i think it was because Trigonometry is not suitable for drill, other Math subjects were based on graph, or matrix, easy to recreate, you can create your own problem, you can practice as much as you can but trigonometry?, sure you can use your protactor, but it took more effort than just creating graph with coordinate, on top of that you need to memorize, sin, arcsin, Identities and unit circle, and their relation to triangle. the problem resources are also very sparse because people think that it was a chore to do all of that visualization then putting algebra on top of that, no wonder it became a complete mess
I am so thankful for your content. You are the very reason why I am pursuing a degree in engineering in my 30s. You have helped me believe in myself and to do hard things. I wish there were more teachers with your talents. Just.. thank you so so much!
Go get it guys!!! It's never too late! Careers last a long time. If you love it, do it. Never too late. Also never impossible to learn/get good. Just gotta be willing to love it - that's it.
I know this sounds weird, but I actually think the real reason most people aren't good at trig is because their geometry is lacking. But a lot of geometry books wouldn't go into further explanations of irregular polygons, infinite series, and roulettes. And I feel those things really helped me understand trig, at least conceptually. Trigonometry is like a more mechanical geometry to me if that makes sense. When I started asking: What curves can I get from rolling shapes, what can I get by breaking other shapes apart at different angles, and what can I get from the intersection of shapes. This is what helped me.
Not really. I studied astrology for years. The old way where you calculate the chart by hand, look up time zones, calculate the exact position of the planets using logarithms, then add and subtract degrees minutes and seconds by hand creating angles. No calculator or computer. It is definitely geometry. Reading the chart is totally separate from calculating it.
@@sr2291 Studied "for years"? Then, I would think you would know the difference between astrology and astronomy. What you're referring to is astronomy. Astrology deals with horoscopes.
For me, real analysis was genuinely easier than most of calculus. Cause calculus is just about memorization and long long computation. Here's a formula, here's an expression, calculate it. It's boring. I'm a human, not a human calculator. Real analysis is special though. You have to prove stuff, you get to be an explorer and a scientist (in a way) yourself, not just a human calculator as it used to be in Calc classes.
Had trig as a high school sophomore back in the dark ages (ie. slide rules and log tables). Actually loved it and highly respected the teacher. We spent days on the unit circle and I can still visualize the 1st quadrant. It really helped learning calc later in college.
This honestly is what would have helped me get an A instead of a B. I was trying so hard to visualize and connect the semiotics of trig with the unit circle, but our course was a half semester course(bad community college decision of course structure btw, that's popular for some reason now) and there was no time for me to actually make that connection, but I know that if we had spent an absurd, but necessary, amount of time connecting the semiotics of trig to the unit circle that I would've excelled like a mf
My tip for every math major is, when something requires you to understand a concept you should try to teach what you're learning to know if you know it. The only thing you know is the thing you can teach. For memorizing things you should embrace tactics that embrace repetition like writing down things or watching a lecture a dozen times
I thought he was going to talk about Algebraic Topology or something of that sort. Trigonometry is not even difficult. Pretty much all the things you need to remember about trigonometry: * what cos(x) and sin(x) mean in the unit circle * what the plots of sin(x) and cos(x) look like * the definition of tan, cotan, cosec and sec (not that anyone uses the last three anywhere) * sin({0,30,45,60,90} degrees) = sqrt({0,1,2,3,4})/2 (a trick my dad taught me when I was 9 or so) * Euler's identity: exp(i*a)=cos(a)+i*sin(a) * complex multiplication multiplies the moduli and sums the arguments You can deduce all the usual sum/difference formulas from the last two points. Using complex numbers for this is tremendously powerful. For instance, if A, B and C are the interior angles of a triangle, consider the product (1+i*tan(A))(1+i*tan(B))(1+i*tan(C)). The three factors are complex numbers with arguments A, B and C, so their product is a complex number with argument pi, which means the imaginary part of our product is zero. From there, you get that tan(A)+tan(B)+tan(C)=tan(A)*tan(B)*tan(C).
I've been taking tirg on UDEMY from Hania Uscka-Wehlou. Good teacher and easy to follow. Also using a Trig workbook by Chris McMullen. Did your College Algebra course on UDEMY and it prepared me for the course. It all comes down to practice and repetition.
I completely agree with you here Sorcerer. Algebra teaches us to think in terms of polynomial functions. Mapping numbers to angles & vice versa takes an entirely different mindset. Then intro calc baby's students back to the concept of power law relations with... polynomials. Trig functions become another chore of memorization with derivative and integral. As far as memory is concerned, symmetries and discernable patterns play a major role, the human mind remembers things that are attractive to it.
I second the advice to know the special angles in the first quadrant and then use reference angles. That's how I was taught back in high school and it seemed to crystallize quite nicely (I've gone on to become an engineer). That coupled with practice problems. Lots of them. Get some blank paper and work out like 100 problems on e.g. sin(7pi/6) until it's second nature. Everyone needs practice, some people get it with 20 questions, others need 200. Let that not discourage you!
This is so accurate. The switch from Algebra to Trigonometry for me in college threw my brain through a loop. I went from straight A's to getting my first C in a math class. I struggled really badly with it, but what was strange was that when I got to Pre-Calculus, all of a sudden I understood everything in the review they covered for Trigonometry toward the end of the class. I feel like the extra learning I got from the concepts of functions in Pre-Calculus helped me a lot with that.
Let me tell you two things: 1 - I’m addicted to your videos! 2 - I envy your collection! 😂 I’m a book collector but don’t have 1/10 of the books you have.
I struggle with math greatly and have issues with basic arithmetic. One of the biggest barriers for myself learning math is that every time I asked what it's for I was never given an answer and was just told to "Do the Math". What is Trig? What does it seek to measure or resolve? Why would I want to learn it? Is there any practical analogy for an asymptote and cosine or are are they just amorphous concepts understandable only through rote memorization. Most importantly, how does this build on/use what I just learned and how is it preparing me for the next step in my math journey? I love your channel. Thank you for doing Math.
My high school had a honors “precalc” course that was sort of a combination of algebra and trig. It was taught by a sort of controversial teacher in the sense that half the class hated the way he taught and half of us loved him. His lectures weren’t really lectures a lot of the time, he’d give us some new bit of information and then have us try to work out more things about it. For example, he showed us how you get the sine function and what it looked like then had us break up and try to answer the question “what does the cosine graph look like and how does it relate to sine”. At the end of class he would get all of our answers and tell us which were right and why and which were a little off and why. It was a hard class, but it was also insanely interesting in part because of how many different approaches and solutions people took. It really felt like we were building these concepts ourselves rather than being taught them which I absolutely loved (but some people clearly hated, and I can kind of under why). Anyways, that class and teacher has always stuck with me and I think I pretty much owe my love of math to him. I understand that kinda approach isn’t for everyone, but I would absolutely love if more teachers taught like this, teaching students to explore and discover things rather than just memorize concepts.
Didn’t know you still made videos, but your ODE videos are literally getting me through my class. I can’t make it to my lectures because I have to stay at home with my son. I just wanted to say thank you so much, I appreciate you 🫶🏽
I thought that trig was pretty easy. You can fit everything that you need to know about trig on one cheat sheet. I never memorized anything except identities. You can always figure out what sin of pi/3 is by drawing a triangle and use Pythagorean theory to figure out length of the sides.
I'm a doctor - virtually *everything* we studied at med school was just a matter of rote learning - take anatomy for instance - nothing conceptually difficult - just lists and lists of facts about viscera , muscles , vasculature , bones & the central & peripheral nervous system . Clinical stuff - so many * lists * - ' list the causes of chest pain , hypercalcaemia , raised white cells , haemolytic anaemia , splenomegaly , thrombocytopaenia , ST elevation , disc lesions on CXR , LMN facial palsy , Horner's , spastic paraparesis , meningitis , synovitis , supraventricular tachyarrhythmias etc etc etc . Pure slog , you don't even need to be particularly clever or smart - really - it's also really not like 'House' either . Maths & physics is proper brain food .
I'm a mathematician. At school I always found biology the hardest science, as there are so many facts to learn. In my research career I have worked with computer scientists and biologists doing computational biology. I still think biology is really hard: although there are general principle, each organism, or organ, or cell needs a model of its own. I have a good memory, but I find memorising facts is harder than understanding broad concepts, which is how I learn mathematics and physics.
I am a math major wanting to become a mathematician. I will say, for me, trig identities and manipulations took an amazing amount of memorization for me. Basically, youre just expected to know from your first semester all the fundamental forms of limits, derivatives, and integrals and trig functions...most students here easily do so without memorization. But i came from a disadvantaged school far away in the US. And mandarin is my third language, so there was a huuuge learning curve. Their country prioritizes math and English from 1st grade on up as the main 2 courses and courses that determine which universities you have access to. Its similar to GaoKao, but more Japanese in style. So i get why because of the cultural focus and training from a young age. But while i intuitively understand a lot of things in our field, manipulating trig functions is definitely something i had to just raw memorize, and quickly. Also the fact that calculators are banned on all exams and generally not part of the academic culture/conscious is something too. But more minor compared to all that
@@readjordan2257if you are good at understanding mathematics then I would recommend you work at learning proofs rather than trying to memorize formulae. If you are faced with a trig identity, ask yourself, why is that true? Then try to figure it out for yourself, or find an explanation in a book or online, and work through the details. Being a real mathematician isn't about how many formulae you can remember, it's about solving problems and building up concepts so you can develop new ones
I did well in trigo and found that keeping it as simple and true to first principles helped a lot. First drilled down all the axiomatic expressions, applied some simple mnemonics to keep them easy to recall and then used a simple cartesian diagram to recall the relevant angles: S: Sin on the Left Top, A: All on the Right Top, T:Tan on the Bottom Left and C:Cos on the Bottom Right.
I think of all the things I learnt in the earliest days of returning to maths the most "mathematically useful" was transformations of functions. (So how to shift the "basic function" by using f(x-2) to go in the + direction, or f(x+2) to shift it left, and then the way functions scale and stretch, invert, reflect, etc.) The idea of a function as an object that can be handled as a single entity was something it was once important for me to get into my head. Or let's say that's what seemed more obviously "mathematical" at the time. The idea that trig was the rule of a set of _numbers_ might have had deeper significance. (I'm digging around in what are now quite old memories, so the impressions are a bit sketchy now, but I think this is how it was.) Why mention it? I suppose it might be a good direction to send other beginners in. Trig was hard-ish, but I have a quite good memory (or had; it doesn't work all that well these days), so the memorization aspect wasn't too bad. Among the strategies was to not be too "mathematically fastidious" about how you remember the things you need to have ready to hand when the exam comes up. If some kind of image of a pork pie covered in Rat Salad is what it takes to make some pi-number stick, just use that. Understand why with one half of your mind, and when necessary, remember by tricking the part that could be trained to sit and fetch a stick. "Don't be too proud to also learn "sit!" and "fetch the ball!" ".
