduces those radians and the unit circle diagram. So that's the type of thing you're going to use on question number one. Question number two, can you convert an angle measurement from radians to degrees or can you convert an angle measurement from degrees to radians? So question number two is about conversion. If you want to convert from radians to degrees, what are you going to do? You're going to multiply whatever the given angle is by 180 degrees over pi. If you want to convert from degrees to radians, then you need to multiply the angle by pi over 180 degrees. And so that's what you're going to want to use in that situation. Angle conversion was talked about in lecture seven when we introduced radian measure. Question number three is going to be a topic coming from lecture one, which was just a review of very basic Cartesian geometry. Specifically, can you compute a distance between two points? The distance between two points? Let's say that p is given as x1, y1, and q is given as x2, y2. Then can you calculate the distance between p and q? This of course is the square root of x2 minus x1 squared plus y2 minus y1 squared as well. So use the distance formula. You might be asked to compute the midpoint of the two points p and q as well. And so of course the midpoint m is going to look like x1 plus x2 over 2 and y1 plus y2 over 2. So can you do these basic calculations of coordinate geometry that we saw in lecture one? Question number four is going to be sort of a continuation of that, but this one's going to focus on circles specifically. As we saw in chapter eight circles are a very important part of trigonometry. So you need to know the equation of a circle which is actually derived from the Pythagorean equation. So is the distance formula as well. And so remember the equation of a circle is given as x minus h squared plus y minus k squared is equal to r squared, where r is the radius of the circle which is half of the diameter and the coordinates h comma k are the coordinates of the center of the circle. So can you either create the equation of a circle satisfying certain conditions or can you read from the equation of a circle information about centers and radii and things like that? So moving down the page here, question number five will be a question to ask you to compute reference angles. So question number one may or may not ask you to compute a reference angle. Question number five will ask you to compute a reference angle and it will be and that's all it's going to ask you to do. So you'll be given an angle measure, like in this case you'll be given 195 degrees. You're asked to compute the angle measure, the reference angle, excuse me. The angle measure of course will always live in the first quadrant, so always be zero degrees to 90 degrees, but it could also be in radians. So this question could ask you to find the reference angle in radians, in which case the answer will then be in radians, so you need something between zero and pi halves. So reference angles that we mentioned before were introduced in lecture six, lecture seven introduces us to radian measure, so you were also asked in the homework to compute some reference angles of angles that weren't radian measure. Question number six is going to be a question about circles, also about circles, but it's going to be things like arc length. So you need to know the arc length formula s equals r theta. You might be asked to compute sector area, the area of a pizza slice, right? And so remember the area of a sector is equal to one half r squared theta. You might be asked to do something with angular velocity, which we've talked about some of those things as well. Angular velocity, remember that the velocity, the change of s with respect to time, is equal to r omega. We're omega here, remember v is the change of position with respect to time, and omega is the angular velocity, which is the change of theta with respect to time as well. Arc length and sector area was introduced in lecture seven. Angular velocity we talked about in lecture eight of course. When you were using these formulas arc length, sector area, or angular velocity here, these angle measurements must be in radians, must be in radians. If they're not in radians, then you will get the wrong answer. And as this is a multiple choice question, you can very likely anticipate that one of the distracting answers will be the number you get if you erroneously don't convert from degrees to angles, assuming that's a possibility. So be very cautious of that. Continuing on with the multiple choice section. Question number seven is going to ask questions about partner angles. What do we mean by that? So we talked about this in lectures two and lecture three. So what are some examples of partner angles? Things like vertical angles, which would be a picture like this. One and three are vertical angles that are going to be congruent to each other. You have supplementary angles when the two angles come together to form a line. Supplementary angles are important. You have complementary angles. What happens when you have two angles that come together to make a right angle? Complimentary angles as well. This picture, this diagram, this is the type of pictures we expect with like the alternate interior angle theorem. You have, for example, alternate interior angles, which are, of course, congruent. You are going to have corresponding angles. You're going to have consecutive interior angles. You have exterior consecutive interior, excuse me, consecutive exterior angles and what is it? Alternating exterior angles. Things like that. So you have these partner angle relationships that you're going to need to know. Of course, another one is like if you have a triangle with angles one, two, and three, the sum of the three angles always adds up to be 180 degrees. So use these partner angle properties to say stuff about what things are congruent, what are not congruent, and solve various problems related to these partner angles. All right, question number eight, which we can see here, this is a question coming from section eight, and this is going to be circle trigonometry, meaning that when you have the unit circle and you have a point on the unit circle P, x, y, you're going to be using the fact that the x-coordinate is cosine of the angle theta and y is the sign of angle theta, and then the other trig ratios are computed, of course, from their usual ratio situation. So understanding how trigonometry is defined using points on the unit circle, that's what you need for question number eight. Question number nine is going to be right triangle trigonometry. You'll be given the diagram of a right triangle, do not assume it's drawn to scale. You have a right triangle where angle is, the angle theta is indicated, but you do not have the measure, but you are still asked to compute things like sine, cosine, tangent, cotangent, secant, and cosecant, given that triangle. To make life easier, you'll probably be given a Pythagorean triple, like three, four, five, whatever, but you just want to know your basic so-catoa relationships. If you know so-catoa, you're going to be just fine on this question, just follow the information from the triangle, like so. Question number 10, you actually will be given an angle measure, but this is going to be one of our special triangles, which led to the special angles. So how do you use like a 30, 60, 90 triangle? How do you find out the missing information? Like this one's a 30, 60, 90 triangle. What about