I've only watched 38 min so far and I'm already thinking that this is exactly how we should be teaching students. Live or not, in class or online. You sir, are a good teacher. You should be like a consultant for some school board of education or something. What the world needs is more teachers who can teach like you.
I could've used Grant about this time in 1974! The nun I had for algebra II and trigonometry drove the class nuts. I haven't enjoyed math this much since Lancelot Hogben's "Mathematics for the Million". Now when I see a pi, I think of a circle.
Grant, Khan, Bozeman Science and there are many more (I wish I could write all the names here), are fantastic teachers. I wish I had teachers like them back in school.
Incredible, thank you very much. I'm a physics teacher, 45, and since 1990 I've never seen such an interesting and imaginative talk about trigonometry!
@@xDomglmao I'm just here cuz I like listening to grant, and sometimes he hints at higher level topics (that I'm tempted to look up and self-study)... also it never hurts to have a refresher, especially in the styles he likes to teach in
Oh, I have the Same whish. I AM 15 and I want to learn,no I want to understand Everything about Math, from the Basics to........oh it Never ends. It‘s easy to find the „how“ and to KNOW something but it‘s hard for me to find the „Why“ and to UNDERSTAND something. Do you know a good Way to understand something in Math? I Mean,I just hate to know something without understanding it. For example : Why rotates e^(xi) at the Circle by exatly x?( Why gives me x the angle of the number (radian)?) Why is n->endless (1+(pi*i/4n))^n = i^(1/2)? Why (1+(1/n))^n ?? Why e???
Hey Grant. I very much enjoyed your streaming. As a math professor, trig is always one of the topics where most of the students struggle with the most especially for high school or even college students. What I focused on the most is how to visually make them understand especially the relationship among sin, cosin, and tangent as they can be used and manipulated to figure out csc or others. Your manim seems to be a great way to visually make students understand trig more than any other tools available now. Good job and I very much enjoyed!
I learned sine by means of a diagram with a large circle (radius 1 unit), inside of which were the right triangles for 10°, 20°, 30°, and so forth, all drawn with one leg horizontal. I believe that the sine was explained as the length of the vertical leg (negative if downward). It encouraged me to think of sine in terms of the vertical position of a point on a rotating wheel. I put that knowledge to use when animating clock faces: sine and cosine tell you where to draw the end of the hand.
@@robertlozyniak3661 Exactly! If you have a right idea of sine and cosine, everything else would come very easily. Hope I can make a video of trig some other time as well!
The tan part was extremely fascinating. Sad that they never teach such stuff at school. Would have had a much better understanding of trig if these things are taught. Edit: Now also managed to figure out sec and cosec with how you found tan. At last the names make sense. You're a godsend Grant!
Imagine how interesting, and in some ways easier, it would be to learn mathematics the way they were discovered instead of in some random "easier to understand by politicians" order.
i was taught about tan(x) in a different way, using a straight line perpedicular to the x axis and tangent to the unit circle, projecting the radius on top of it. it's much more intuitive this way tho. i love those lectures!
Start (he appears on camera) 9:14 12:16 Housekeeping and preliminary question finished 14:00 Question 1 16:10 Answer reveal, 16:36 further explanation 17:40, Question 2 20:10 Answer reveal, 20:40 further explanation 22:02 Pen and paper, (cos(x))^2 = (1+cos(2x))/2 23:02 Question 3 23:48 (Accidental answer reveal) 25:26 "You think it's about triangles, but really it's about circles." 26:09 Sin(x) animation 27:30 Cos(x) animation 28:51 Question 4 30:45 Answer reveal 33:00 Pen and paper, Soh Cah Toa 34:45 Question 5 37:02 Answer reveal and explanation 39:51 Pen and paper, Unit circle 41:50 Radians and degrees connection 43:00 How do you compute these values? 44:12 Special right triangles 45:32 Question 6 46:15 3b1b answers, Do you use pen or pencil? 47:20 Q6 Reveal 48:30 Question 7 49:41 Answer reveal 51:14 Pythagorean theorem connection 54:50 New page, Question 8 57:25 Answer reveal 59:43 Back to cos^2 (x) 1:02:00 Question 9 1:04:25 Answer reveal 1:09:00 Where is tan(x)? Where is cos^2 (x)? 1:11:46 Tangent animation 1:12:30 Final Question 1:14:34 The 'hackerman comment' 1:15:16 Final question, answer reveal 1:16:41 Animation 1:17:45 Back to the tower problem 1:22:00 Fully labeled and explained triangle 1:23:54 Textbook formula connection 1:25:08 Goodbye, Patreon supporter screen Fin
I (just like most viewers) know the topics he explains already, but I'm still learning new things; 1:17:45 was an interesting take, watch it even if you think you can't learn anything new!
Thanks! So sorry to do this, but I've trimmed the start of the video so that the intro screen and housekeeping isn't part of the final video, which sometimes takes YT a day to properly process.
I am in absolute awe of this lesson. It’s structured so perfectly and I enjoyed every moment of it. Thank you for dedicating yourself to raising the bar of math education. I have no doubt thousands of teachers for years to come will use this and your other lessons as a model to teach this material. And no, I don’t think I’m being hyperbolic here.
When you got to about the midpoint of the video and started to talk about the Pythagorean Identity of the Trig Functions, I would like to elaborate on this and add to it. We know that the Pythagorean Theorem is: A^2+ B^2 = C^2. Let's keep this in mind. I will show and prove 4 - 5 different things that most math classes never fully express and these are the following: * The Pythagorean Theorem and the Equation to a Circle are symbolically similar, also The equation to the Unit Circle centered at the origin is, in fact, the Pythagorean Theorem! It's just that one is in terms of right triangles and the other is in terms of a circle with a radius of 1. * That there is a direct relationship of the Trigonomic Functions and linear equations in regards to their slopes. * Without considering the use of limits and applying them, the Tangent Function is, in contrast, the definition of the slope function that is used to define a derivative within Calculus. * That vector notation is symbolic of both linear and trigonometric calculations. For example ⟨a,b⟩=∥a∥∥b∥cosθ which states that the dot product between two vectors is the product of their magnitudes multiplied by the cosine of the angle between them. So if you know the lengths of two vectors you can find the angle! After you calculate the dot product, then you would have to take the arccos of that value, however, make sure you are using the right system (degrees, radians) to get the correct result you are looking for. * All of this is rooted in the simplest of all mathematical expressions, not even an equation or a function, just a simple expression, being the very first one we are ever taught: (1+1) and that the operation of adding one to itself satisfies the construction of both the Unit Circle and defines the Pythagorean Theorem. Also, when we turn this into an equation (1+1) = 2 we will see that there is perfect symmetry, reflection, and rotation that is embedded within this. The comparison of the Pythagorean Theorem and the Equation to a Circle: The equation to an arbitrary circle is defined by (x - h)^2 + (y - k)^2 = r^2 where (h,k) is the 2D coordinate of the center of the circle. Let's center this at the origin (0,0) and refer to the unit circle with a radius of 1. We now end up with: (x-0)^2 + (y-0)^2 + 1^2 = x^2 + y^2 = 1. Let's compare this to the Pythagorean Theorem. x^2 + y^2 = 1 == A^2 + B^2 + C^2 when C = 1. So for any circle that is centered at the origin, it's equation is the Pythagorean Theorem. The direct relationship of the Trigonometric Functions and Linear Equations: We will use the slope-intercept form of a linear equation: y = mx+b. We know that m is defined by rise/run or (y2 - y1)/(x2 - x1) where can use the coordinate points to find the slope. The value of this slope is a proportion of how much change in height over the change in the horizontal. All angles will be relative to the line y = mx+b and the x-axis. In other words, it is the angle that is above the X-axis or the Horizontal axis up to the line itself. Here I mentioned the rate of change. We can take rise/run or (y2-y1)/(x2-x1) and rewrite this as dy/dx. We know that we can make a right triangle from the x-axis up to the line in question. By doing so we can see that the dy is also sin(t) and dx is cos(t). We can see that the slope of the line m is also the tangent of the angle above the horizon to that line. So we can rewrite the slope-intercept form y = mx+b to y = tan(t)x + b or sin(t)/cos(t)x + b. This will lead us to the next section about derivatives! The Tangent Function and the Derivative: We know that when we have a curve that its slope is not constant. We can take two points on that curve that are relatively close and get a good approximation of its slope near that point, however, the farther the gap the more margin of error this is. This approach is what is referred to as finding the secant slope. If we make smaller and smaller incremental steps where we get closer and closer to that point where the limit of the size of that step approaches 0, we then end up with a line that has the slope of that point which is tangent to that curve at that point. This is seen in the difference quotient in Calculus to find a derivative. f'(x) = lim dx->0 (f(x+dx) - f(x))/dx Which is basically taking the slope form of a line (y2 - y1/(x2-x1) and rewriting it from point notation into function notation with respect to x, changing the difference in points to rate of change by using delta x and delta y, which is still all algebra, and the only part that is Calculus is when you actually apply the limit! So in a sense, we can see a direct relationship and similarity of (f(x+dx) - f(x)) / dx and tan(t). If we substitute tan(t) as sin(t)/cos(t). We can see that cos(t) = dx which is straight forward. However, if we look at f(x+dx) - f(x) it isn't quite obvious but this would be equivalent to sin(t). Where t is the angle above the x-axis and up to the point on the curve whose slope is the tangent of that angle produced by that linear equation. The vector portion should be self-explanatory as I had already mentioned the most important piece and that is the dot product in relation to the cosine of the angle between those two vectors. Not much more needs to be expressed about this here, but will be referred in the last section. (...continued in a reply to this post)