I teach Integrated Math 2 and Integrated Math 3 where they split trig into smaller parts of each class and my students do well. In IM2 we explore the Law of Sines and Law of Cosines as well as how to use SohCahToa. In IM3 we explore functions, and periodic functions come toward the end. I do the unit circle in both years. I definitely teach the first quadrant, then I show how to find the other values using similar triangles (major theme in IM2). In IM3 we talk about those common angles and plot them to create their graphs. From there we apply transformations (major theme in IM3). Splitting this up for my students seems to make it a lot easier for them since they follow themes in each class so we're building off ideas they are mastering rather than referring back to them. Most of my students are on a track to take Calculus their 11th grade year. So this won't be the case for everyone.
I found trig easy as well. But then, I memorize numbers, formulas and such easily. Agree on the quadrant method re. evaluating "must know" trig function values - that's how I was taught.
This is awesome. Thanks for the Trig textbook recommendation. Do you plan on doing a video like this on geometry? I took pre-calc and I would love to dive deeper into Trig and Geometry. I feel like I missed so many good tidbits that would be helpful in Calculus courses.
Real Analysis is the hardest for me .. i dont know what to do about it and same with its relatives - complex, metric, topology ... so i always focus on other courses like algebra , differential equations etc 😅 ..
This brings me back to my undergraduate academic advisor telling me that real analysis was a "watershed course" (repeatedly). Then I took it and found that proof writing and the epsilon-delta formalism were the most natural things, at least for me. I aced the course - literally a 100 average! - and never heard my advisor doubting me again (take that!). What makes it hard for most is that it is the first time they move completely from solving problems to proof. It's a different mindset: the level of abstraction rises sharply.
@5:15 I am not sure if we are talking about the same algebra but in my time and my country, Algebra was the hardest subject for Math students; in particular algebraic structures, vector spaces, function spaces, groups, rings, fields. This was the type of questions we used to get in our exams. "Exam Question: Consider a field F and the ring of polynomials over F, denoted by R = F[x]. Demonstrate that the ideal generated by a non-constant polynomial p(x) in R is a maximal ideal if and only if p(x) is irreducible over F. Good luck!"😇
I know that I am quite late here, but from what i've seen the US seems to put a lot of maths that would be considered highschool level in some other countries as university classes. So the "college algebra" class he is talking about doesn't go really much into the "abstract algebra" part of algebra, which would be talked abt in more advanced seperate classes.
Trig was the first math class that clicked for me. I felt like I had a cheat code. On your advice I bought Swokowski's textbook, mainly for analytic geometry. I think trig was easy for me because I was working on radar data concurrently, as an adult. All of the radar geometries, as well as wave functions, were expressed the same week I was trying to understand radar vectors, so it was just amazing timing to have an applied example. Made things much easier. Meanwhile, I'm having the hardest time ever, just sitting trying to memorize the equations for conics. Edit: I did work my butt off though.
Other countries teach algebra and trig to younger students. As you mentioned that trig is often taught in college. That is true in the United States. I hosted Chinese and Japanese high school students. And they loved taking math in the United States, because it is all review for them. These two students said they did algebra, trig and other pre-calculus courses in middle school.
we don’t even have pre-calculus/calculus classes in brazilian schools. we’re too behind everyone and that makes me feel less capable than people from other countries
I love that book!! I used that in high school it was great. Trig was the class that convinced me to give math and physics a try. Now I’m finishing a masters in math lol
I was almost sure you were going to talk about trigonometry. :) Thank you for that. Btw you had a book/textbook about spherical trigonometry - would you please make a video about it?
When I was in high-school I did calculus ab and I didn't get a high enough score to get cal 1 credit and decided to do a dual corse pre-cal and trig and I thought it sucked how I had to go backwards and how I would be doing things I already knew. On the first day, we were already taking notes. It really taught me how to study as before in calculus the teacher's homework would be notes and I would just not write them down and try to pick things up from class the next.
When doing trigonometry exams, we were often allowed to use sheets with the trig identities printed on them as they were too numerous to remember. As for sines, cosines and tangents, we were permitted to use trig tables, and later calculators to obtain them. We were almost never required to memorize them.
I can tell you that trig was definitely a hard subject for those who have never taken any sort of geometry in high school. My teacher in college particularly was extremely hard. Most of the stuff I had to teach myself, but i don’t regret having self taught myself trig. Self teaching made me have a really good foundation in math principles such as strong algebraic background. One thing that really helped me pass this class was interpreting cos(x) as x value and sin(x) as the y value. In addition this, you only need to remember two types of triangle in this course: 30,60,90. And the 45,45 90. The hardest part about this course was definitely the identities. If I would have known how they were created back then, and how to just created them from eulers formula, it would have made it so much easier to do problems.
I agree wholeheartedly though, I think people underestimate just how much of maths really boils down to just memorising certain things, and conversely underestimate how much memorising stuff can make a real difference.
When I was younger I would always hear people talk about how difficult trig is and feel discouraged because I struggled so much with math back then. Earlier this year I took a trig course and it was the most fun math course I had taken up to that point. Identities and equations were a bit rough, but vectors, imaginary numbers, and polar coordinates were so exciting. I get that isn't everyone's experience. But trig is the first mathematics course anyone takes that had direct and immediate real world applications. And I wish more people could have a positive experience with trig.
Thank you, Math Sorcerer for your inspiration over the last few years. I remember writing the unit circle at the start of each class and on every page of homework. You just get used to seeing it. Yesterday I was using trig and calculus to help solve quantum mechanics problems like momentum eigenfunctions of a particle. (ħ/i)(d/(dx))(√(2/L))sin((nπx)/L) Looking back, it helps to use your imagination and RUclips to give you a reason. Now, I see trig and all of math as a tool. In physics these are like your screwdrivers or wrenches, to help solve other problems.
Woah, trigonometry was my absolute favorite and had a great time with it. HOWEVER, trig definitely was more challenging in my calculus classes. I definitely thought this was going to be about real analysis or topology. But there’s no shame in not getting this. I’m gifted at memorization.
Here's how I managed to memorize the values of trig functions. Hopefully this advice will help you. I memorise the trig function values by visualising the angles on the unit circle. I visualize what the angle is and the length of the sides of the triangle that would be drawn on that unit circle with that angle in my head. For sin(pi/6) or cos(pi/6) for example, I visualize that the cos part is bigger than the sin part. Thus, since I know that they are either sqrt(3)/2 or 1/2, I know that cos(pi/6) is sqrt(3)/2 and sin(pi/6) is 1/2, as sqrt(3)/2 is greater than 1/2. For sin(pi/4) or cos(pi/4), I know that each side is the same, thus they are both sqrt(2)/2. If it helps, we know that the sin part of the triangle is larger than sin(pi/6) and the cos part is smaller than cos(pi/6), and so both sides are sqrt(2)/2, as this is larger than 1/2, and smaller than sqrt(3)/2. Finally for cos and sin of pi/3, I see that the sin part is greater than the cos part. Therefore sin(pi/3) is sqrt(3)/2, and cos(pi/3) is 1/2. sin(pi/2) is 1 and cos(pi/2) is 0 since there is no cos component to the triangle, and so the sin part length is one. sin(0) is 1 and cos(0) is one for similar reasons, except the line does not have any sin component. You can also do this with angles outside the 1st quadrant of the unit circle. For example, for sin(5pi/3), I start from the angle of an integer multiple of pi closest to our angle (in this case 2pi), and then imagine a triangle with a clockwise angle to our angle (in this case an angle of pi/3) from or original angle. We see the answer is -sqrt(3)/2, since our y values (or sin values) for our unit circle are negative as 5pi/3 is on the fourth quadrant, and our triangle has an angle of pi/3, so from our knowledge of the first quadrant, we know that this angle is either positive or negative sqrt(3)/2. To find tan values, I don't memorize them, rather I use that fact that tan(x) = sin(x)/cos(x) and sub in sin(x) and cos(x). For cot and sec and cosec, I memorize the equations in terms of sin and cos. cot is 1/tan since it has a t in the end. sec is 1/cos since there is a c on the end. cosec=1/sin since there is an s as the third letter.
Finally understanding how Euler's Formula works was one of my happiest moments. I wish our teachers had taken a few hours to show us the infinite series calculations instead of just using those functions without understanding them properly.
Yes, absolutely! What fascinates me is that you can also do it with a special class of 2x2 matrices instead of complex numbers (both are isomorph structures), so that you - at least appearently - don't need complex numbers for Euler's formula. The imaginary unit in this model then becomes a special rotation matrix, which rotates the first unit vector of R^2 by pi/2.
In my college math classes I would always just use imaginary exponentials in my integrals instead of sines, cosines, and tangents, eg cos x = (e^ix +e^-ix)/2, sin x = (e^ix - e^-ix)/2i, etc. that way I'd never have to remember trig identities for my calculus, PDE, physics, etc courses. Exponentials are so straightforward to work with and require no cleverness like trig does.
I think that understanding what each function represents on a circle really helps for the rest. There is a video on RUclips, with a circle and a slowly rotating radius, showing in real time what cosinus, sinus, tangent and their friends represent on that circle.
Memorization & Understanding need to be treated like 2 sides of the same coin. You remember new information by connecting it to what you already know. And you understand a new idea by connecting it to what you already understand. But each of these processes reinforces the other. So make connections. If you're struggling to do that, then ask yourself what you're not understanding or not remembering. But you need to constantly go back and forth between what you remember and what you understand. Use it to test yourself. Use the logic to build bridges between memory and understanding. Use what you understand to recall what you need to remember. Use what you remember to figure out what you need to understand. Logic is connection and connection is logic!