like a 45, 45, 90 triangle? How do you use that? So basically, you need to know sine and cosine of 30, 45, and 60 degrees, and you can do those various things. So like, for example, if you take the short side of a 30, 60, 90 triangle, the hypotenuse is double that. Can you do trigonometry in those basic situations, those I should say the special situations, 30, 45, and 60? That then brings us to the end of the multiple choice section. So we are now entering the free response section of the test. This part of the test will be taken separately from the multiple choice section. On this portion, you will be allowed a calculator and you are allowed a 3x5 note card. You won't have the formula sheet, but you can put whatever formula is on said formula sheet that you desire if you need those for these types of questions. There's going to be five questions in this section and on average, there are about 10 points each. Question number 11, you'll be asked to solve some type of triangle diagram. So like you see in this picture, there's two unknowns. There's this number H and there's this number X that we do not know, but we do know some things about distance. We do know some things about angle measure. So using right triangle trigonometry, maybe involving some circles too, but using right triangle trigonometry, find the missing unknowns like find X, find H. That's what you're asked to do right here. I do want to see the answers exact on this question. So do write this as like X is five times cosine of 17 degrees plus, you know, tangent of 50 degrees whatever it turns out to be. You will notice that in this section, the angles are not the special angles, 55 degrees, 62 degrees. You are not anticipated to know those trigonometric ratios. Give me the exact answers, but also give me a solution that's approximate to three decimal places. That would be very appropriate for question number 11 right here. And so you'll be asked to solve triangle diagrams. So triangle diagrams, we first saw in section four, lecture four, when we first introduced right triangle trigonometry, but we also more likely should look to question number five. Question number lecture number five, excuse me. Lecture number four did sort of very basic ones, very, very basic ones that was just basically a right triangle. Number five about story problems involving right triangles, those ones have these more interconnected diagrams and that's what you should anticipate on question number 11. It's also worth mentioning you can also talk about six special angles could come into play, but most likely you don't need any special angles on these type of questions. So moving on to question number 12, which is right here. Question number 12, you'll be given a diagram of the unit circle and you'll be given a picture looking very similar to this. You might actually get this exact same picture or you might get something that looks similar to it. So you have the unit circle you see right here, there's some other points like O, A, P, R, B, other labeled and again this is the unit circle. So what you're going to be asked to do is you'll be asked to prove some type of statement like this one, it asks you to prove that the line segment O, R right here is equal in length to secant of theta. So that's what you're asked to be able to do. This is something we did of course in lecture eight, we did things very, very similar to this. Prove this line segment is secant, prove this one's tangent, prove this one's cotangent. You'll be asked to do something like that and the basis of the argument comes from similar triangles. So from the diagram we should be able to extrapolate maybe two triangles that are similar to each other or something like that and use a similar triangle argument to do this. Similar triangles were discussed in detail in lecture three but lecture eight is the type of similar triangle argument you're going to do, something using the unit circle like you see right here. Question number 13 will be a story problem involving right triangle trigonometry. So some more of stuff like section five. Question number 11, there won't be a story problem. You'll be given a diagram and you're asked to find the missing pieces of the diagram. Question 13 though, no picture will be given to you. You'll be given the story like so. You are by all means encouraged, encouraged to draw a picture to not just to help you process it but also to help the greater know what you're thinking. But this answer, this question will have an answer, some type of numerical quantity. Give it to me in exact form and approximate it to three decimal places. That would be appropriate for this question number 13. All of these questions so far, let's see, question, I'll take that back, question 11 was worth 10, 13 was worth 10, but question 12 I believe was worth 8. So let's move down here. Question number 14, this will be another question involving sort of like partner angles and things like that. But unlike what we saw in the multiple choice section where we just had identified whether these angles were congruent or supplementary or something like that, this will ask us to solve for unknown quantities. So you see like there's a triangle right here with some angles we can solve for some of those angles. We have some supplementary angles, things like that. So use partner angle relationships to help you solve the unknown values x and y. So do a little bit of algebra based upon the geometric relationships. This is type of stuff we did in lectures two and three. That's where it was very, very heavy in our lecture series so far. Question number 15, hi Pac-Man, it's good to see you right here. But question number 15, this is going to be a story problem involving circles. So remember question 13 was a story problem involving right triangles. Question 15 will involve story problems of circles. So this will be things about area of circles or sector area of circles. That's what this question is asking. What's the area of Pac-Man that you see right here? That's kind of a fun question. You could talk about arc length. Like what's the circumference of Pac-Man? That'd be kind of a fun question as an alternative. Angular velocity, things like that. So those type of story problems we did with circles, particularly in chapter three, about lecture seven and eight, that's where we saw these type of questions. So be prepared to answer some type of story problem involving these properties about circles. And then question number 16 down here is just a reminder to turn in your no card. I do put your name on it. Sometimes the no cards get detached from the test by mistake. And so having your name on it is a good way to make sure you get the credit for making the no card, which you definitely do want to make a no card. And so that then gets us through the exam. I've illustrated the types of question types that you will see. If you want to, I would recommend if you haven't already, to take a look at the practice exam and try the problems out on your own. There are solutions attached to them so that you can check your answers or if you get stuck to get some help. But as you're studying for this test, please, please, please work with partners in the class, work with tutors in the tutoring center, include me, your instructor, you'll come by office hours and be emails, just ask me questions when appropriate. I'm here to help you as much as possible. Just let me know. And I think this test is going to be okay for you all. I'll see you next time, everyone. Bye.