(...continued) How (1+1) ties all of this together! Consider the value 1 to be an arbitrary unit length, it could be 1 inch, 1 mile, 1 cm, it doesn't matter! Now, take a piece of square paper and mark a point near the center of the paper, then draw a line from there in any direction and stop a little less than 1/2 across the paper and make another point where you stopped. Now go back to the center and mark this as 0. Then mark the endpoint as 1 and make a slight arrowhead at that point. Direction here is completely arbitrary and agnostic to the length or magnitude of that line segment or vector that you had just drawn. Now this will be a unit vector. The starting point labeled 0 will be the 0 vector, this will be important later on. Now rotate this paper until the line segment is parallel with your body and that the 1 is to the right side. What we are going to do next is to apply the operation of addition to this unit vector with itself. By convention addition in the positive sense will be towards the right. So what you now need to do is visualize that entire line from 0 to 1 sliding across itself so that the 0 point is now at 1 and the new endpoint of 1 is near the edge of the paper you can mark this with a 2 and put another arrowhead there. By drawing these two vectors you have physically done (1+1) = 2. Okay, so we saw this being done through vectors. I had you do this to express the importance that Addition is a Linear Transformation. It is Translation to be exact. Now without knowing it, by performing that addition you also simultaneously performed multiplication. You see Addition is a 1 Dimensional Operation, but it introduces us into 2 Dimensional Space. It doesn't seem apparent yet, but I will get to that! We know that (1+1) = 2 and we know that 1*2 = 2. So we can easily see that (1+1) = 1*2 = 2*1. Multiplication is embedded within addition, however, multiplication can be both a 1 and 2 Dimensional Transformations. A 1D Transformation of multiplication is scaling or shearing in a specific direction and a 2D transformation is a Rotation or defining Area. Now, how does this relate to the Unit Circle, Triangles, Trigonometry, The Pythagorean Theorem, and even Calculus? It's quite simple... let's take a step back and remember when I said to draw a line in any arbitrary direction... then I had you rotate that paper to position the line to be parallel with your body and the line pointing to the right. Now, we denoted this as being +1 which could be symbolic with the direction East. Nothing stopped us from being able to go West by 1 unit. So now let's make another unit vector that goes from the origin 0, and mark this Western or Left point with a 1. Now, when it comes to vectors, the magnitudes of the first vector, and this new one are the same, but their directions are Opposite. So we can use the negative sign to represent the opposite direction. So we can now add a - to this 1. If we add these two values (1) + (-1) we end up with 0 which leads us back to the starting point or the 0 vector. However, their combined value |1| + |-1| = 2. These three values of 0, 1 and 2 are all related to each other. Now we said that addition is a 1D transformation translation to be exact. We also said that 1+1 =2 is also multiplication. Well we have a line and there is no area, so the 2D transformation here must be rotation. Let's take the first line 0 to 1 and rotate that about the 0 point and see what happens when we end up at -1. We did a 1/2 rotation of a full circle. We know that the arc length is PI and that the central angle is 180 degrees. We also know that a line has 180 degrees of rotation which is also the addition of all 3 interior angles of any triangle. We know that the unit circle has a radius of 1 and a diameter of 2. Its circumference is 2PI and its area is PI unit^2. Yes, even the value PI is embedded in the expression (1+1). This expression has a perfect symmetry and complete rotation embedded within it. It also has perfect reflection too. And if we understand what Derivatives and Integrals are, we also know that (1+1) doesn't just introduce multiplication but it also introduces powers because (1+1) = 2^1 which is a 3D transformation provided there are three components to the vector. For each component of a vector , , , etc... the number of elements or the size of that vector is the dimensional space we are working in. Each dimensional space has specific attributes or properties. 0D = an arbitrary point. 1D = length, magnitude, line. 2D = rotation, area. 3D = multiple rotations and volume. etc... We also know that when we look at the polynomials in algebra: f(x) = x, f(x) = x^2, f(x) = x^3 that x is just x which is a scalar quantity and is linear and that x^2 represents the area of all squares, and that x^3 represents the volume of all cubes provided x is +, there is a case where x can be - in to define a - volume, but this involves complex numbers and is completely abstract! We also know that x, x^2, and x^3 are partially derivatives and integrals to each other so to speak. F(x) = xdx = 1/2x^2 + c. which states that the integral of f(x) = x is the family of functions f(x) 1/2x^2 + c. and so on, and the reverse is true too. All of this is embedded in (1+1) = 2. All of your irrational numbers are embedded within it. Your logarithmic and exponential functions are embedded within it! Everything is integrated or derived from (1+1)!
@@Kr3nkt Maybe, maybe not, but that's not the point or the reason I put it out there. It's there for any who would like to. No one has to and I'm not expecting anyone to. I just like sharing what I know and understand. It's not my problem that others may be too lazy to read it. Yet there are 29 likes on the first part, and an additional 10 on the second part. So it does appear a few have read it after all!
Compare this with my University lectures and there is no comparison. We need to have this type of fun in the University and have the time to ponder and think instead of rushing at the speed of light and cramming for a test.
*Cries in dropping out of university because of light speed lectures and coming from an highschool perspective, which is light years away from what university is*
Nathanael Kuechenberg Age doesnt matter but I am truly glad to see how motivated you are to pass the classes. Thats what matters the most and you seem to be determined. Let me know if you need any help with math!
Grant, thank you for the BEAUTIFUL knowledge you said in the live stream. I’ve never been so intrigued by trigonometry until I saw this. Never really understood how trig worked in circles, the relationship between Pythagora’s theorem and Trig identities. This video has surpassed any trig class I have taken in my life. Thank you 😁
Knowing that sin and cos relate to the coordinates of a point in the unit circle has unlocked a subject that was opaque to me for so long. Facts like sohcahtoa and the rotation matrix make sense now. It's especially satisfying that I can now combine what I learned from you about linear algebra with this to have a solid understanding of what the 2D rotation matrix does.
This series is a gem. I really hope you can continue this sort of livestream after the covid 19 lockdown has passed, because it is truly a wonderful experience.
After the last one, I went and watched the entire imaginary numbers series by Welch Labs that Grant recommended, and seriously had my mind blown. Highly recommend
LOVE relating the trig functions to circles. I realised that myself about 3 years ago and have been teaching it that way ever since. I approach it as a rotating radius around the circle and that sine and cosine tell you the ratio of how much of that radius vector is vertical or horizontal. It helps build that intuition for where sine and cosine are -1, 0, or 1... and also what sign to expect the value to be in which quadrant. I then extend it to tangent as being the slope/gradient of that rotating radius vector which explains why the rotation by pi has the same tangent value, because it's part of the same line that goes through the circle's centre. :)
My high school teacher taught trigonometry from circles, and I am so thankful. It is so much clearer. Then again, he was pretty overqualified for that job and I believe he may have also taught at universities.
the way you explain things so good, always coming to that mind blowing connection/context at the end and missing out difficult things till they get necessary and mixing fun and good jokes in-between reminds me of Gilbert Strang. The only thing you updated is instead of using a board and chalk you use cool new tools that really help to visualize something. thank you a lot, that really helped me.
When Grant pulled out the second printed sheet I imagined him stockpiling trigonometric circles at night next to his desk to make sure he has enough for his live Great video.
I'm doing a mega math review because I want to undergraduate in math, and this is an absolutely amazing second introduction to trig. Absolutely beautiful.
21:00 my math teacher in Israel had a way dumber (and easier) way to remember (though it relies a bit on the different pronunciation of the function names): Sin -> seeing (so its the opposite one that the angle "sees") Cos -> cousin (so it's adjacent) Tan -> tango (so you invite someone you see to dance closer to you). Nothing beats that intuition for me
I have a method to correlate trigonometry functions with the angles too! Here’s the way to memorise it: Cos: The arc of the angle we usually draw to indicate theta Tan: The capital T has a right angle Sin: The remaining angle, or I remember it by s for the (usually) smallest angle Although it’s very intuitive, this method doesn’t tell you what sides are on the numerator and the denominator. So this is a merit of the Soh Cah Toa method. I still prefer the method I mentioned since I work better with shapes. Hope this helps!
Once more thing I would like to add is students seem to struggle understanding the concept of distance when we deal with trig, and your manim also seems to be a great tool to help them understand those concepts. Once again, great job, and I very much enjoyed.