When I found out how simple a radian was, everything was so much easier. Not single teacher told me why "1 radian length" goes specifically at a spot in the unit circle. I had to find out MYSELF that once you hit about 3 radian lengths it never really touched halfway across the circle and then I told myself "it looks like it needs an extra like 1/10th of the way there........ohhhhhhh" that is where pi comes from! So now I understand why "pi/2" is where it needs to be, and why 11pi/6 is where it has to be, or where pi/4 has to be, it wreally was a game changer. made me feel like I didnt have to memorize that unit circle anymore Every teacher in my opinion needs to state at the start of the semester why those radians are there
I always found trig very easy. One of my favourite parts of math class but I have a photographic memory. And although that is not proven to be a thing, I can look at a map for 10 seconds and still draw it for you a year later.
As a working mathematician, I totally agree with the advice about using the first quadrant and then extending. But I don't even try to memorize those special angles: I just remember that the angle sum of a triangle is pi, then draw a picture of an equilateral triangle or a right-angle isosceles triangle, then apply Pythagoras. In the UK, students learn the equivalent of US college trigonometry and calculus in the final two years of high school (and some of it earlier). However, that doesn't guarantee that they understand it properly, and it is one of the stumbling blocks that makes many UK students struggle when they do calculus, or analysis courses, at university.
Exactly. My first struggle was WTF all was so smooth before but now function is piecewise and in additional to usual math struggle I MUST taking care in what quadrant it is and change signs. Late in hi school polar form of complex numbers was best invention, no more dreadful ugly tracking of signs.
I did enjoy the trig class but it was easy for our teacher printed all the equations and the circle we used them in exams, I do believe that understanding how trig equations work is a big part of making it easy
I taught myself trig over the summer in an independent precal class, and yes, I agree with the video. Trig was the hardest part of that class. Mostly due to the graphing and the time it takes to solve.
One of the reasons I became a mathematics major was because I hated memorization. If you understand the “why” behind your mathematics, you’ll be able to derive anything you forget in the future.
Memorization and calculation was the reason I hated trigonometry in high school. On the other hand, nobody needed it nor taught it at university (I've a math masters degree), let alone in a postgraduate context (phd in math). So... I don't really understand all these USA people talking about trig at uni... Trig is a high school thing!
@@marik8770 Not really, don't get me wrong I quite enjoyed both topology and analysis, but differential equations are the foundation upon which our understanding of the world is built. Getting at least through differential equations is essential for anyone who is to claim to be educated, regardless of their field of expertise, it's just too fundamental to pass over.
Trigonometry is much easier if you consider everything quadratically instead of linearly. The pythagorean theorem becomes a + b =c. Angles become opposite^2/hypotenuse^2 = sin^2 where 1 is equivalent to 90 degrees. Then you get less approximations because you dodge the square roots and you wont need a calculator for trig. So you push the sqrt calculation out to the last possible step which usually leaves it as part of your answer and you dont need to evaluate it.
There is a lot of stuff to memorize but at the same time it also gives structure to the subject. I feel like a lot of problems has a very straight forward solution that you either know or you don't. It just comes down to practice, practice, and practice.
I think that one reason that trigonometry was easy for me was High School Geometry. For an entire school semester we're thinking about triangles and four-sided figures, goin' through those Side-Angle-Side, etc. theorems and doing simple proofs. And understanding proportion, and Euclidian geometry. From what you are saying, that geometry class helped me much more than I realized. In trig it was challenging simplifying expressions. I thought that the teacher wouldn't like this because he never mentioned it, but I would convert everything to sines, and cosines, oftentimes replacing cos^2(angle) with 1 - sin^2(angle). Then simplifying, and converting things back again.
I took it once and didnt do well enough to move on so had to retake it. I got an A in it the second time. The reason is because I had all my old exams and knew exactly what it was I was going to need to know. A study guide is the answer to making students do better. A good study guide. One that is an actual representation of the exam that you are going to make for them and give them. What I hate are semesters and the time frame we have as students to learn stuff. I'm 34 years old and a non traditional student. College needs an overhaul. It would be nice if the time frame that students had to learn something just didn't exist. Imagine if we were actually building students up and teaching them. Instead, we give them 4 months to learn something and they have to just move on and then have low GPAs and can't get into the grad programs they want. As an old student that knows better... I would much rather you teach me what I need to know to move on and I shouldn't be allowed to move on until I show that, even if it takes longer. If we look to technology, AI can do away with professors all together. It will be able to teach a class and explain topics in multiple different ways better than any human. It will write tests and practice questions in minutes all with new variables. Students could take classes as long as they needed to for basically no cost... the cost of some software. It will do it better than the best colleges in the entire world. The goal needs to be to TEACH people and build them up.. no matter what that takes. People need longer than others and they shouldn't be denied or have to wear that bad grade around for the rest of their life that says they aren't good enough when they have ALL THE TIME in the world to prove they can learn it and they ARE good enough. We've built a system that keeps people out!!! If professors give a shit about their jobs and their professions they should be the ones at the forefront of this change... or AI will come in and take it away from all of you. You hear a lot of college professors say "I don't do study guides. You all should just be old enough and responsible enough to do it on your own now!" F*** that. That is the biggest cancer in college system right now. Students are paying 1000's and 1000's for something they just end up youtubeing anyways. It's rediculous. You would think the very least they could do for the students paying so much money is give them a damn study guide. Then they fail all the classes and the colleges surely don't give a shit but want to wonder why? Because ur not even doing the obvious things people need to pass. If your 18 years old and have to go to college for 6 years for ur 4 year degree... so what? We should have a system that can accommodate both people. If people can come in and show they know what they need to know in 2 months then let them go. If it takes them 8 months then fine. But lets build the future of our nation and stop building barriers that don't let them do what they want in life... but then the grad schools will end up just have some arbitrary cut off of 4 years and if you didn't complete it in that time then you just automatically get rejected. Sorry for the rant. I'm sure ur a great professor that has to do exactly what they tell you. I feel like if you started turning out classes full of A+ students they would tell you that you need to increase the rigors of your course to be more challenging. That they some how can't allow that. My aim is dental school and I never once cared about how long it takes me. I just want them to build me into the person I want to be. I'm willing to do whatever it takes.. but the college can't help you. You can only help yourself. I should just be on a track through college that leads me to being a dentist whether it takes 8 years or 12 years. Some times it feels like I have a masters degree in common sense or something. Why nobody talks about any of this? Somebody please help me create an artificial intelligence college! I know what ur thinking... o like an online college? No. I mean a college with desks that the students go to and they sign into their desk for attendance and an AI professor teaches the class and then sends an email to the students that were present. Ur AI professor will literally be standing over ur shoulder during ur homework assignments. It's going to teach people better than any human ever could. Students simply pay a monthly subscription fee to take any path they want in any amount of time they want. We prepare for DAT and MCAT and every other test you can think of. I could create a utopian college that promises to prepare people for the exact life they want to have. Not a system that is going to destroy their dreams and leave them with 1000's of debt they have no way to pay back other than doing something they don't love. We can program an AI to care about you 1000x more than any human professor actually cares about you. Some of the human professors are going to assign you work that is going to make you cry... and they think that is funny. O also, at my school you don't have to pay $400 for parking. We don't charge for that..
Cuesta College San Luis Obispo, CA. College is close to the Pacific Coast Highway (PCH). Put this on your bucket list for places to see. My daughter and I drove from Monterey CA to Long Beach. Views are breathtaking.
What you need to is think in pictures and numbers at the same time connecting the two. Algebra is connected to Geometry through pictures on graph paper. All trig and trig functions are is names for the pictures you draw. If you remember the pictures you can draw them and then reconstruct the reasoning needed to solve problems you forgot the answers too. That way you don't have to worry about remembering everything.
Anki is what saved me. I’m in med school. My first run in with struggling to memorize content was in biochemistry. Anki is a flashcard program that spaces out your repetition and it helped me stay on track.
I think that key moment in trigonometry is just trig functions and theyre relationships between them. Every angle has all six values of trig functions.
I feel like Trig may be the hardest for non-math majors having to take the course for their science degree; for math majors, I've never seen a class complained about so much as Real Analysis personally.
My old EE professor told me in high school back in the day, they didn't have calculus and AP math classes. Trigonometry was senior level math. Which they really learned in detail.
Trig wasn’t difficult but it wasn’t easy either because I like to work from first principles and it would be hard to get tests done that way when they are timed (took trig in high school). Also, using the first quadrant is the way to go. Funny enough my knowledge of hyperbolic functions actually made more sense once I did some complex analysis and special relativity. Everyone has their struggles and this is definitely one of them.
There are a few tricks needed in order to kickass at Trig. You need to invent some mnemonic devices for this. If you just create a few mnemonic devices ...you will NEVER forget anything, you will rock the exams, and you will absolutely ACE calculus. You MUST create some mnemonic devices for trig, you'll thank me later. You only need a couple of these and you're golden. Truly excellent video. I have seen some very smart people who crumble in Calculus because their trig was not so polished. Don't be that guy. If you can ace trig, you can ace Calc II. Also - to reinforce your trig ... solve every single problem on your own time before you ever go to class. Then go back and solve the same problems several times to build up some speed. If you solve the same problem 4 or 5 times you start getting really fast. I would be happy to share my personal mnemonic devices with the world if anyone wants to see them. I recently showed a few of these to a co-worker and he instantly understood some trig, and he's not even a math person.
I teach pre-calc at the university level and I agree that students find trigonometry hard. My own take on the trig values is not to memorize any trig values at all. I think you need just a little memorization from geometry that shouldn't be rote memorization. Most students already know the Pythaorean theorem by heart. Add to this the 30/60/90 and 45/45/90 triangles add the unit circle idea and you can figure out any of the values that come up.
The only trig things i ever memorized are formula for sine of sum, formula for cosine of sum and then i remember that in a 30 60 90 triangle with hypotenuse of 1 the shorter cathete is 1/2 and the longer one is √3/2, then i fit the triangle onto the unit circle to find the values for sine or cosine of π/3, π/6
I love trigonometry it was the reason I was able to do well in other math classes, currently I’m in Differential Geometry and woah do I find it difficult lol so far it’s harder to grasp than Real Analysis.
I've found that in physics & engineering classes, most of the trig you learned, you never use, but the little you DO use (mostly the trig definitions), you use a lot.
I was fully prepared for this to be a discussion on analysis. While it's more abstract, I thought it was more enjoyable then trigonometry, and most likely because of this, I found analysis (real) much easier. I had to take trigonometry twice.
