Trigonometry fundamentals | Ep. 2 Lockdown live math

[@geoc2140]

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

[@noonesperfect]

if they copied in description then new KZbin features will automatically make it as a marker of each segment in timeline ... thanks for this

[@blauesserpiroyal2887]

I have made a timestamp too, but this is obviously better.

[@That_One_Guy...]

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

[@blauesserpiroyal2887]

@@That_One_Guy... he put away the biginning of the video, where nothing happens

[@geoc2140]

@@That_One_Guy... I fixed it now so it should be correct

[@goofyrice]

When people were voting on 4 options, the most voted bar should have been brown, so there were 3 blue and 1 brown bars.

[@alinanto7013]

savage

[@arturkarlgren6845]

314 likes, nice

[@sofia.eris.bauhaus]

@@arturkarlgren6845 314 blaze it

[@sermuns]

YES

[@thasyashetty3797]

Genius.

[@AalapShah12297]

It's amazing how this livestream has better editing than so many non-live videos. Live quizzes, Q&A, seamless switching between so many screens - all after already delivering top notch content.

[@thecuriousgorilla6005]

Yeah it feels almost non-live

[@donegal79]

yes its great...but remember he has a team around him....at least 4 others.......

[@Zhung36]

@@donegal79 so...........?

[@HansFlamme]

My profs just stream on zoom with shitty audio lmao

[@armorsmith43]

Zhung36 so if you aspire to make videos with such quality, you should be aware you will need to collaborate with others.

[@anteconfig5391]

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.

[@BuddyNovinski]

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.

[@kbm-iy9tj]

69:nice

[@pseudorealityisreal]

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.

[@panchovilla5400]

Amen. This guy is like a champion in shining armor!!!!!!!!!!!!!!!!!!

[@realcomments]

And to think of my maths teachers... 🤮

[@xristiano_talimpan]

KZbin's new feauture allows for timestams to be shown in the video progress bar, so a nice idea would be to add timestamps in the description so we wouldn't have to scroll.

[@JayLooney]

Countour-Integral Incredible, I didn’t realize KZbin had this update, but you’re right it’s really nice for informational stuff like this, especially for getting back to your place if you took a break for a while.

[@randyollmann4202]

I like the fifth letter of this guy's name

[@artvandalay5804]

3b1b: makes a video on high school math topic Comment section: *grown adults with full time jobs*

[@unknownChungus]

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.

[@baobaomeow]

Hi. I'm in 7th Grade. You?

[@xDomglmao]

Prolly ppl who want to see if it's rly them or the HS teachers who fucked up D:

[@firebrain2991]

@@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

[@namavoid3266]

Baobaomeow Oh boy, you have a fun journey ahead. don’t die!

[@p0kenon]

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

[@bee_irl]

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!

[@3blue1brown]

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.

[@Maazin5]

Oh no everything is off now. This must've taken you a long time :(

[@KodamatiUdayaPrabhuKiran]

Oh Grant! 😏

[@EirikAnd99]

Luckily all timestamps are equally moved. Just figure out how much the first timestamp is off, then adjust every one accordingly.

[@jlpsinde]

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!

[@squibble311]

thats the power of 3Blue1Brown :)

[@MM6700-o4w]

Ok

[@ri-lj5ev]

This man is literally the Bob Ross of math 🤣

[@joseorozco8164]

We need to make this a tshirt!

[@ri-lj5ev]

Jose Orozco absolutely agree lol

[@Utopianwinds]

"There are no mistakes, just happy little accidents " ✨😂

[@TaylorGolmon]

He is good.

[@staticcactus6029]

@Honest no Tibees is the Ross of physics

[@HexerPsy]

Can i just say - im approaching 30, I havent touched these formulas for about 12 years. Much of it I have remembered, much of it is a new approach for me. The teaching method, the music, the setting, the ease to following this.... OUTSTANDING CONTENT :D

[@Ikantspell4]

I wish I was 15 again and could relearn everything with your instruction. These lectures are beautifully crafted, clear and easy to follow.

[@emilysha418]

I'm 32, I'm doing just that. No need to go back in time.

[@lealebold1865]

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???

[@hoogreen]

@@lealebold1865 thats the beauty of us being young, we are less cynical and ask more questions, which can be a good thing

[@AhilyaKash]

​@@lealebold1865So am I. Though I was never brilliant in the subject, I'm willing to put in , all that it takes!

[@sidarth632]

@@hoogreen Absolutely.I am pretty cynical despite being only 14 but I am extremely curious and extremely gratefull for ppl like 3b1b for showing these concepts in an intuitive way.