About 33:40 - I know everyone now swears by SohCahToa but I've remembered for nearly 50 years what my father told me, and he must have learned it in the 1930s. It uses the same diagram as yours, but has a Base instead of an Adjacent and a Perpendicular instead of an Opposite. I found it mildly amusing, which helped me remember it: Some People Have Curly Brown Hair Till Painted Black.
I love your videos, your music, and ways of breaking things down to the smallest possible explanation in math. Most times, because the way we learn maths (or related subjects) is to ace an exam rather than understand what we're doing both visually and numerically, we learn the different aspects separately as opposed to relating them this way, right up till university. Thank you and your team so much, for putting up content worth while on youtube!
there's an animation bug in your itempool thing: it's revealing the answer to the next question. eg watch @45:35 at slow speed: the green bar indicates 'B', the correct answer to the tower question, but then it switches to the next question, moves the green bar to the correct answer and _then_ flips the card to a version with the answer hidden.
Yeah, this happens several times (also: 49:50). Upvoting and commenting in hopes to drive this comment up into 3b1b's radar! (And/or the radar of the folks working with him on that stuff.)
I've learned more watching this than I have in all my other math classes! I might have made it through Calc 2 if I'd understood trig at this depth! Thanks!
My trig intuition went from triangles to circles pretty quick once I started playing around with 3d modelling software and needed to get radial symmetry in my vertices.
I'm honestly so glad I watched this, even though I am pretty far beyond Trig. I skipped Pre-Calculus in High-School and I've always just implicitly trusted the identities without ever thinking about the underlying geometry.
damn i miss these lockdown math videoes. i learned so much by them and it was so nice to see it live and paticipate. wish some youtubers would do this more :) and in such a nice pace as you did so i could write it all down too!
Back in high school, I used a simple table to remember the sin/cos of the angles 0, 30, 45, 60 and 90 degrees: "sin (Nth friendly angle) = sqrt (N)/2, for N = 0 .. 4; cos() goes in the opposite direction". With the friendly angles being 0, 30, 45, 60 and 90 degrees. Using reflection and remembering sin() is non-negative for angles 0 .. 180, and cos() is non-negative for angles -90 .. 90 gives the values in the other 3 quadrants.
Timestamps: 9:15 Grant comes to the stream 12:17 What will we talk about today 14:36 Graph of cos(x)² 17:59 equation for cos(x)² 19:40 the new music 25:59 trigonometry basis 34:49 leaning tower of pisa 39:45 the functions on the unit-circle 43:04 computing special values 55:00 negative angles 56:10 reality breaks 59:30 using cos²(x) for computing values 1:03:31 reality breaks (again) 1:09:03 where is tan(θ) on the unit-circle? 1:16:20 where is cos²(θ) on the unit-circle?
You really care about people who want to learn something and give your best. I am not going to talk about your teaching style and animations, which are world-class. No words are enough for them in my opinion and there are thousands of us appreciating these. It is really heartwarming for me to see that you have a pre-printed unit circle for us (and make small jokes about it). I have worse handwriting than you and have had even worse writing professors than me (yours is way better than many people I've worked with). Simple details like this unit circle, your care for us are carrying your video quality even higher. Kind regards and stay healthy :)
I love it that the bookshelf in the back contains what is in all probability the three-volume Calvin & Hobbes collection. Unless the three hefty brown spines are just eerily resembling said work
Best part of watching this video is that this guy seems like a regular human. His animation-only videos are just so elegantly done you kind of wondered who (or what) was making them. 🙂
I've seen many people in the comments who are already in designated positions like engineering but even they say that some of the things Grant says does widen their eyes sometimes. So, for people like me who just saw trigonometry for the first time in their textbook on January 1st, 2020, I can't thank you enough Grant for making these videos at a fundamental level. It helps me more more than I could imagine. Thanks a lot, Grant. Love these streams
"Instead they want some sort of natural units, something where you imagine if you talk to an alien civilisation about 'math' they would have the same convention." *stares in tau*
Let's suppose someone hands you a cylinder and asks you to give it's circumference. The thing you can measure (say with a caliper) is the diameter. So the circumference would be pi*diameter, what you could easily measure. Obviously you can calculate it by tau*radius, but then you have to change the diameter, one more mathematical operation, one more chance for a typo or another type of problem. So for something that promises simplification, tau does a really bad job at that. Or let's say you have the radius because it was given for you, but you need the area. you could just square it and multiply by pi, but no, we need the extra operation because tau. Moral of the story is, people like to argue over numb nothings lol.
Actually, pi/2 is more generalizable to higher dimensions, so it is more natural. Tau is a nice circle constant, but only in 2-D (and not actually much nicer than pi). So there is a good chance aliens would use 1.57 instead of 3.14 or 6.28.
I do not like tau because it is already used as a measure of period, torques, dummy variables for time, etc. Atleast when you use pi it only means 3.141... except in advanced number theory.
@@hassanakhtar7874 And also in economics for profit. The important part is the value, not the name, since we are discussing which value aliens would use.
I imagine aliens would go with the simpler of the two numbers for ease of operation, like when simplifying fractions. Tau is just twice Pi, so Pi is the smaller number. Smaller is simpler in math terms. Simpler math = better math Pi > Tau...in a manner of speaking😎
I still think that thinking of tan as "slope" is generally the best approach. This ties really well with sine = height, cos = width ie horizontal distance (with negatives for both)
If you think about tan as the magical function that converts angles to slopes, it makes a lot more sense. Why is it undefined at regular intervals? Because that's when the slope is pointing straight up! Why is tan's period twice that of sin and cos? Because 45° and 225° technically point at the same slope! Why does arctan have asymptotes? Because increasing larger slopes approach but do not reach 90°!
When he pulled out the compass, I had nightmare flashbacks to Euclid Elements. It’s VERY old school geometry where you can only use a compass and straight edge, but NOT a ruler.
All the trig identities are what really killed me in 2nd semester calculus. I had no problems with the concepts of integrating, but remembering AND figuring out which identities I needed to actually solve an integration problem was torture. Learning via this approach in trig class would have made things so much better.
@@bananastuff2840 JSON is a way of storing data so that it's easy to send to the Q&A program from the browsers. The live Q&A feature was apparently thrown together in a quick time. So apparently didn't quite take into account users "having fun" and sending in different answers than expected. That's why there's the '@'s. At least one person sent in data which had a comment in the data, which avoided the livestream slow mode chat. (Trimmed timestamp now at 1:01:21 or so).
They taught us trigonometry only trough triangles and only barely mentioned circles once, I didn’t even know about the whole pi thing until now, thank you
1:11:45 Wow, that's a great picture to keep in mind. Since the legs of the larger triangle are sec and csc we get a funny trig ID: sec^2 + csc^2 = (tan + cot)^2
im in integral calculus and i still struggle conceptually with trig sometimes. thankyou so much, i actually love starting from ground zero regardless of how much i know already because it allows you to feel confident you haven't overlooked anything
You should probably change the code for the pop-up transition cards of your questions because it reveals the correct answer card before the question card segways into place.
I am a final year Physics undergrad, and after watching your video (literally on any topic), I feel like we were living in darkness and this man showed us light. At first I was like, trig fundamentals, what could be there that I don't know and when he said imagine you know nothing about trig, that thing came out to be pretty true after watching video.
tanget: just draw vertical line straight via rightmost point of the unit circle, and height of the point that laying on intersection of hypothenuse and the vertical line is tangent.
Honestly, being a physics student, SOHCAHTOA, quadratic formula, and Pythagorean theorem have been super useful for me quite consistently and I use them all the time. Never thought I'd use those things I learned so many years ago still!
Sir you defined tan∅ as the distance between tangent point and intersection of tangent with X axis. But how do we explain negative values of tan∅ in that case? We we simply marked it negative on left side of graph when it's a distance. It was understandable for sin and cos because even though you started then as distance quantities, you quickly took it as coordinates represented on different quadrants. But we can't do that thing with tan∅ right? So we just define it that if it's on the left side, we will use negative sign? Also, I got very confused in the beginning when you used sin(x)² and (sin(x))² interchangeably. After 8 years I finally understood why we shifted to representing angles in radians from degrees. Thank you so much ♥️ P.S. Loved this class and all other videos. Keep doing the great work!
It is refreshing to learn trigonometry each time , i think visualizations are quiet elegant to learn in 3D coordinates if we practice more out of these amazing lectures.Also adding application oriented problem is creative way to learn.
Perfect way to refresh my memory on trig concepts before moving on to calculus. This channel is amazing for providing this level of instruction for free.
Just one livestreaming detail - your audio is consistently ahead of your video (maybe between 50 and 100ms?), so you might consider measuring this and delaying your audio to sync with the video.
Yeah, I was going to make a comment about the syncing problem in editing, but then I remembered this was a livestream, and that made me unsure of what to even suggest... But yeah, this might be the thing.
Amazing stream! Will watch all of them! NOTE: There's a bug with the software (pointed out by many I'm sure) that reveals the answer of the next question as the flip animation runs.