We take collagae level algebra and trig in highschool thanks to Türkiye's education system(I graduated from hs in 2016).We are thought to referance first quadrant by default.If you can get a grasp of referancing since all trig functions are periodic functions,it becomes much easier leaving a few formulas to memorize here and there which can easily be derived if you get your basics right.
I found trig to be very easy, but I think it was because I was into electronics from the time I was in elementary school and took electronics classes starting in 9th grade. We were working with complex numbers, trig functions, phasors, etc long before I set foot in a formal trig class.
That unit circle and those trig definitions -- I always thought that we have it backwards -- sine and cosine. The x-axis is first, and the y-axis is second. So if anything is co- it should be the y-axis. And we always say, "sine and cosine", which is like saying, "y and x".
I had no problem with math until I got to integrals in calculus. No problems with algebra, geometry, trig, solid geometry, and derivatives. Integrals requires you to remember how to play with equalities/replacements and my mind just does not seem to work that way. I was a CS major whose mind works in very straight lines. Almost failed partial differential equations, but I think the professor gave me a pass cause I worked hard at it. I also took group theory and I loved the course - it was easy.
Trig is underlooked. Many even myself mistake as plug n chug, but the identities and relation to 2pi is the true buttkicker. Its daunting, or too niche. Calc comes - you gotta know it. Practice problems are a must if just saying "memorize" as it becomes memorable as pythagorean
I remember learning the sine and cosine values of common angles, and the positive/negative signs of sin, cos, tan around the unit circle, in what was the equivalent of 9th/10th grade (age 14 or so). So when it came to learn more advanced trig in 11th and 12th grade high school years (age 16-18) like the double angle/half angle formulas, and how to prove trig identities using them, the basic understanding was already there. I think the reason "trig is hard" is that you are not meant to learn it in one or two semesters in college. You are meant to learn it gradually over a period of years, starting early if possible, so it has time to sink in. Anything is hard if you get required to learn it quickly over a short period of time.
I found trig easy, but I took it after calc 2. I tested into calculus not having a solid algebra 2 foundation because my ghetto high school didn't teach very well... All my friends went to rehab for drug use, so I had a lot of free time, and so I dual majored in math and physics getting a 4.0, outstanding senor award, and excellence in undergraduate research award. Didn't even know I liked studying and learning like that till college!
I was surprised at how well I took to Trig in college. I was inept in my math classes in HS: only took geometry and applied algebra classes. But was able to get a math minor in college. Built my skill from the ground up!
As a physical scientist, it is unsurprising that I like math. I agree that trigonometry requires a fair bit of memorization, but what I learned later is that complex numbers and a thorough understanding of how to use Euler's relation and the unit circle, converts a lot memorization - especially trig identities - into easy algebra problems. Yeah, there's a barrier to getting past the abstractness of exp(iθ) = cosθ + isinθ, but once you get used to complex plane, trig gets pretty easy. Chapter 8 is underemphasized or even skipped when it's actually taught.
Trig was easy for me. Everything had a relationship to geometry and could be drawn on paper. It was stats that I really struggled with. I still struggle with it today... confusion with different aspects with probability (like combinations and permutations), certain types of distributions, how to calculate certain values correctly (like quartiles).
As someone who teaches calc I'm curious about how you feel about professors providing "formula booklets" with trig identities /trig vals or letting students bring a cheat sheet to calc exams ? I genuinely don't get whats the difference between being able to solve a question becauee you mugged up a trig fact vs being able to solve a question because you consulted a list of trig facts. In fact even when I was first learning trig ( as part of IGCSE math in my freshman snd sophomore year, followed by IB math during my junior and senior year) we had formula booklets for our math examinations and were assesed using 2 different final exams (a calculator paper and a non-calculator paper). This did not "trivialize" trig but managed to make it challenging in more interesting ways , because you were solely tested in your ability to use these trig facts as tools to solve a more complicated equation or model a problem.
When I attended high school many decades (5) ago I had no issues with algebra 1 and 2. My algebra 2 class my teacher told me I didn't even need to take the final exam. Trigonometry & Geometry was just the opposite. I blame myself for not grasping or studying. Some classes you love the math and the teachers are excellent. My older brother was the math whiz. By the way do we still use a slide rule in Trig?
I had an easy time with trig, but I've always been a very visual person. And I think that's what makes trig so difficult for many students--there's so much of it you have to understand visually. You mentioned reference angles, and that's a great example... A student asks, "what's the formula for a reference angle?" The answer, "There's not one... You have to draw a picture and connect the angle back to the x-axis. Writing down a "formula" would involve too many cases depending on where the angle lands." You really need to understand it visually rather than in terms of a formula, but students are often taught that math is all about formulas. Just to give one more example, "What's the formula for the polar angle?" Well, "It's theta = arctan(y/x), except that that formula always gives an angle in QI or QIV. If your point is in QII or QIII then you need to add pi. Also, for a point in QIV the formula gives a negative angle; so if your book or teacher wants a positive angle you need to add 2pi." That's a lot of caveats to just remember. They are easy to understand if you really have that graph of the tangent function in your head... The main connected component goes from (-pi/2) to (pi/2), which means QIV to QI on the unit circle. The period is pi, so if you need an answer in QII or QIII you can just shift over by pi, or by 2pi if you want a positive answer in QIV. But that understanding requires you to both have the image of the unit circle and also the image of the graph of tanx in your head... Most students, if you ask them what the graph of tanx looks like, they hit a button on their graphing calculator because that's what they've been taught to do. They don't have that solid mental image to help them understand why there are those caveats to the formulae and how to work around them. In more mathematical terms, the real issue is that the trig functions are horribly not one-to-one, and so any process that requires finding an angle is going to require careful adjustment. In algebra, the only not one-to-one thing students do is squaring... So the only experience they have with adjustment is putting a +/- when they take the square root of both sides. Trig is quite a bit more complicated. One last thing, when I teach calculus I find the background material gives students more difficulty than calculus itself. That's why now in a 16 week calculus course I spend the first week just on algebra review, then the next two weeks just on trig review. The first exam is just over the algebra and trig review. Finally at week 4 we start actually talking about calculus itself.
I believe Trig is difficult to teach because it's never broken into subset semesters: Trig 1, 2, and maybe even 3. As with many math subjects, there's not enough time in a semester to teach everything thoroughly -- and there aren't enough semesters in an undergraduate education to divide Trig like that. Yet thorough knowledge of trig is essential to understanding in depth most math subjects to follow. I myself think trig is fun, poetry with numbers; but my learning technique doesn't involve classrooms. Thank you for another excellent video.
I totally agree. I taught Trig at out local J.C. several times but it was only a 3 unit class. I used exactly the same book as The Math Sorcerer and it is an excellent textbook. Unfortunately, the best of the 8 chapters is Chapter 8 where Trig and College Algebra are blended together into a beautiful topic called Complex Numbers and Polar Coordinates. Generating the neat graphs in polar coordinates and understanding the complex numbers for learning fractals is why Chapter 8 is so important. I had a difficult time getting to and covering that chapter thoroughly because of the breadth of material in the book and the little time 3 units gives you.
Trigonometry is full of facts and formulas and rules and identies that are interrelated in all sorts of ways, so that many of the things you "have to memorize" are things that you could re-derive or figure out every time you need them, except that having them memorized is so much faster and easier. I think eventually, if you work with it long enough, things fall into three categories: (1) Stuff you remember because you've used it enough, (2) Stuff you remember how to easily figure out, and (3) Stuff you know exists and you can look it up if you need it.
This is so true. Trig identities, even if you can derive them, still need to be memorised to be recalled quickly enough in exams. It just sucks as the failure to memorise doesn't demonstrate a lack of skill, maybe just a lack of memory. Why not make students memorise log tables while we're at it.
sin(0°) = ½√0 = cos(90°) sin(30°)= ½√1 = cos(60°) sin(45°)= ½√2 = cos(45°) sin(60°)= ½√3 = cos(30°) sin(90°)= ½√4 = cos(0°) Trigonometry is such an irrelevant artificially overcomplicated subject. You are right about having to learn too much. So for Standard Angles, I came up with the above.
Hello, Basically trig wasn't hard in the beginning. But when started with systems of trig equations where you need to use some identities that are rarely used to find the solution, knowing we had to memorise all those identities... Well, this part wasn't easy. As you said, the problem was mainly memorisation and a bit of "finding the trick" to rewrite some equation using a good combination of identities...
Not sure what I expected this to be, maybe calculus, but I never had a problem with trig. I went through a few of my father's old math books around 8th grade and it didn't take long to figure out "what trig is," it's basically functions of a right triangle, essentially the ratios of different sides based on the angle. How tall is a flagpole? On a sunny day you can measure the length of its shadow and the angle above the horizon of the sun, and find out from that info. This helped later in high school with complex numbers. and in college with AC circuits (involving sine waves at different phases, more complex numbers, but with j, because i is used for current in EE). Cotangent, secant and cosecant are easily defined as the reciprocals of tangent, sine and cosine, but I was never sure why these named things exist. There aren't a lot of named functions that also have their reciprocals named. I'm sure there's historical reasons for this, but I never knew about math history to learn that.
Trig was the big bully with a dark past, who had a big heart and alot of good things to say. But getting past its demeanor is a real doozy.
i think it was because Trigonometry is not suitable for drill,
other Math subjects were based on graph, or matrix, easy to recreate,
you can create your own problem,
you can practice as much as you can
but trigonometry?, sure you can use your protactor, but it took more effort
than just creating graph with coordinate,
on top of that you need to memorize, sin, arcsin, Identities and unit circle, and their relation to triangle.
the problem resources are also very sparse because people think that it was a chore to do all of that visualization
then putting algebra on top of that, no wonder it became a complete mess
I am so thankful for your content. You are the very reason why I am pursuing a degree in engineering in my 30s. You have helped me believe in myself and to do hard things. I wish there were more teachers with your talents. Just.. thank you so so much!
Bro, I'm with you. Going after CompSci at 37.
Go get it guys!!! It's never too late! Careers last a long time. If you love it, do it. Never too late. Also never impossible to learn/get good. Just gotta be willing to love it - that's it.
I know this sounds weird, but I actually think the real reason most people aren't good at trig is because their geometry is lacking. But a lot of geometry books wouldn't go into further explanations of irregular polygons, infinite series, and roulettes. And I feel those things really helped me understand trig, at least conceptually. Trigonometry is like a more mechanical geometry to me if that makes sense. When I started asking: What curves can I get from rolling shapes, what can I get by breaking other shapes apart at different angles, and what can I get from the intersection of shapes. This is what helped me.