Yeah, there's are several of these. In the list of comments I'm seeing, yours is sandwiched between two of them. ;) (But only yours, among those 3, uses the word "reveal", which was what I looked for to find them...) Example: 49:50... though it's more telling at an earlier one... too lazy to dig back up the timestamp. :)
What a delight... I am a graduate student in engineering who has worked with trog functions for years now. There were still some insights in the video that were entirely new to me! Was a treat. Thanks 👍
So, about halfway through you mentioned that f2(x) usually means do the function twice, and that trigonometry ignores that. So I pulled up desmos and did cos(cos(x)) and it looks a little flatter. Then I did cos(cos(cos(cos(cos(x))))) and it's even flatter. In fact it seems to be tending toward the value 1/root(2). Why is that? And does it have something to do with the rms value of a Cosine function with an amplitude of a half being 1/root(2)?
1/sqrt(2)~0.71 is close (but not equal to) the fixed point of cosine (solution to cos(x)=x) which is ~.73. what's happening is that cos carries [0,2pi] (or even all of R) to [0,1], so it's "squishing" the interval. Applying cos over and over will continue to squish the number line to the unique point such that cos(x) = x, i.e. the unique fixed point of cos. If you want to learn more search "Banach's fixed point theorem". It is a general statement that maps which "squish" space everywhere converge to a unique fixed point. BFPT has very important applications to differential equations, topology, multivariable calculus and more. Good on you for discovering a special case of this phenomenon!
I'm 20 years old, in second year of mechanical engineering, and the tangent part caught me surprised. Can't wait to see the class on imaginary numbers; unfortunately, teachers in high school didn't told us and uni teachers take it for granted. Very good videos, hope they enlighten some of the younger minds
In highschool, I thought trig was triangles and waves. Now I think of the unit circle and theta, which can be applied to so many more real-world issues. Understanding the circle is a must
I can't stress this enough, at how amazing of a mentor you are! Though I missed the live stream (by three years haha) I found myself patiently and closely watching your 1 hour masterpiece! I'm currently in high school and will be graduating in 2 months. Though I don't always understand everything when watching the first time, it gradually becomes clearer and clearer the more time I ponder on it. One of my dreams are to actually meet you in real life ( I know it sounds corny) and at least give you one hug! I'm truly grateful of the work you do, and hope that you know it! Thanks Grant!
I've definitely seen that proof of Pythagoras before, but done without the trig functions, just using the similarity of the triangles formed by the altitude of a right triangle (which is, itself, easy to prove).
mr i hope that you create a spacial channel to explain this lessons because your way to exepressing of defferent concepts is obsolutly incridible thanks
Knowing how (cos(x))^2 is equal to 1/2(1+cos(2x)) made integrating trigonometric functions a lot easier for me. Admire the effort you do to present math in a new light.
I'll have to remember that. I've always been confused by being taught the order is sine-then-cosine, but then they correspond to distances on y-then-x axes.
9:00 The best way I find to think about is here is the following: "just like multiplying a function by two DOUBLES the height of the function, so too does multiplying the independent variable, x, prior to placing it into the function, DOUBLES the slope of y=x to y-2x, and hence DOUBLES the velocity crossing the interval comprising the period of the function. Since period is, informally, the length of the interval required for you to "start over" the repeated pattern a periodic function has, this means that the graph needs half as long of an interval to begin repeating with 2x as the input. Similarly, if you have x+1 (say) as your input, your input's y-intercept is now 1, so that 1 is placed into the function if you put 0 into the independent variable (that's the definition of y-intercept). In fact, the function receives now a value that is always 1 greater than you intended to put in when you set the variable x, the same as if your roommate moves the thermometer up one degree each night after you set it to your perfect temp. So, just as you would eventually figure out to "trick" your roommate by putting in one *less* degree than your perfect temp in anticipation of your roommate moving the thermometer one degree higher, so too must you shift the input variable one degree to the left on the number line to compensate for the x+1 adding 1 to the independent value you originally intended for the function. Thus, "counterintuitive" shifting right with x-1 and left with x+1 is in fact quite intuitive if you realize that the independent is messed with FIRST, BEFORE the base function is executed!
Hello 3b1b, I just wanna say that because I'm on the opposite side of the earth I can't tune in to watch the livestream, but I always watch the stream as a video later on. Keep making these, I love them a lot!
Consider a real function f(x) and imagine its graph in the plane. Then the graph of f(x+2) is simply the graph of f shifted to the left 2 units while the graph of f(x−2) is that of f shifted to the right 2 units. Also, the graph of f(2x) is the graph of f shrunk towards the y-axis by a factor of 1/2 and the graph of f(1/2x) is that of f stretched outwards from the y-axis by a factor of 2. These transformations operate in the opposite way than one would initially think in so much as that the transformations that operate on the whole function (such as f(x)+2 , 2f(x), and 1/2f(x)) act exactly as one would expect. How does one explain that transformation 'inside' a function opertate in the opposite direction than intuition suggest? I'm struggling to understand this 😖
Oh god that leaning tower example where you set up h=1 just blew up my mind. You are like that dude in Civilization when a Great Person gets born. That one educator brings the education level of the whole civilization to a new level.
Start of Lecture - 0:00 (Fixed to trimmed video)
Q1 Graph of (cos θ)² - 2:14
Q2 Translations of cos θ to (cos θ)² - 5:34
Q3 f(2x) = f(x)² - 10:54
Intro to Trig - 13:14
Q4 sin(3) & cos(3) - 16:44
SohCahToa - 20:44
Q5 Leaning Tower - 22:29
How to Compute Trig Functions? - 30:59
Q6 sin(π/6) - 33:34
Q7 cos(π/6) - 36:19
Q8 Trig of -θ - 43:34
Computing Trig Functions - 47:44
Q9 cos(π/12) - 0:49:54
Adv Trig Functions - 0:56:44
Q10 Graph of tan(θ) - 1:00:30
JSON comment - 1:02:24
“The most exciting part of the lecture” - Grant 1:05:34
if they copied in description then new UA-cam features will automatically make it as a marker of each segment in timeline ... thanks for this
I have made a timestamp too, but this is obviously better.
Idk whether it's only me, but the video only goes until 1:13:17
edit: it's not me, but 3b1b trimmed some part
@@That_One_Guy... he put away the biginning of the video, where nothing happens
@@That_One_Guy... I fixed it now so it should be correct
When people were voting on 4 options, the most voted bar should have been brown, so there were 3 blue and 1 brown bars.
savage
314 likes, nice
@@arturkarlgren6845 314 blaze it
YES
Genius.
I've only watched 38 min so far and I'm already thinking that this is exactly how we should be teaching students. Live or not, in class or online. You sir, are a good teacher. You should be like a consultant for some school board of education or something. What the world needs is more teachers who can teach like you.
I could've used Grant about this time in 1974! The nun I had for algebra II and trigonometry drove the class nuts. I haven't enjoyed math this much since Lancelot Hogben's "Mathematics for the Million". Now when I see a pi, I think of a circle.
69:nice
Grant, Khan, Bozeman Science and there are many more (I wish I could write all the names here), are fantastic teachers. I wish I had teachers like them back in school.
Amen. This guy is like a champion in shining armor!!!!!!!!!!!!!!!!!!
And to think of my maths teachers... 🤮
Incredible, thank you very much.
I'm a physics teacher, 45, and since 1990 I've never seen such an interesting and imaginative talk about trigonometry!
thats the power of 3Blue1Brown :)
3b1b: makes a video on high school math topic
Comment section: *grown adults with full time jobs*
This is so true!!
I am a 3rd year College student but I enjoyed this so much. Got a really different perspective on this topic.
Hi. I'm in 7th Grade. You?
Prolly ppl who want to see if it's rly them or the HS teachers who fucked up D:
@@xDomglmao I'm just here cuz I like listening to grant, and sometimes he hints at higher level topics (that I'm tempted to look up and self-study)... also it never hurts to have a refresher, especially in the styles he likes to teach in
Baobaomeow Oh boy, you have a fun journey ahead. don’t die!
I wish I was 15 again and could relearn everything with your instruction. These lectures are beautifully crafted, clear and easy to follow.
I'm 32, I'm doing just that. No need to go back in time.
Oh, I have the Same whish. I AM 15 and I want to learn,no I want to understand Everything about Math, from the Basics to........oh it Never ends.
It‘s easy to find the „how“ and to KNOW something but it‘s hard for me to find the „Why“ and to UNDERSTAND something.
Do you know a good Way to understand something in Math?
I Mean,I just hate to know something without understanding it.
For example : Why rotates e^(xi) at the Circle by exatly x?( Why gives me x the angle of the number (radian)?)
Why is n->endless (1+(pi*i/4n))^n = i^(1/2)?
Why (1+(1/n))^n ??
Why e???
@@lealebold1865 thats the beauty of us being young, we are less cynical and ask more questions, which can be a good thing
@@lealebold1865So am I. Though I was never brilliant in the subject, I'm willing to put in , all that it takes!
This man is literally the Bob Ross of math 🤣
We need to make this a tshirt!
Jose Orozco absolutely agree lol
"There are no mistakes, just happy little accidents " ✨😂
He is good.
@Honest no Tibees is the Ross of physics
Hey Grant. I very much enjoyed your streaming. As a math professor, trig is always one of the topics where most of the students struggle with the most especially for high school or even college students. What I focused on the most is how to visually make them understand especially the relationship among sin, cosin, and tangent as they can be used and manipulated to figure out csc or others. Your manim seems to be a great way to visually make students understand trig more than any other tools available now. Good job and I very much enjoyed!
I learned sine by means of a diagram with a large circle (radius 1 unit), inside of which were the right triangles for 10°, 20°, 30°, and so forth, all drawn with one leg horizontal. I believe that the sine was explained as the length of the vertical leg (negative if downward). It encouraged me to think of sine in terms of the vertical position of a point on a rotating wheel. I put that knowledge to use when animating clock faces: sine and cosine tell you where to draw the end of the hand.
@@robertlozyniak3661 Exactly! If you have a right idea of sine and cosine, everything else would come very easily. Hope I can make a video of trig some other time as well!
The tan part was extremely fascinating. Sad that they never teach such stuff at school. Would have had a much better understanding of trig if these things are taught.
Edit: Now also managed to figure out sec and cosec with how you found tan. At last the names make sense. You're a godsend Grant!
Imagine how interesting, and in some ways easier, it would be to learn mathematics the way they were discovered instead of in some random "easier to understand by politicians" order.
I agree with ya!
i was taught about tan(x) in a different way, using a straight line perpedicular to the x axis and tangent to the unit circle, projecting the radius on top of it. it's much more intuitive this way tho. i love those lectures!
DDC that should be how we learn tangent for the first time
@@9nikolai As a high school teacher... I agree.
Start (he appears on camera) 9:14
12:16 Housekeeping and preliminary question finished
14:00 Question 1
16:10 Answer reveal, 16:36 further explanation
17:40, Question 2
20:10 Answer reveal, 20:40 further explanation
22:02 Pen and paper, (cos(x))^2 = (1+cos(2x))/2
23:02 Question 3
23:48 (Accidental answer reveal)
25:26 "You think it's about triangles, but really it's about circles."
26:09 Sin(x) animation
27:30 Cos(x) animation
28:51 Question 4
30:45 Answer reveal
33:00 Pen and paper, Soh Cah Toa
34:45 Question 5
37:02 Answer reveal and explanation
39:51 Pen and paper, Unit circle
41:50 Radians and degrees connection
43:00 How do you compute these values?
44:12 Special right triangles
45:32 Question 6
46:15 3b1b answers, Do you use pen or pencil?
47:20 Q6 Reveal
48:30 Question 7
49:41 Answer reveal
51:14 Pythagorean theorem connection
54:50 New page, Question 8
57:25 Answer reveal
59:43 Back to cos^2 (x)
1:02:00 Question 9
1:04:25 Answer reveal
1:09:00 Where is tan(x)? Where is cos^2 (x)?
1:11:46 Tangent animation
1:12:30 Final Question
1:14:34 The 'hackerman comment'
1:15:16 Final question, answer reveal
1:16:41 Animation
1:17:45 Back to the tower problem
1:22:00 Fully labeled and explained triangle
1:23:54 Textbook formula connection
1:25:08 Goodbye, Patreon supporter screen
Fin
I (just like most viewers) know the topics he explains already, but I'm still learning new things; 1:17:45 was an interesting take, watch it even if you think you can't learn anything new!
Thanks! So sorry to do this, but I've trimmed the start of the video so that the intro screen and housekeeping isn't part of the final video, which sometimes takes YT a day to properly process.
Oh no everything is off now. This must've taken you a long time :(
Oh Grant! 😏
Luckily all timestamps are equally moved. Just figure out how much the first timestamp is off, then adjust every one accordingly.
I am in absolute awe of this lesson. It’s structured so perfectly and I enjoyed every moment of it. Thank you for dedicating yourself to raising the bar of math education. I have no doubt thousands of teachers for years to come will use this and your other lessons as a model to teach this material. And no, I don’t think I’m being hyperbolic here.
When you got to about the midpoint of the video and started to talk about the Pythagorean Identity of the Trig Functions, I would like to elaborate on this and add to it. We know that the Pythagorean Theorem is: A^2+ B^2 = C^2. Let's keep this in mind.
I will show and prove 4 - 5 different things that most math classes never fully express and these are the following:
* The Pythagorean Theorem and the Equation to a Circle are symbolically similar, also The equation to the Unit Circle centered at the origin is, in fact, the Pythagorean Theorem! It's just that one is in terms of right triangles and the other is in terms of a circle with a radius of 1.
* That there is a direct relationship of the Trigonomic Functions and linear equations in regards to their slopes.
* Without considering the use of limits and applying them, the Tangent Function is, in contrast, the definition of the slope function that is used to define a derivative within Calculus.
* That vector notation is symbolic of both linear and trigonometric calculations. For example ⟨a,b⟩=∥a∥∥b∥cosθ which states that the dot product between two vectors is the product of their magnitudes multiplied by the cosine of the angle between them. So if you know the lengths of two vectors you can find the angle! After you calculate the dot product, then you would have to take the arccos of that value, however, make sure you are using the right system (degrees, radians) to get the correct result you are looking for.
* All of this is rooted in the simplest of all mathematical expressions, not even an equation or a function, just a simple expression, being the very first one we are ever taught: (1+1) and that the operation of adding one to itself satisfies the construction of both the Unit Circle and defines the Pythagorean Theorem. Also, when we turn this into an equation (1+1) = 2 we will see that there is perfect symmetry, reflection, and rotation that is embedded within this.
The comparison of the Pythagorean Theorem and the Equation to a Circle:
The equation to an arbitrary circle is defined by (x - h)^2 + (y - k)^2 = r^2 where (h,k) is the 2D coordinate of the center of the circle.
Let's center this at the origin (0,0) and refer to the unit circle with a radius of 1. We now end up with: (x-0)^2 + (y-0)^2 + 1^2 = x^2 + y^2 = 1.
Let's compare this to the Pythagorean Theorem. x^2 + y^2 = 1 == A^2 + B^2 + C^2 when C = 1. So for any circle that is centered at the origin, it's equation is the Pythagorean Theorem.
The direct relationship of the Trigonometric Functions and Linear Equations:
We will use the slope-intercept form of a linear equation: y = mx+b. We know that m is defined by rise/run or (y2 - y1)/(x2 - x1) where can use the coordinate points to find the slope. The value of this slope is a proportion of how much change in height over the change in the horizontal. All angles will be relative to the line y = mx+b and the x-axis. In other words, it is the angle that is above the X-axis or the Horizontal axis up to the line itself. Here I mentioned the rate of change. We can take rise/run or (y2-y1)/(x2-x1) and rewrite this as dy/dx. We know that we can make a right triangle from the x-axis up to the line in question. By doing so we can see that the dy is also sin(t) and dx is cos(t). We can see that the slope of the line m is also the tangent of the angle above the horizon to that line. So we can rewrite the slope-intercept form y = mx+b to y = tan(t)x + b or sin(t)/cos(t)x + b. This will lead us to the next section about derivatives!
The Tangent Function and the Derivative: We know that when we have a curve that its slope is not constant. We can take two points on that curve that are relatively close and get a good approximation of its slope near that point, however, the farther the gap the more margin of error this is. This approach is what is referred to as finding the secant slope. If we make smaller and smaller incremental steps where we get closer and closer to that point where the limit of the size of that step approaches 0, we then end up with a line that has the slope of that point which is tangent to that curve at that point. This is seen in the difference quotient in Calculus to find a derivative. f'(x) = lim dx->0 (f(x+dx) - f(x))/dx Which is basically taking the slope form of a line (y2 - y1/(x2-x1) and rewriting it from point notation into function notation with respect to x, changing the difference in points to rate of change by using delta x and delta y, which is still all algebra, and the only part that is Calculus is when you actually apply the limit! So in a sense, we can see a direct relationship and similarity of (f(x+dx) - f(x)) / dx and tan(t). If we substitute tan(t) as sin(t)/cos(t). We can see that cos(t) = dx which is straight forward. However, if we look at f(x+dx) - f(x) it isn't quite obvious but this would be equivalent to sin(t). Where t is the angle above the x-axis and up to the point on the curve whose slope is the tangent of that angle produced by that linear equation.
The vector portion should be self-explanatory as I had already mentioned the most important piece and that is the dot product in relation to the cosine of the angle between those two vectors. Not much more needs to be expressed about this here, but will be referred in the last section.