Not really. I studied astrology for years. The old way where you calculate the chart by hand, look up time zones, calculate the exact position of the planets using logarithms, then add and subtract degrees minutes and seconds by hand creating angles. No calculator or computer. It is definitely geometry. Reading the chart is totally separate from calculating it.
Heavily agree!
@@sr2291 Studied "for years"? Then, I would think you would know the difference between astrology and astronomy. What you're referring to is astronomy. Astrology deals with horoscopes.
I can potentially see this.
@@victoza9232 “Calculate the chart” is an astrology concept. Maybe you should learn the difference between astrology and astronomy.
Haha, based on the intro I thought you were going to say “real analysis”!
For me, real analysis was genuinely easier than most of calculus. Cause calculus is just about memorization and long long computation. Here's a formula, here's an expression, calculate it. It's boring. I'm a human, not a human calculator.
Real analysis is special though. You have to prove stuff, you get to be an explorer and a scientist (in a way) yourself, not just a human calculator as it used to be in Calc classes.
Had trig as a high school sophomore back in the dark ages (ie. slide rules and log tables). Actually loved it and highly respected the teacher. We spent days on the unit circle and I can still visualize the 1st quadrant. It really helped learning calc later in college.
This honestly is what would have helped me get an A instead of a B. I was trying so hard to visualize and connect the semiotics of trig with the unit circle, but our course was a half semester course(bad community college decision of course structure btw, that's popular for some reason now) and there was no time for me to actually make that connection, but I know that if we had spent an absurd, but necessary, amount of time connecting the semiotics of trig to the unit circle that I would've excelled like a mf
It's pretty normal to have the unit circle memorized. I'm a 22 year old college student and we memorized unit circles in like grade 11
@@muhammadmustafa2946 that’s cool, not everyone has access to these math courses just because you did, especially not standard track students.
@@pichirisu It was a standard track course, literally everyone had access to it.
@@muhammadmustafa2946 For wherever you are, sure, but that is not the case for all. Good luck with learning how the national education system works.
My tip for every math major is, when something requires you to understand a concept you should try to teach what you're learning to know if you know it. The only thing you know is the thing you can teach. For memorizing things you should embrace tactics that embrace repetition like writing down things or watching a lecture a dozen times
😂 drive the adage home
Those who can't do teach 😂
I thought he was going to talk about Algebraic Topology or something of that sort. Trigonometry is not even difficult.
Pretty much all the things you need to remember about trigonometry:
* what cos(x) and sin(x) mean in the unit circle
* what the plots of sin(x) and cos(x) look like
* the definition of tan, cotan, cosec and sec (not that anyone uses the last three anywhere)
* sin({0,30,45,60,90} degrees) = sqrt({0,1,2,3,4})/2 (a trick my dad taught me when I was 9 or so)
* Euler's identity: exp(i*a)=cos(a)+i*sin(a)
* complex multiplication multiplies the moduli and sums the arguments
You can deduce all the usual sum/difference formulas from the last two points.
Using complex numbers for this is tremendously powerful. For instance, if A, B and C are the interior angles of a triangle, consider the product (1+i*tan(A))(1+i*tan(B))(1+i*tan(C)). The three factors are complex numbers with arguments A, B and C, so their product is a complex number with argument pi, which means the imaginary part of our product is zero. From there, you get that tan(A)+tan(B)+tan(C)=tan(A)*tan(B)*tan(C).
I've been taking tirg on UDEMY from Hania Uscka-Wehlou. Good teacher and easy to follow. Also using a Trig workbook by Chris McMullen. Did your College Algebra course on UDEMY and it prepared me for the course. It all comes down to practice and repetition.
I completely agree with you here Sorcerer. Algebra teaches us to think in terms of polynomial functions. Mapping numbers to angles & vice versa takes an entirely different mindset. Then intro calc baby's students back to the concept of power law relations with... polynomials. Trig functions become another chore of memorization with derivative and integral.
As far as memory is concerned, symmetries and discernable patterns play a major role, the human mind remembers things that are attractive to it.
I second the advice to know the special angles in the first quadrant and then use reference angles. That's how I was taught back in high school and it seemed to crystallize quite nicely (I've gone on to become an engineer). That coupled with practice problems. Lots of them. Get some blank paper and work out like 100 problems on e.g. sin(7pi/6) until it's second nature. Everyone needs practice, some people get it with 20 questions, others need 200. Let that not discourage you!
This is so accurate. The switch from Algebra to Trigonometry for me in college threw my brain through a loop. I went from straight A's to getting my first C in a math class. I struggled really badly with it, but what was strange was that when I got to Pre-Calculus, all of a sudden I understood everything in the review they covered for Trigonometry toward the end of the class. I feel like the extra learning I got from the concepts of functions in Pre-Calculus helped me a lot with that.
Let me tell you two things:
1 - I’m addicted to your videos!
2 - I envy your collection! 😂 I’m a book collector but don’t have 1/10 of the books you have.
I struggle with math greatly and have issues with basic arithmetic. One of the biggest barriers for myself learning math is that every time I asked what it's for I was never given an answer and was just told to "Do the Math".
What is Trig? What does it seek to measure or resolve? Why would I want to learn it? Is there any practical analogy for an asymptote and cosine or are are they just amorphous concepts understandable only through rote memorization. Most importantly, how does this build on/use what I just learned and how is it preparing me for the next step in my math journey?
I love your channel. Thank you for doing Math.
My high school had a honors “precalc” course that was sort of a combination of algebra and trig. It was taught by a sort of controversial teacher in the sense that half the class hated the way he taught and half of us loved him.
His lectures weren’t really lectures a lot of the time, he’d give us some new bit of information and then have us try to work out more things about it. For example, he showed us how you get the sine function and what it looked like then had us break up and try to answer the question “what does the cosine graph look like and how does it relate to sine”. At the end of class he would get all of our answers and tell us which were right and why and which were a little off and why. It was a hard class, but it was also insanely interesting in part because of how many different approaches and solutions people took. It really felt like we were building these concepts ourselves rather than being taught them which I absolutely loved (but some people clearly hated, and I can kind of under why).
Anyways, that class and teacher has always stuck with me and I think I pretty much owe my love of math to him. I understand that kinda approach isn’t for everyone, but I would absolutely love if more teachers taught like this, teaching students to explore and discover things rather than just memorize concepts.
Didn’t know you still made videos, but your ODE videos are literally getting me through my class. I can’t make it to my lectures because I have to stay at home with my son. I just wanted to say thank you so much, I appreciate you 🫶🏽
I thought that trig was pretty easy. You can fit everything that you need to know about trig on one cheat sheet. I never memorized anything except identities. You can always figure out what sin of pi/3 is by drawing a triangle and use Pythagorean theory to figure out length of the sides.
I'm a doctor - virtually *everything* we studied at med school was just a matter of rote learning - take anatomy for instance - nothing conceptually difficult - just lists and lists of facts about viscera , muscles , vasculature , bones & the central & peripheral nervous system . Clinical stuff - so many * lists * - ' list the causes of chest pain , hypercalcaemia , raised white cells , haemolytic anaemia , splenomegaly , thrombocytopaenia , ST elevation , disc lesions on CXR , LMN facial palsy , Horner's , spastic paraparesis , meningitis , synovitis , supraventricular tachyarrhythmias etc etc etc . Pure slog , you don't even need to be particularly clever or smart - really - it's also really not like 'House' either . Maths & physics is proper brain food .
I'm a mathematician. At school I always found biology the hardest science, as there are so many facts to learn. In my research career I have worked with computer scientists and biologists doing computational biology. I still think biology is really hard: although there are general principle, each organism, or organ, or cell needs a model of its own. I have a good memory, but I find memorising facts is harder than understanding broad concepts, which is how I learn mathematics and physics.
I am a math major wanting to become a mathematician.
I will say, for me, trig identities and manipulations took an amazing amount of memorization for me. Basically, youre just expected to know from your first semester all the fundamental forms of limits, derivatives, and integrals and trig functions...most students here easily do so without memorization. But i came from a disadvantaged school far away in the US. And mandarin is my third language, so there was a huuuge learning curve. Their country prioritizes math and English from 1st grade on up as the main 2 courses and courses that determine which universities you have access to. Its similar to GaoKao, but more Japanese in style. So i get why because of the cultural focus and training from a young age. But while i intuitively understand a lot of things in our field, manipulating trig functions is definitely something i had to just raw memorize, and quickly.
Also the fact that calculators are banned on all exams and generally not part of the academic culture/conscious is something too. But more minor compared to all that
@@readjordan2257if you are good at understanding mathematics then I would recommend you work at learning proofs rather than trying to memorize formulae. If you are faced with a trig identity, ask yourself, why is that true? Then try to figure it out for yourself, or find an explanation in a book or online, and work through the details. Being a real mathematician isn't about how many formulae you can remember, it's about solving problems and building up concepts so you can develop new ones
I did well in trigo and found that keeping it as simple and true to first principles helped a lot. First drilled down all the axiomatic expressions, applied some simple mnemonics to keep them easy to recall and then used a simple cartesian diagram to recall the relevant angles: S: Sin on the Left Top, A: All on the Right Top, T:Tan on the Bottom Left and C:Cos on the Bottom Right.
I think of all the things I learnt in the earliest days of returning to maths the most "mathematically useful" was transformations of functions. (So how to shift the "basic function" by using f(x-2) to go in the + direction, or f(x+2) to shift it left, and then the way functions scale and stretch, invert, reflect, etc.) The idea of a function as an object that can be handled as a single entity was something it was once important for me to get into my head.
Or let's say that's what seemed more obviously "mathematical" at the time. The idea that trig was the rule of a set of _numbers_ might have had deeper significance.
(I'm digging around in what are now quite old memories, so the impressions are a bit sketchy now, but I think this is how it was.)
Why mention it? I suppose it might be a good direction to send other beginners in.
Trig was hard-ish, but I have a quite good memory (or had; it doesn't work all that well these days), so the memorization aspect wasn't too bad. Among the strategies was to not be too "mathematically fastidious" about how you remember the things you need to have ready to hand when the exam comes up. If some kind of image of a pork pie covered in Rat Salad is what it takes to make some pi-number stick, just use that. Understand why with one half of your mind, and when necessary, remember by tricking the part that could be trained to sit and fetch a stick.