(...continued in a reply to this post)
(...continued)
How (1+1) ties all of this together! Consider the value 1 to be an arbitrary unit length, it could be 1 inch, 1 mile, 1 cm, it doesn't matter! Now, take a piece of square paper and mark a point near the center of the paper, then draw a line from there in any direction and stop a little less than 1/2 across the paper and make another point where you stopped. Now go back to the center and mark this as 0. Then mark the endpoint as 1 and make a slight arrowhead at that point. Direction here is completely arbitrary and agnostic to the length or magnitude of that line segment or vector that you had just drawn. Now this will be a unit vector. The starting point labeled 0 will be the 0 vector, this will be important later on. Now rotate this paper until the line segment is parallel with your body and that the 1 is to the right side. What we are going to do next is to apply the operation of addition to this unit vector with itself. By convention addition in the positive sense will be towards the right. So what you now need to do is visualize that entire line from 0 to 1 sliding across itself so that the 0 point is now at 1 and the new endpoint of 1 is near the edge of the paper you can mark this with a 2 and put another arrowhead there. By drawing these two vectors you have physically done (1+1) = 2. Okay, so we saw this being done through vectors. I had you do this to express the importance that Addition is a Linear Transformation. It is Translation to be exact. Now without knowing it, by performing that addition you also simultaneously performed multiplication. You see Addition is a 1 Dimensional Operation, but it introduces us into 2 Dimensional Space. It doesn't seem apparent yet, but I will get to that! We know that (1+1) = 2 and we know that 1*2 = 2. So we can easily see that (1+1) = 1*2 = 2*1.
Multiplication is embedded within addition, however, multiplication can be both a 1 and 2 Dimensional Transformations. A 1D Transformation of multiplication is scaling or shearing in a specific direction and a 2D transformation is a Rotation or defining Area. Now, how does this relate to the Unit Circle, Triangles, Trigonometry, The Pythagorean Theorem, and even Calculus? It's quite simple... let's take a step back and remember when I said to draw a line in any arbitrary direction... then I had you rotate that paper to position the line to be parallel with your body and the line pointing to the right. Now, we denoted this as being +1 which could be symbolic with the direction East. Nothing stopped us from being able to go West by 1 unit. So now let's make another unit vector that goes from the origin 0, and mark this Western or Left point with a 1. Now, when it comes to vectors, the magnitudes of the first vector, and this new one are the same, but their directions are Opposite. So we can use the negative sign to represent the opposite direction. So we can now add a - to this 1. If we add these two values (1) + (-1) we end up with 0 which leads us back to the starting point or the 0 vector. However, their combined value |1| + |-1| = 2. These three values of 0, 1 and 2 are all related to each other. Now we said that addition is a 1D transformation translation to be exact. We also said that 1+1 =2 is also multiplication. Well we have a line and there is no area, so the 2D transformation here must be rotation. Let's take the first line 0 to 1 and rotate that about the 0 point and see what happens when we end up at -1. We did a 1/2 rotation of a full circle.
We know that the arc length is PI and that the central angle is 180 degrees. We also know that a line has 180 degrees of rotation which is also the addition of all 3 interior angles of any triangle. We know that the unit circle has a radius of 1 and a diameter of 2. Its circumference is 2PI and its area is PI unit^2. Yes, even the value PI is embedded in the expression (1+1). This expression has a perfect symmetry and complete rotation embedded within it. It also has perfect reflection too. And if we understand what Derivatives and Integrals are, we also know that (1+1) doesn't just introduce multiplication but it also introduces powers because (1+1) = 2^1 which is a 3D transformation provided there are three components to the vector. For each component of a vector , , , etc... the number of elements or the size of that vector is the dimensional space we are working in. Each dimensional space has specific attributes or properties. 0D = an arbitrary point. 1D = length, magnitude, line. 2D = rotation, area. 3D = multiple rotations and volume. etc... We also know that when we look at the polynomials in algebra: f(x) = x, f(x) = x^2, f(x) = x^3 that x is just x which is a scalar quantity and is linear and that x^2 represents the area of all squares, and that x^3 represents the volume of all cubes provided x is +, there is a case where x can be - in to define a - volume, but this involves complex numbers and is completely abstract! We also know that x, x^2, and x^3 are partially derivatives and integrals to each other so to speak. F(x) = xdx = 1/2x^2 + c. which states that the integral of f(x) = x is the family of functions f(x) 1/2x^2 + c. and so on, and the reverse is true too. All of this is embedded in (1+1) = 2. All of your irrational numbers are embedded within it. Your logarithmic and exponential functions are embedded within it! Everything is integrated or derived from (1+1)!
@@skilz8098 im sorry, but probably no one read that fully
@@Kr3nkt Maybe, maybe not, but that's not the point or the reason I put it out there. It's there for any who would like to. No one has to and I'm not expecting anyone to. I just like sharing what I know and understand. It's not my problem that others may be too lazy to read it. Yet there are 29 likes on the first part, and an additional 10 on the second part. So it does appear a few have read it after all!
you deserve more likes
@@Kr3nkt You may now, though there are many of us who find this very useful and will read all of it.
Compare this with my University lectures and there is no comparison. We need to have this type of fun in the University and have the time to ponder and think instead of rushing at the speed of light and cramming for a test.
I agree with ya!
*Cries in dropping out of university because of light speed lectures and coming from an highschool perspective, which is light years away from what university is*
DDC True~ I see that
@@drpkmath12345 I dropped out twice. Now I'm back for a 3rd time. I'm 23 and finally mature enough to focus and pass my classes.
Nathanael Kuechenberg Age doesnt matter but I am truly glad to see how motivated you are to pass the classes. Thats what matters the most and you seem to be determined. Let me know if you need any help with math!
Grant, thank you for the BEAUTIFUL knowledge you said in the live stream. I’ve never been so intrigued by trigonometry until I saw this. Never really understood how trig worked in circles, the relationship between Pythagora’s theorem and Trig identities. This video has surpassed any trig class I have taken in my life. Thank you 😁
Knowing that sin and cos relate to the coordinates of a point in the unit circle has unlocked a subject that was opaque to me for so long.
Facts like sohcahtoa and the rotation matrix make sense now. It's especially satisfying that I can now combine what I learned from you about linear algebra with this to have a solid understanding of what the 2D rotation matrix does.
When he explained the tangent part, everything came together. A whole new perspective.
Welcome out from flatland. ;)
This series is a gem. I really hope you can continue this sort of livestream after the covid 19 lockdown has passed, because it is truly a wonderful experience.
After the last one, I went and watched the entire imaginary numbers series by Welch Labs that Grant recommended, and seriously had my mind blown. Highly recommend
Same.
Same . Mind blown , it is super concise and tightly packed .
Same story here. I thought I understood complex numbers quite well after about 5 years of working with them. I was wrong.
Definitely. One of the best playlists out there
LOVE relating the trig functions to circles. I realised that myself about 3 years ago and have been teaching it that way ever since. I approach it as a rotating radius around the circle and that sine and cosine tell you the ratio of how much of that radius vector is vertical or horizontal. It helps build that intuition for where sine and cosine are -1, 0, or 1... and also what sign to expect the value to be in which quadrant. I then extend it to tangent as being the slope/gradient of that rotating radius vector which explains why the rotation by pi has the same tangent value, because it's part of the same line that goes through the circle's centre. :)
My high school teacher taught trigonometry from circles, and I am so thankful. It is so much clearer. Then again, he was pretty overqualified for that job and I believe he may have also taught at universities.
the way you explain things so good, always coming to that mind blowing connection/context at the end and missing out difficult things till they get necessary and mixing fun and good jokes in-between reminds me of Gilbert Strang. The only thing you updated is instead of using a board and chalk you use cool new tools that really help to visualize something.
thank you a lot, that really helped me.
When Grant pulled out the second printed sheet I imagined him stockpiling trigonometric circles at night next to his desk to make sure he has enough for his live
Great video.
He did say "infinite supply"
Did you fall in love with Grant?
I'm doing a mega math review because I want to undergraduate in math, and this is an absolutely amazing second introduction to trig. Absolutely beautiful.
I am finishing my bachelor's in mathematics and this is like the first time I ever really understand the stuff ...
Seeing how excited you are about all kinds of Mathematics and how you're able to get other people excited about it is very satisfying and beautiful.
21:00 my math teacher in Israel had a way dumber (and easier) way to remember (though it relies a bit on the different pronunciation of the function names):
Sin -> seeing (so its the opposite one that the angle "sees")
Cos -> cousin (so it's adjacent)
Tan -> tango (so you invite someone you see to dance closer to you).
Nothing beats that intuition for me
I would use this to teach my daughter
lol
I have a method to correlate trigonometry functions with the angles too! Here’s the way to memorise it:
Cos: The arc of the angle we usually draw to indicate theta
Tan: The capital T has a right angle
Sin: The remaining angle, or I remember it by s for the (usually) smallest angle
Although it’s very intuitive, this method doesn’t tell you what sides are on the numerator and the denominator. So this is a merit of the Soh Cah Toa method. I still prefer the method I mentioned since I work better with shapes. Hope this helps!
I missed initial lectures of Trigonometry in my 9th standard. And then it was roller coaster ride till I finished my engineering.
Once more thing I would like to add is students seem to struggle understanding the concept of distance when we deal with trig, and your manim also seems to be a great tool to help them understand those concepts. Once again, great job, and I very much enjoyed.
About 33:40 - I know everyone now swears by SohCahToa but I've remembered for nearly 50 years what my father told me, and he must have learned it in the 1930s. It uses the same diagram as yours, but has a Base instead of an Adjacent and a Perpendicular instead of an Opposite. I found it mildly amusing, which helped me remember it: Some People Have Curly Brown Hair Till Painted Black.
noice
I love your videos, your music, and ways of breaking things down to the smallest possible explanation in math. Most times, because the way we learn maths (or related subjects) is to ace an exam rather than understand what we're doing both visually and numerically, we learn the different aspects separately as opposed to relating them this way, right up till university. Thank you and your team so much, for putting up content worth while on youtube!
my 8 year old son is your fan, I suspicious whether he really understands, but seems he get the most multiple choice correct.
He must be a genius
I have a sister who is 9 and is yet to learn division ._.
And also hearing about your son gives me Gaussian vibes.
he do 😼
there's an animation bug in your itempool thing: it's revealing the answer to the next question. eg watch @45:35 at slow speed: the green bar indicates 'B', the correct answer to the tower question, but then it switches to the next question, moves the green bar to the correct answer and _then_ flips the card to a version with the answer hidden.
this
That's now at 36:15
that's the previous question
Yeah, this happens several times (also: 49:50). Upvoting and commenting in hopes to drive this comment up into 3b1b's radar! (And/or the radar of the folks working with him on that stuff.)
iirc people realized that in stream 3 or 4 and used that xD
I've learned more watching this than I have in all my other math classes! I might have made it through Calc 2 if I'd understood trig at this depth! Thanks!
My trig intuition went from triangles to circles pretty quick once I started playing around with 3d modelling software and needed to get radial symmetry in my vertices.
I'm honestly so glad I watched this, even though I am pretty far beyond Trig. I skipped Pre-Calculus in High-School and I've always just implicitly trusted the identities without ever thinking about the underlying geometry.
damn i miss these lockdown math videoes. i learned so much by them and it was so nice to see it live and paticipate. wish some youtubers would do this more :) and in such a nice pace as you did so i could write it all down too!
Back in high school, I used a simple table to remember the sin/cos of the angles 0, 30, 45, 60 and 90 degrees: "sin (Nth friendly angle) = sqrt (N)/2, for N = 0 .. 4; cos() goes in the opposite direction". With the friendly angles being 0, 30, 45, 60 and 90 degrees. Using reflection and remembering sin() is non-negative for angles 0 .. 180, and cos() is non-negative for angles -90 .. 90 gives the values in the other 3 quadrants.
Timestamps:
9:15 Grant comes to the stream
12:17 What will we talk about today
14:36 Graph of cos(x)²
17:59 equation for cos(x)²
19:40 the new music
25:59 trigonometry basis
34:49 leaning tower of pisa
39:45 the functions on the unit-circle
43:04 computing special values
55:00 negative angles
56:10 reality breaks
59:30 using cos²(x) for computing values
1:03:31 reality breaks (again)
1:09:03 where is tan(θ) on the unit-circle?
1:16:20 where is cos²(θ) on the unit-circle?
You really care about people who want to learn something and give your best. I am not going to talk about your teaching style and animations, which are world-class. No words are enough for them in my opinion and there are thousands of us appreciating these. It is really heartwarming for me to see that you have a pre-printed unit circle for us (and make small jokes about it). I have worse handwriting than you and have had even worse writing professors than me (yours is way better than many people I've worked with). Simple details like this unit circle, your care for us are carrying your video quality even higher. Kind regards and stay healthy :)
No wonder every time I've gone to the store they're out of unit circles. Grant's been hoarding them!
I love it that the bookshelf in the back contains what is in all probability the three-volume Calvin & Hobbes collection. Unless the three hefty brown spines are just eerily resembling said work
This is absolutely incredible. I love your approach to math, Grant.
Best part of watching this video is that this guy seems like a regular human. His animation-only videos are just so elegantly done you kind of wondered who (or what) was making them. 🙂
44:54 Ramanujan's Constant - Reaction
I've seen many people in the comments who are already in designated positions like engineering but even they say that some of the things Grant says does widen their eyes sometimes. So, for people like me who just saw trigonometry for the first time in their textbook on January 1st, 2020, I can't thank you enough Grant for making these videos at a fundamental level. It helps me more more than I could imagine. Thanks a lot, Grant. Love these streams
"Instead they want some sort of natural units, something where you imagine if you talk to an alien civilisation about 'math' they would have the same convention."
*stares in tau*
Let's suppose someone hands you a cylinder and asks you to give it's circumference. The thing you can measure (say with a caliper) is the diameter. So the circumference would be pi*diameter, what you could easily measure. Obviously you can calculate it by tau*radius, but then you have to change the diameter, one more mathematical operation, one more chance for a typo or another type of problem. So for something that promises simplification, tau does a really bad job at that. Or let's say you have the radius because it was given for you, but you need the area. you could just square it and multiply by pi, but no, we need the extra operation because tau.
Moral of the story is, people like to argue over numb nothings lol.
Actually, pi/2 is more generalizable to higher dimensions, so it is more natural. Tau is a nice circle constant, but only in 2-D (and not actually much nicer than pi). So there is a good chance aliens would use 1.57 instead of 3.14 or 6.28.
I do not like tau because it is already used as a measure of period, torques, dummy variables for time, etc.
Atleast when you use pi it only means 3.141... except in advanced number theory.
@@hassanakhtar7874 And also in economics for profit.
The important part is the value, not the name, since we are discussing which value aliens would use.
I imagine aliens would go with the simpler of the two numbers for ease of operation, like when simplifying fractions. Tau is just twice Pi, so Pi is the smaller number. Smaller is simpler in math terms. Simpler math = better math
Pi > Tau...in a manner of speaking😎
I still think that thinking of tan as "slope" is generally the best approach. This ties really well with sine = height, cos = width ie horizontal distance (with negatives for both)
If you think about tan as the magical function that converts angles to slopes, it makes a lot more sense. Why is it undefined at regular intervals? Because that's when the slope is pointing straight up! Why is tan's period twice that of sin and cos? Because 45° and 225° technically point at the same slope! Why does arctan have asymptotes? Because increasing larger slopes approach but do not reach 90°!
When he pulled out the compass, I had nightmare flashbacks to Euclid Elements. It’s VERY old school geometry where you can only use a compass and straight edge, but NOT a ruler.
I bet you know the size of all your fingers
It is still taught in our school.
In Geometry we still learned how to construct a 30 degree angle, 45 degree angle, perpendicular bisector, etc.
All the trig identities are what really killed me in 2nd semester calculus. I had no problems with the concepts of integrating, but remembering AND figuring out which identities I needed to actually solve an integration problem was torture. Learning via this approach in trig class would have made things so much better.
Lmao the JSON comment at 1:14:37
Can you explain what he meant?
@@bananastuff2840 JSON is a way of storing data so that it's easy to send to the Q&A program from the browsers. The live Q&A feature was apparently thrown together in a quick time. So apparently didn't quite take into account users "having fun" and sending in different answers than expected. That's why there's the '@'s.
At least one person sent in data which had a comment in the data, which avoided the livestream slow mode chat. (Trimmed timestamp now at 1:01:21 or so).
They taught us trigonometry only trough triangles and only barely mentioned circles once, I didn’t even know about the whole pi thing until now, thank you
1:11:45 Wow, that's a great picture to keep in mind. Since the legs of the larger triangle are sec and csc we get a funny trig ID:
sec^2 + csc^2 = (tan + cot)^2
@wise ol' man b-but Algebra ≅Geometry!
im in integral calculus and i still struggle conceptually with trig sometimes. thankyou so much, i actually love starting from ground zero regardless of how much i know already because it allows you to feel confident you haven't overlooked anything
You should probably change the code for the pop-up transition cards of your questions because it reveals the correct answer card before the question card segways into place.
So hard not to look at it once you notice it...
I am a final year Physics undergrad, and after watching your video (literally on any topic), I feel like we were living in darkness and this man showed us light.
At first I was like, trig fundamentals, what could be there that I don't know and when he said imagine you know nothing about trig, that thing came out to be pretty true after watching video.
tanget: just draw vertical line straight via rightmost point of the unit circle, and height of the point that laying on intersection of hypothenuse and the vertical line is tangent.
r75shell yes right. Good point!
This was how I was taught it. I think it's better because the measure is on a line that doesn't move.
Makes sense. However I feel it might get more confusing once you get to angles outside of the first quadrant of the circle.
Honestly, being a physics student, SOHCAHTOA, quadratic formula, and Pythagorean theorem have been super useful for me quite consistently and I use them all the time. Never thought I'd use those things I learned so many years ago still!
Sir you defined tan∅ as the distance between tangent point and intersection of tangent with X axis.
But how do we explain negative values of tan∅ in that case? We we simply marked it negative on left side of graph when it's a distance.