"Don't be too proud to also learn "sit!" and "fetch the ball!" ".
I teach Integrated Math 2 and Integrated Math 3 where they split trig into smaller parts of each class and my students do well. In IM2 we explore the Law of Sines and Law of Cosines as well as how to use SohCahToa. In IM3 we explore functions, and periodic functions come toward the end. I do the unit circle in both years. I definitely teach the first quadrant, then I show how to find the other values using similar triangles (major theme in IM2). In IM3 we talk about those common angles and plot them to create their graphs. From there we apply transformations (major theme in IM3).
Splitting this up for my students seems to make it a lot easier for them since they follow themes in each class so we're building off ideas they are mastering rather than referring back to them.
Most of my students are on a track to take Calculus their 11th grade year. So this won't be the case for everyone.
I found trig easy as well. But then, I memorize numbers, formulas and such easily.
Agree on the quadrant method re. evaluating "must know" trig function values - that's how I was taught.
This is awesome. Thanks for the Trig textbook recommendation. Do you plan on doing a video like this on geometry? I took pre-calc and I would love to dive deeper into Trig and Geometry. I feel like I missed so many good tidbits that would be helpful in Calculus courses.
great idea I should do that!
Real Analysis is the hardest for me .. i dont know what to do about it and same with its relatives - complex, metric, topology ... so i always focus on other courses like algebra , differential equations etc 😅 ..
This brings me back to my undergraduate academic advisor telling me that real analysis was a "watershed course" (repeatedly). Then I took it and found that proof writing and the epsilon-delta formalism were the most natural things, at least for me. I aced the course - literally a 100 average! - and never heard my advisor doubting me again (take that!). What makes it hard for most is that it is the first time they move completely from solving problems to proof. It's a different mindset: the level of abstraction rises sharply.
@@peterbenoit5886 Great .. i want the same to happen to me too..
@5:15 I am not sure if we are talking about the same algebra but in my time and my country, Algebra was the hardest subject for Math students; in particular algebraic structures, vector spaces, function spaces, groups, rings, fields. This was the type of questions we used to get in our exams. "Exam Question: Consider a field F and the ring of polynomials over F, denoted by R = F[x]. Demonstrate that the ideal generated by a non-constant polynomial p(x) in R is a maximal ideal if and only if p(x) is irreducible over F. Good luck!"😇
I know that I am quite late here, but from what i've seen the US seems to put a lot of maths that would be considered highschool level in some other countries as university classes.
So the "college algebra" class he is talking about doesn't go really much into the "abstract algebra" part of algebra, which would be talked abt in more advanced seperate classes.
Trig was the first math class that clicked for me. I felt like I had a cheat code. On your advice I bought Swokowski's textbook, mainly for analytic geometry. I think trig was easy for me because I was working on radar data concurrently, as an adult. All of the radar geometries, as well as wave functions, were expressed the same week I was trying to understand radar vectors, so it was just amazing timing to have an applied example. Made things much easier. Meanwhile, I'm having the hardest time ever, just sitting trying to memorize the equations for conics.
Edit: I did work my butt off though.
Other countries teach algebra and trig to younger students. As you mentioned that trig is often taught in college. That is true in the United States. I hosted Chinese and Japanese high school students. And they loved taking math in the United States, because it is all review for them. These two students said they did algebra, trig and other pre-calculus courses in middle school.
we don’t even have pre-calculus/calculus classes in brazilian schools. we’re too behind everyone and that makes me feel less capable than people from other countries
In the Army we learned land nav and I had no clue but when I started astrophtograohy and learning astrology and astrophysics that Trig is beautfil
I love that book!! I used that in high school it was great. Trig was the class that convinced me to give math and physics a try. Now I’m finishing a masters in math lol
After reading a bits of Epps book I kind of am excited to look at trig again because now I have a much deeper understanding of what functions are.
I was almost sure you were going to talk about trigonometry. :) Thank you for that. Btw you had a book/textbook about spherical trigonometry - would you please make a video about it?
When I was in high-school I did calculus ab and I didn't get a high enough score to get cal 1 credit and decided to do a dual corse pre-cal and trig and I thought it sucked how I had to go backwards and how I would be doing things I already knew. On the first day, we were already taking notes. It really taught me how to study as before in calculus the teacher's homework would be notes and I would just not write them down and try to pick things up from class the next.
When doing trigonometry exams, we were often allowed to use sheets with the trig identities printed on them as they were too numerous to remember. As for sines, cosines and tangents, we were permitted to use trig tables, and later calculators to obtain them. We were almost never required to memorize them.
I can tell you that trig was definitely a hard subject for those who have never taken any sort of geometry in high school. My teacher in college particularly was extremely hard. Most of the stuff I had to teach myself, but i don’t regret having self taught myself trig. Self teaching made me have a really good foundation in math principles such as strong algebraic background. One thing that really helped me pass this class was interpreting cos(x) as x value and sin(x) as the y value. In addition this, you only need to remember two types of triangle in this course: 30,60,90. And the 45,45 90. The hardest part about this course was definitely the identities. If I would have known how they were created back then, and how to just created them from eulers formula, it would have made it so much easier to do problems.
You sound like an exceptional teacher. It wouldn't hurt to bring a video out about maths study tips & tricks.
I agree wholeheartedly though, I think people underestimate just how much of maths really boils down to just memorising certain things, and conversely underestimate how much memorising stuff can make a real difference.
Thanks. Trig was never an issue for me; just another math class.
When I was younger I would always hear people talk about how difficult trig is and feel discouraged because I struggled so much with math back then. Earlier this year I took a trig course and it was the most fun math course I had taken up to that point. Identities and equations were a bit rough, but vectors, imaginary numbers, and polar coordinates were so exciting.
I get that isn't everyone's experience. But trig is the first mathematics course anyone takes that had direct and immediate real world applications. And I wish more people could have a positive experience with trig.
Thank you, Math Sorcerer for your inspiration over the last few years. I remember writing the unit circle at the start of each class and on every page of homework. You just get used to seeing it. Yesterday I was using trig and calculus to help solve quantum mechanics problems like momentum eigenfunctions of a particle.
(ħ/i)(d/(dx))(√(2/L))sin((nπx)/L)
Looking back, it helps to use your imagination and RUclips to give you a reason. Now, I see trig and all of math as a tool. In physics these are like your screwdrivers or wrenches, to help solve other problems.
Woah, trigonometry was my absolute favorite and had a great time with it. HOWEVER, trig definitely was more challenging in my calculus classes. I definitely thought this was going to be about real analysis or topology. But there’s no shame in not getting this. I’m gifted at memorization.
Here's how I managed to memorize the values of trig functions. Hopefully this advice will help you.
I memorise the trig function values by visualising the angles on the unit circle. I visualize what the angle is and the length of the sides of the triangle that would be drawn on that unit circle with that angle in my head. For sin(pi/6) or cos(pi/6) for example, I visualize that the cos part is bigger than the sin part. Thus, since I know that they are either sqrt(3)/2 or 1/2, I know that cos(pi/6) is sqrt(3)/2 and sin(pi/6) is 1/2, as sqrt(3)/2 is greater than 1/2. For sin(pi/4) or cos(pi/4), I know that each side is the same, thus they are both sqrt(2)/2. If it helps, we know that the sin part of the triangle is larger than sin(pi/6) and the cos part is smaller than cos(pi/6), and so both sides are sqrt(2)/2, as this is larger than 1/2, and smaller than sqrt(3)/2. Finally for cos and sin of pi/3, I see that the sin part is greater than the cos part. Therefore sin(pi/3) is sqrt(3)/2, and cos(pi/3) is 1/2. sin(pi/2) is 1 and cos(pi/2) is 0 since there is no cos component to the triangle, and so the sin part length is one. sin(0) is 1 and cos(0) is one for similar reasons, except the line does not have any sin component.
You can also do this with angles outside the 1st quadrant of the unit circle. For example, for sin(5pi/3), I start from the angle of an integer multiple of pi closest to our angle (in this case 2pi), and then imagine a triangle with a clockwise angle to our angle (in this case an angle of pi/3) from or original angle. We see the answer is -sqrt(3)/2, since our y values (or sin values) for our unit circle are negative as 5pi/3 is on the fourth quadrant, and our triangle has an angle of pi/3, so from our knowledge of the first quadrant, we know that this angle is either positive or negative sqrt(3)/2.
To find tan values, I don't memorize them, rather I use that fact that tan(x) = sin(x)/cos(x) and sub in sin(x) and cos(x). For cot and sec and cosec, I memorize the equations in terms of sin and cos. cot is 1/tan since it has a t in the end. sec is 1/cos since there is a c on the end. cosec=1/sin since there is an s as the third letter.
Finally understanding how Euler's Formula works was one of my happiest moments. I wish our teachers had taken a few hours to show us the infinite series calculations instead of just using those functions without understanding them properly.
Yes, absolutely! What fascinates me is that you can also do it with a special class of 2x2 matrices instead of complex numbers (both are isomorph structures), so that you - at least appearently - don't need complex numbers for Euler's formula. The imaginary unit in this model then becomes a special rotation matrix, which rotates the first unit vector of R^2 by pi/2.
I think the only time I struggled with trigonometry was when I had to graph the transformation of the reciprocal function
In my college math classes I would always just use imaginary exponentials in my integrals instead of sines, cosines, and tangents, eg cos x = (e^ix +e^-ix)/2, sin x = (e^ix - e^-ix)/2i, etc. that way I'd never have to remember trig identities for my calculus, PDE, physics, etc courses. Exponentials are so straightforward to work with and require no cleverness like trig does.
I think that understanding what each function represents on a circle really helps for the rest. There is a video on RUclips, with a circle and a slowly rotating radius, showing in real time what cosinus, sinus, tangent and their friends represent on that circle.
Memorization & Understanding need to be treated like 2 sides of the same coin. You remember new information by connecting it to what you already know. And you understand a new idea by connecting it to what you already understand. But each of these processes reinforces the other.
So make connections. If you're struggling to do that, then ask yourself what you're not understanding or not remembering. But you need to constantly go back and forth between what you remember and what you understand. Use it to test yourself.
Use the logic to build bridges between memory and understanding. Use what you understand to recall what you need to remember. Use what you remember to figure out what you need to understand. Logic is connection and connection is logic!
When I found out how simple a radian was, everything was so much easier.
Not single teacher told me why "1 radian length" goes specifically at a spot in the unit circle. I had to find out MYSELF that once you hit about 3 radian lengths it never really touched halfway across the circle and then I told myself "it looks like it needs an extra like 1/10th of the way there........ohhhhhhh" that is where pi comes from!
So now I understand why "pi/2" is where it needs to be, and why 11pi/6 is where it has to be, or where pi/4 has to be, it wreally was a game changer. made me feel like I didnt have to memorize that unit circle anymore
Every teacher in my opinion needs to state at the start of the semester why those radians are there
I LOVE trig, but I'm terrible at calculus.
I always found trig very easy. One of my favourite parts of math class but I have a photographic memory. And although that is not proven to be a thing, I can look at a map for 10 seconds and still draw it for you a year later.
As a working mathematician, I totally agree with the advice about using the first quadrant and then extending. But I don't even try to memorize those special angles: I just remember that the angle sum of a triangle is pi, then draw a picture of an equilateral triangle or a right-angle isosceles triangle, then apply Pythagoras. In the UK, students learn the equivalent of US college trigonometry and calculus in the final two years of high school (and some of it earlier). However, that doesn't guarantee that they understand it properly, and it is one of the stumbling blocks that makes many UK students struggle when they do calculus, or analysis courses, at university.
Exactly. My first struggle was WTF all was so smooth before but now function is piecewise and in additional to usual math struggle I MUST taking care in what quadrant it is and change signs. Late in hi school polar form of complex numbers was best invention, no more dreadful ugly tracking of signs.
I did enjoy the trig class but it was easy for our teacher printed all the equations and the circle we used them in exams, I do believe that understanding how trig equations work is a big part of making it easy
I remembered only sin and cos values for Q1 of unit circle. It was enough to derive all other values for all trig functions quickly when needed.
I didn't think trig was so bad, and I actually enjoyed it. Proofs and abstract algebra was tough for me.
I taught myself trig over the summer in an independent precal class, and yes, I agree with the video. Trig was the hardest part of that class. Mostly due to the graphing and the time it takes to solve.
One of the reasons I became a mathematics major was because I hated memorization. If you understand the “why” behind your mathematics, you’ll be able to derive anything you forget in the future.
Memorization and calculation was the reason I hated trigonometry in high school. On the other hand, nobody needed it nor taught it at university (I've a math masters degree), let alone in a postgraduate context (phd in math). So... I don't really understand all these USA people talking about trig at uni... Trig is a high school thing!
@@rv706in France we do lot of analysis and topology with some algebra! Much better than calculus
@@marik8770 Not really, don't get me wrong I quite enjoyed both topology and analysis, but differential equations are the foundation upon which our understanding of the world is built. Getting at least through differential equations is essential for anyone who is to claim to be educated, regardless of their field of expertise, it's just too fundamental to pass over.
Trigonometry is much easier if you consider everything quadratically instead of linearly. The pythagorean theorem becomes a + b =c. Angles become opposite^2/hypotenuse^2 = sin^2 where 1 is equivalent to 90 degrees. Then you get less approximations because you dodge the square roots and you wont need a calculator for trig. So you push the sqrt calculation out to the last possible step which usually leaves it as part of your answer and you dont need to evaluate it.
There is a lot of stuff to memorize but at the same time it also gives structure to the subject. I feel like a lot of problems has a very straight forward solution that you either know or you don't. It just comes down to practice, practice, and practice.
I think that one reason that trigonometry was easy for me was High School Geometry. For an entire school semester we're thinking about triangles and four-sided figures, goin' through those Side-Angle-Side, etc. theorems and doing simple proofs. And understanding proportion, and Euclidian geometry. From what you are saying, that geometry class helped me much more than I realized. In trig it was challenging simplifying expressions. I thought that the teacher wouldn't like this because he never mentioned it, but I would convert everything to sines, and cosines, oftentimes replacing cos^2(angle) with 1 - sin^2(angle). Then simplifying, and converting things back again.
I took it once and didnt do well enough to move on so had to retake it. I got an A in it the second time. The reason is because I had all my old exams and knew exactly what it was I was going to need to know. A study guide is the answer to making students do better. A good study guide. One that is an actual representation of the exam that you are going to make for them and give them. What I hate are semesters and the time frame we have as students to learn stuff. I'm 34 years old and a non traditional student. College needs an overhaul. It would be nice if the time frame that students had to learn something just didn't exist. Imagine if we were actually building students up and teaching them. Instead, we give them 4 months to learn something and they have to just move on and then have low GPAs and can't get into the grad programs they want. As an old student that knows better... I would much rather you teach me what I need to know to move on and I shouldn't be allowed to move on until I show that, even if it takes longer. If we look to technology, AI can do away with professors all together. It will be able to teach a class and explain topics in multiple different ways better than any human. It will write tests and practice questions in minutes all with new variables. Students could take classes as long as they needed to for basically no cost... the cost of some software. It will do it better than the best colleges in the entire world. The goal needs to be to TEACH people and build them up.. no matter what that takes. People need longer than others and they shouldn't be denied or have to wear that bad grade around for the rest of their life that says they aren't good enough when they have ALL THE TIME in the world to prove they can learn it and they ARE good enough. We've built a system that keeps people out!!! If professors give a shit about their jobs and their professions they should be the ones at the forefront of this change... or AI will come in and take it away from all of you. You hear a lot of college professors say "I don't do study guides. You all should just be old enough and responsible enough to do it on your own now!" F*** that. That is the biggest cancer in college system right now. Students are paying 1000's and 1000's for something they just end up youtubeing anyways. It's rediculous. You would think the very least they could do for the students paying so much money is give them a damn study guide. Then they fail all the classes and the colleges surely don't give a shit but want to wonder why? Because ur not even doing the obvious things people need to pass. If your 18 years old and have to go to college for 6 years for ur 4 year degree... so what? We should have a system that can accommodate both people. If people can come in and show they know what they need to know in 2 months then let them go. If it takes them 8 months then fine. But lets build the future of our nation and stop building barriers that don't let them do what they want in life... but then the grad schools will end up just have some arbitrary cut off of 4 years and if you didn't complete it in that time then you just automatically get rejected. Sorry for the rant. I'm sure ur a great professor that has to do exactly what they tell you. I feel like if you started turning out classes full of A+ students they would tell you that you need to increase the rigors of your course to be more challenging. That they some how can't allow that. My aim is dental school and I never once cared about how long it takes me. I just want them to build me into the person I want to be. I'm willing to do whatever it takes.. but the college can't help you. You can only help yourself. I should just be on a track through college that leads me to being a dentist whether it takes 8 years or 12 years. Some times it feels like I have a masters degree in common sense or something. Why nobody talks about any of this? Somebody please help me create an artificial intelligence college! I know what ur thinking... o like an online college? No. I mean a college with desks that the students go to and they sign into their desk for attendance and an AI professor teaches the class and then sends an email to the students that were present. Ur AI professor will literally be standing over ur shoulder during ur homework assignments. It's going to teach people better than any human ever could. Students simply pay a monthly subscription fee to take any path they want in any amount of time they want. We prepare for DAT and MCAT and every other test you can think of. I could create a utopian college that promises to prepare people for the exact life they want to have. Not a system that is going to destroy their dreams and leave them with 1000's of debt they have no way to pay back other than doing something they don't love. We can program an AI to care about you 1000x more than any human professor actually cares about you. Some of the human professors are going to assign you work that is going to make you cry... and they think that is funny. O also, at my school you don't have to pay $400 for parking. We don't charge for that..
What is Math Goggles? at the comfort of your space
Cuesta College San Luis Obispo, CA. College is close to the Pacific Coast Highway (PCH). Put this on your bucket list for places to see. My daughter and I drove from Monterey CA to Long Beach. Views are breathtaking.
What you need to is think in pictures and numbers at the same time connecting the two. Algebra is connected to Geometry through pictures on graph paper. All trig and trig functions are is names for the pictures you draw. If you remember the pictures you can draw them and then reconstruct the reasoning needed to solve problems you forgot the answers too. That way you don't have to worry about remembering everything.
Anki is what saved me. I’m in med school. My first run in with struggling to memorize content was in biochemistry. Anki is a flashcard program that spaces out your repetition and it helped me stay on track.
Anki is a godsend
I think that key moment in trigonometry is just trig functions and theyre relationships between them. Every angle has all six values of trig functions.
I feel like Trig may be the hardest for non-math majors having to take the course for their science degree; for math majors, I've never seen a class complained about so much as Real Analysis personally.
My old EE professor told me in high school back in the day, they didn't have calculus and AP math classes. Trigonometry was senior level math. Which they really learned in detail.
You only gotta remember a few sines (pi/6, pi/4, pi/3) and you can derive the rest on the fly as needed.
Trig wasn’t difficult but it wasn’t easy either because I like to work from first principles and it would be hard to get tests done that way when they are timed (took trig in high school). Also, using the first quadrant is the way to go. Funny enough my knowledge of hyperbolic functions actually made more sense once I did some complex analysis and special relativity. Everyone has their struggles and this is definitely one of them.
There are a few tricks needed in order to kickass at Trig. You need to invent some mnemonic devices for this. If you just create a few mnemonic devices ...you will NEVER forget anything, you will rock the exams, and you will absolutely ACE calculus. You MUST create some mnemonic devices for trig, you'll thank me later. You only need a couple of these and you're golden. Truly excellent video. I have seen some very smart people who crumble in Calculus because their trig was not so polished. Don't be that guy. If you can ace trig, you can ace Calc II.
Also - to reinforce your trig ... solve every single problem on your own time before you ever go to class. Then go back and solve the same problems several times to build up some speed. If you solve the same problem 4 or 5 times you start getting really fast. I would be happy to share my personal mnemonic devices with the world if anyone wants to see them. I recently showed a few of these to a co-worker and he instantly understood some trig, and he's not even a math person.
I teach pre-calc at the university level and I agree that students find trigonometry hard. My own take on the trig values is not to memorize any trig values at all. I think you need just a little memorization from geometry that shouldn't be rote memorization. Most students already know the Pythaorean theorem by heart. Add to this the 30/60/90 and 45/45/90 triangles add the unit circle idea and you can figure out any of the values that come up.