It was understandable for sin and cos because even though you started then as distance quantities, you quickly took it as coordinates represented on different quadrants. But we can't do that thing with tan∅ right? So we just define it that if it's on the left side, we will use negative sign?
Also, I got very confused in the beginning when you used sin(x)² and (sin(x))² interchangeably.
After 8 years I finally understood why we shifted to representing angles in radians from degrees. Thank you so much ♥️
P.S. Loved this class and all other videos. Keep doing the great work!
tanθ is also the slope of the radius from (0, 0) to (cosθ, sinθ); its sinθ/cosθ, the rise/run
It is refreshing to learn trigonometry each time , i think visualizations are quiet elegant to learn in 3D coordinates if we practice more out of these amazing lectures.Also adding application oriented problem is creative way to learn.
Grant looks like a young Conan O’Brien
Mixed with Matthew mcconaughey
He's definitely Irish, I believe he said as much in the Ben Ben Blue podcast a couple years ago.
Grant looks like if Conan O'Brien went to Harvard....oh wait
You all are wrong, Grant looks like Eddie Redmayne. 😄
More like a young Norm Macdonald
Like most people I'm sure, I watch you video just for fun, that's how you know you are doing something right...you are a blessing to youtube
Perfect way to refresh my memory on trig concepts before moving on to calculus.
This channel is amazing for providing this level of instruction for free.
Just one livestreaming detail - your audio is consistently ahead of your video (maybe between 50 and 100ms?), so you might consider measuring this and delaying your audio to sync with the video.
Yeah, I was going to make a comment about the syncing problem in editing, but then I remembered this was a livestream, and that made me unsure of what to even suggest... But yeah, this might be the thing.
casual math talk, gaining interest and insight into various topics. love it
Amazing stream! Will watch all of them! NOTE: There's a bug with the software (pointed out by many I'm sure) that reveals the answer of the next question as the flip animation runs.
Yeah, there's are several of these. In the list of comments I'm seeing, yours is sandwiched between two of them. ;) (But only yours, among those 3, uses the word "reveal", which was what I looked for to find them...)
Example: 49:50... though it's more telling at an earlier one... too lazy to dig back up the timestamp. :)
What a delight... I am a graduate student in engineering who has worked with trog functions for years now. There were still some insights in the video that were entirely new to me! Was a treat. Thanks 👍
Please do an essence on trigonometry, show us more intrinsic beauty of it!!!
So, about halfway through you mentioned that f2(x) usually means do the function twice, and that trigonometry ignores that. So I pulled up desmos and did cos(cos(x)) and it looks a little flatter. Then I did cos(cos(cos(cos(cos(x))))) and it's even flatter. In fact it seems to be tending toward the value 1/root(2). Why is that? And does it have something to do with the rms value of a Cosine function with an amplitude of a half being 1/root(2)?
1/sqrt(2)~0.71 is close (but not equal to) the fixed point of cosine (solution to cos(x)=x) which is ~.73. what's happening is that cos carries [0,2pi] (or even all of R) to [0,1], so it's "squishing" the interval. Applying cos over and over will continue to squish the number line to the unique point such that cos(x) = x, i.e. the unique fixed point of cos.
If you want to learn more search "Banach's fixed point theorem". It is a general statement that maps which "squish" space everywhere converge to a unique fixed point. BFPT has very important applications to differential equations, topology, multivariable calculus and more. Good on you for discovering a special case of this phenomenon!
I'm 20 years old, in second year of mechanical engineering, and the tangent part caught me surprised. Can't wait to see the class on imaginary numbers; unfortunately, teachers in high school didn't told us and uni teachers take it for granted. Very good videos, hope they enlighten some of the younger minds
23:00 also the towers can "learn" 🙈
I really love the encouragement to pause & ponder, It really did give me a better grasp over the concepts
I am not here for course, but to appreciate your hard work. Your channel is one of the few that provide quality knowledge to the viewers for free.
Gave a new perspective of thinking about trigonometry.
please do continue this series.
In highschool, I thought trig was triangles and waves. Now I think of the unit circle and theta, which can be applied to so many more real-world issues. Understanding the circle is a must
I love how you don't just teach us math facts, you build it in front of us and make us help
Thank you so much for the stream, it really helped to think deeper. Hope, this type of videos won't disappear!
so much appreciation for u, your videos make quarantine a little less bland
The treble levels of your mic are kinda low so it's hard to hear you articulate words.
i suspect its your hearing.thats the problem
Man, we can never thank you enough. please never stop this
I can't stress this enough, at how amazing of a mentor you are! Though I missed the live stream (by three years haha) I found myself patiently and closely watching your 1 hour masterpiece! I'm currently in high school and will be graduating in 2 months. Though I don't always understand everything when watching the first time, it gradually becomes clearer and clearer the more time I ponder on it. One of my dreams are to actually meet you in real life ( I know it sounds corny) and at least give you one hug! I'm truly grateful of the work you do, and hope that you know it! Thanks Grant!
It's 2:24am...I'm trying the last proof myself and it's hurting my brain. It seems I'm going round and round.
I've definitely seen that proof of Pythagoras before, but done without the trig functions, just using the similarity of the triangles formed by the altitude of a right triangle (which is, itself, easy to prove).
52:47 grant looks genuinely impressed at first, then surprised, then confused lol
mr i hope that you create a spacial channel to explain this lessons because your way to exepressing of defferent concepts is obsolutly incridible thanks
if it could just be louder it's amazing but a little bit more of volume would have been good
44:33 Reject the false quachotomy, embrace a fifth position.
That is a sentence I never knew brain brain already understood. Well done.
@@okuno54 I try
Knowing how (cos(x))^2 is equal to 1/2(1+cos(2x)) made integrating trigonometric functions a lot easier for me. Admire the effort you do to present math in a new light.
My trig prof is so dry and confusing that this video is basically saving my grade.
Big thank you
same tho
Hihi.
In french, SohCahToa is rather Cah Soh Toa, which sounds like "Casse-toi" which means "Go away !"
Very French indeed :) In Quebec we learned SohCahToa, maybe because we don't use Casse-toi as an expression
I'll have to remember that. I've always been confused by being taught the order is sine-then-cosine, but then they correspond to distances on y-then-x axes.
Wow, I’m so glad I got over my trig trauma.
I love math! *π* ❤️
"This is the one thing you should stick around for" he says 1 hr and 17 mins in
And he's right, you should
but the video is 1h13m
@@minewarz Video length was changed to remove the first 10 minutes or so
9:00 The best way I find to think about is here is the following: "just like multiplying a function by two DOUBLES the height of the function, so too does multiplying the independent variable, x, prior to placing it into the function, DOUBLES the slope of y=x to y-2x, and hence DOUBLES the velocity crossing the interval comprising the period of the function. Since period is, informally, the length of the interval required for you to "start over" the repeated pattern a periodic function has, this means that the graph needs half as long of an interval to begin repeating with 2x as the input.
Similarly, if you have x+1 (say) as your input, your input's y-intercept is now 1, so that 1 is placed into the function if you put 0 into the independent variable (that's the definition of y-intercept). In fact, the function receives now a value that is always 1 greater than you intended to put in when you set the variable x, the same as if your roommate moves the thermometer up one degree each night after you set it to your perfect temp. So, just as you would eventually figure out to "trick" your roommate by putting in one *less* degree than your perfect temp in anticipation of your roommate moving the thermometer one degree higher, so too must you shift the input variable one degree to the left on the number line to compensate for the x+1 adding 1 to the independent value you originally intended for the function. Thus, "counterintuitive" shifting right with x-1 and left with x+1 is in fact quite intuitive if you realize that the independent is messed with FIRST, BEFORE the base function is executed!
I don't like recaps and lives. BUT this high quality one really took my eye! fantastic!
Hello 3b1b, I just wanna say that because I'm on the opposite side of the earth I can't tune in to watch the livestream, but I always watch the stream as a video later on. Keep making these, I love them a lot!
I cant find any word to thank,you are literally the best math you tube
Consider a real function f(x) and imagine its graph in the plane. Then the graph of f(x+2) is simply the graph of f shifted to the left 2 units while the graph of f(x−2) is that of f shifted to the right 2 units. Also, the graph of f(2x) is the graph of f shrunk towards the y-axis by a factor of 1/2 and the graph of f(1/2x) is that of f stretched outwards from the y-axis by a factor of 2. These transformations operate in the opposite way than one would initially think in so much as that the transformations that operate on the whole function (such as f(x)+2 , 2f(x), and 1/2f(x)) act exactly as one would expect. How does one explain that transformation 'inside' a function opertate in the opposite direction than intuition suggest? I'm struggling to understand this 😖
Oh god that leaning tower example where you set up h=1 just blew up my mind. You are like that dude in Civilization when a Great Person gets born. That one educator brings the education level of the whole civilization to a new level.
The problem solving music sounds like Minecraft Music.
this is really great for developing an intuitive sense and familiarity with the unit circle and trig functions. an amazing talent.
THANK YOU VERY MUCH, Grant Sanderson. You made my understanding of trig into a whole new level.