The only trig things i ever memorized are formula for sine of sum, formula for cosine of sum and then i remember that in a 30 60 90 triangle with hypotenuse of 1 the shorter cathete is 1/2 and the longer one is √3/2, then i fit the triangle onto the unit circle to find the values for sine or cosine of π/3, π/6
Ironically I’m starting to work through the Trig workbook by McMullen off of your recommendation today, and I see you upload this!! Lol
I love trigonometry it was the reason I was able to do well in other math classes, currently I’m in Differential Geometry and woah do I find it difficult lol so far it’s harder to grasp than Real Analysis.
Learn to fail and don't fear the maths start from the basics of geometry start approaching In different ways give time and see the magic
I've found that in physics & engineering classes, most of the trig you learned, you never use, but the little you DO use (mostly the trig definitions), you use a lot.
Knowing the history on how some of these concepts originated has helped me understand them, and easily recall them to memory. Instead of memorising
Math competition problems are surprisingly really hard. Reading someone else's solution is difficult
I was fully prepared for this to be a discussion on analysis. While it's more abstract, I thought it was more enjoyable then trigonometry, and most likely because of this, I found analysis (real) much easier. I had to take trigonometry twice.
We take collagae level algebra and trig in highschool thanks to Türkiye's education system(I graduated from hs in 2016).We are thought to referance first quadrant by default.If you can get a grasp of referancing since all trig functions are periodic functions,it becomes much easier leaving a few formulas to memorize here and there which can easily be derived if you get your basics right.
I found trig to be very easy, but I think it was because I was into electronics from the time I was in elementary school and took electronics classes starting in 9th grade. We were working with complex numbers, trig functions, phasors, etc long before I set foot in a formal trig class.
That unit circle and those trig definitions -- I always thought that we have it backwards -- sine and cosine. The x-axis is first, and the y-axis is second. So if anything is co- it should be the y-axis. And we always say, "sine and cosine", which is like saying, "y and x".
I had no problem with math until I got to integrals in calculus. No problems with algebra, geometry, trig, solid geometry, and derivatives. Integrals requires you to remember how to play with equalities/replacements and my mind just does not seem to work that way. I was a CS major whose mind works in very straight lines. Almost failed partial differential equations, but I think the professor gave me a pass cause I worked hard at it. I also took group theory and I loved the course - it was easy.
Trig is underlooked. Many even myself mistake as plug n chug, but the identities and relation to 2pi is the true buttkicker.
Its daunting, or too niche. Calc comes - you gotta know it. Practice problems are a must if just saying "memorize" as it becomes memorable as pythagorean
I remember learning the sine and cosine values of common angles, and the positive/negative signs of sin, cos, tan around the unit circle, in what was the equivalent of 9th/10th grade (age 14 or so). So when it came to learn more advanced trig in 11th and 12th grade high school years (age 16-18) like the double angle/half angle formulas, and how to prove trig identities using them, the basic understanding was already there. I think the reason "trig is hard" is that you are not meant to learn it in one or two semesters in college. You are meant to learn it gradually over a period of years, starting early if possible, so it has time to sink in. Anything is hard if you get required to learn it quickly over a short period of time.
I just started to self study trig today, so this upload is a bit of a funny coincidence
I found trig easy, but I took it after calc 2. I tested into calculus not having a solid algebra 2 foundation because my ghetto high school didn't teach very well...
All my friends went to rehab for drug use, so I had a lot of free time, and so I dual majored in math and physics getting a 4.0, outstanding senor award, and excellence in undergraduate research award. Didn't even know I liked studying and learning like that till college!
If you can draw the unit circle then you can make sense of all the graphs of functions, just picturing different right triangles in the unit circle.
I was surprised at how well I took to Trig in college. I was inept in my math classes in HS: only took geometry and applied algebra classes. But was able to get a math minor in college. Built my skill from the ground up!
As a physical scientist, it is unsurprising that I like math. I agree that trigonometry requires a fair bit of memorization, but what I learned later is that complex numbers and a thorough understanding of how to use Euler's relation and the unit circle, converts a lot memorization - especially trig identities - into easy algebra problems. Yeah, there's a barrier to getting past the abstractness of exp(iθ) = cosθ + isinθ, but once you get used to complex plane, trig gets pretty easy. Chapter 8 is underemphasized or even skipped when it's actually taught.
The most directly useful math I know.
Trig was easy for me. Everything had a relationship to geometry and could be drawn on paper. It was stats that I really struggled with. I still struggle with it today... confusion with different aspects with probability (like combinations and permutations), certain types of distributions, how to calculate certain values correctly (like quartiles).
As someone who teaches calc I'm curious about how you feel about professors providing "formula booklets" with trig identities /trig vals or letting students bring a cheat sheet to calc exams ?
I genuinely don't get whats the difference between being able to solve a question becauee you mugged up a trig fact vs being able to solve a question because you consulted a list of trig facts.
In fact even when I was first learning trig ( as part of IGCSE math in my freshman snd sophomore year, followed by IB math during my junior and senior year) we had formula booklets for our math examinations and were assesed using 2 different final exams (a calculator paper and a non-calculator paper).
This did not "trivialize" trig but managed to make it challenging in more interesting ways , because you were solely tested in your ability to use these trig facts as tools to solve a more complicated equation or model a problem.
When I attended high school many decades (5) ago I had no issues with algebra 1 and 2. My algebra 2 class my teacher told me I didn't even need to take the final exam. Trigonometry & Geometry was just the opposite. I blame myself for not grasping or studying. Some classes you love the math and the teachers are excellent. My older brother was the math whiz. By the way do we still use a slide rule in Trig?
I had an easy time with trig, but I've always been a very visual person. And I think that's what makes trig so difficult for many students--there's so much of it you have to understand visually. You mentioned reference angles, and that's a great example... A student asks, "what's the formula for a reference angle?" The answer, "There's not one... You have to draw a picture and connect the angle back to the x-axis. Writing down a "formula" would involve too many cases depending on where the angle lands." You really need to understand it visually rather than in terms of a formula, but students are often taught that math is all about formulas.
Just to give one more example, "What's the formula for the polar angle?" Well, "It's theta = arctan(y/x), except that that formula always gives an angle in QI or QIV. If your point is in QII or QIII then you need to add pi. Also, for a point in QIV the formula gives a negative angle; so if your book or teacher wants a positive angle you need to add 2pi." That's a lot of caveats to just remember. They are easy to understand if you really have that graph of the tangent function in your head... The main connected component goes from (-pi/2) to (pi/2), which means QIV to QI on the unit circle. The period is pi, so if you need an answer in QII or QIII you can just shift over by pi, or by 2pi if you want a positive answer in QIV. But that understanding requires you to both have the image of the unit circle and also the image of the graph of tanx in your head... Most students, if you ask them what the graph of tanx looks like, they hit a button on their graphing calculator because that's what they've been taught to do. They don't have that solid mental image to help them understand why there are those caveats to the formulae and how to work around them.
In more mathematical terms, the real issue is that the trig functions are horribly not one-to-one, and so any process that requires finding an angle is going to require careful adjustment. In algebra, the only not one-to-one thing students do is squaring... So the only experience they have with adjustment is putting a +/- when they take the square root of both sides. Trig is quite a bit more complicated.
One last thing, when I teach calculus I find the background material gives students more difficulty than calculus itself. That's why now in a 16 week calculus course I spend the first week just on algebra review, then the next two weeks just on trig review. The first exam is just over the algebra and trig review. Finally at week 4 we start actually talking about calculus itself.
I believe Trig is difficult to teach because it's never broken into subset semesters: Trig 1, 2, and maybe even 3. As with many math subjects, there's not enough time in a semester to teach everything thoroughly -- and there aren't enough semesters in an undergraduate education to divide Trig like that. Yet thorough knowledge of trig is essential to understanding in depth most math subjects to follow. I myself think trig is fun, poetry with numbers; but my learning technique doesn't involve classrooms. Thank you for another excellent video.
I totally agree. I taught Trig at out local J.C. several times but it was only a 3 unit class. I used exactly the same book as The Math Sorcerer and it is an excellent textbook. Unfortunately, the best of the 8 chapters is Chapter 8 where Trig and College Algebra are blended together into a beautiful topic called Complex Numbers and Polar Coordinates. Generating the neat graphs in polar coordinates and understanding the complex numbers for learning fractals is why Chapter 8 is so important. I had a difficult time getting to and covering that chapter thoroughly because of the breadth of material in the book and the little time 3 units gives you.
Trigonometry is full of facts and formulas and rules and identies that are interrelated in all sorts of ways, so that many of the things you "have to memorize" are things that you could re-derive or figure out every time you need them, except that having them memorized is so much faster and easier.
I think eventually, if you work with it long enough, things fall into three categories: (1) Stuff you remember because you've used it enough, (2) Stuff you remember how to easily figure out, and (3) Stuff you know exists and you can look it up if you need it.
I just started trig and I love it
This is so true. Trig identities, even if you can derive them, still need to be memorised to be recalled quickly enough in exams. It just sucks as the failure to memorise doesn't demonstrate a lack of skill, maybe just a lack of memory. Why not make students memorise log tables while we're at it.
sin(0°) = ½√0 = cos(90°)
sin(30°)= ½√1 = cos(60°)
sin(45°)= ½√2 = cos(45°)
sin(60°)= ½√3 = cos(30°)
sin(90°)= ½√4 = cos(0°)
Trigonometry is such an irrelevant artificially overcomplicated subject. You are right about having to learn too much. So for Standard Angles, I came up with the above.
Hello,
Basically trig wasn't hard in the beginning. But when started with systems of trig equations where you need to use some identities that are rarely used to find the solution, knowing we had to memorise all those identities... Well, this part wasn't easy. As you said, the problem was mainly memorisation and a bit of "finding the trick" to rewrite some equation using a good combination of identities...
Not sure what I expected this to be, maybe calculus, but I never had a problem with trig. I went through a few of my father's old math books around 8th grade and it didn't take long to figure out "what trig is," it's basically functions of a right triangle, essentially the ratios of different sides based on the angle. How tall is a flagpole? On a sunny day you can measure the length of its shadow and the angle above the horizon of the sun, and find out from that info. This helped later in high school with complex numbers. and in college with AC circuits (involving sine waves at different phases, more complex numbers, but with j, because i is used for current in EE).
Cotangent, secant and cosecant are easily defined as the reciprocals of tangent, sine and cosine, but I was never sure why these named things exist. There aren't a lot of named functions that also have their reciprocals named. I'm sure there's historical reasons for this, but I never knew about math history to learn that.