[@skilz8098]

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)

[@skilz8098]

(...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]

@@skilz8098 im sorry, but probably no one read that fully

[@skilz8098]

@@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!

[@heleme4335]

you deserve more likes

[@mrmunchkin2181]

@@Kr3nkt You may now, though there are many of us who find this very useful and will read all of it.

[@sachingiyer]

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!

[@9nikola]

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.

[@drpkmath12345]

I agree with ya!

[@ddcddc_]

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!

[@drpkmath12345]

DDC that should be how we learn tangent for the first time

[@MrMusicM67]

​@@9nikola​ As a high school teacher... I agree.

[@agg42]

13:15 I'm a machinist, I use trig all the time to figure out lengths of angles or angles from lengths, so it's always been a triangle thing for me. Now you've shown me how it's really about circles! Thank you 3Blue1Brown! I wish google's search algorithm puts you on top.

[@shub]

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.

[@NathanaelKuechenberg]

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.

[@drpkmath12345]

I agree with ya!

[@ddcddc_]

*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*

[@drpkmath12345]

DDC True~ I see that

[@NathanaelKuechenberg]

@@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.

[@drpkmath12345]

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!

[@drpkmath12345]

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!

[@robertlozyniak3661]

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.

[@drpkmath12345]

@@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!

[@basantachaulagain6104]

When he explained the tangent part, everything came together. A whole new perspective.

[@DavidLindes]

Welcome out from flatland. ;)

[@deltanebula8622]

Your content is SUCH HIGH QUALITY. Those transitions and seamless movements from incredible animations to quiz questions to paper all while maintaining the video feed just makes these lessons such a joy to watch

[@massimila5303]

I totally agree. He is simply amazing!

[@adrianfernandez4305]

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 😁

[@blauesserpiroyal2887]

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?

[@expropogul1719]

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.

[@capilover1023]

Grant: ok so there're 4 possible answers for the question so choose wis... People who broke down the poll: *REALITY CAN BE WHATEVER I WANT*

[@jackalopegaming4948]

To quote Adam Savage: "I reject your reality and substitute my own."

[@jingermcblabbersnitch7162]

To be fair, reality is often disappointing

[@skpcboy]

"i go by my own conventions"

[@heartles_xyz]

Grant: "Don't @ me"

[@tylern6060]

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.

[@ack_]

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.

[@leonerduk]

He did say "infinite supply"

[@howardOKC]

Did you fall in love with Grant?

[@odedsayar4345]

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

[@howardOKC]

I would use this to teach my daughter

[@squibble311]

lol

[@ant7723]

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!

[@meowl2396]

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.

[@FB-no4lr]

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. :)

[@rageprod]

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.

[@randomNunber]

Oh my god, the last cos2theta pythagorian interpretation totally blow my mind off, you are great man.

[@dmsalomon]

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.

[@danielzastrau4571]

I am finishing my bachelor's in mathematics and this is like the first time I ever really understand the stuff ...

[@RePuLseHQKing]

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.

[@marcinzych6011]

Hands down, this is The best Math related youtube channel, clear winner !

[@grayjphys]

I'd love to see some advanced topics like fractional calculus explained in your way of simplification. You're really amazing at translating math into a simple understandable set of ideas.

[@stevelk1329]

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. 🙂

[@thobiaslarsen8336]

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!

[@ulugvarnal6061]

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 :)

[@mariovrpereira]

I'll save these classes to my future children. amazing stuff!

[@prodbyKamikaZ]

The way you made me understand ai and complex numbers visually before i even knew trig is impressive

[@lewismassie]

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

[@green0563]

Same.

[@suryakiran3085]

Same . Mind blown , it is super concise and tightly packed .

[@comradefeo]

Same story here. I thought I understood complex numbers quite well after about 5 years of working with them. I was wrong.

[@vma011]

Definitely. One of the best playlists out there

[@seblingtonw]

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

[@Spongman]

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.

[@ezeposada1276]

this

[@blinkybool]

That's now at 36:15

[@UnderscoreZeroLP]

that's the previous question

[@DavidLindes]

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.)

[@squibble311]

iirc people realized that in stream 3 or 4 and used that xD

[@Eterrath]

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

[@NilesMontblair]

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.

[@drpkmath12345]

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.

[@martynasvenckus423]

my teacher was complaining about not being able to control a class of 30 children. I would love to show her how Grant keeps thousands of people engaged in style!

[@aliasks6559]

Martynas Venckus well for stareters everybody here WANTS to be here...

[@carlostdied1184]

@@aliasks6559 thats a good point. it's a false equivalency comparing 30 students, most of which im sure dont love math, to an arbitrary amount of numberphiles who decide to tune into grant's stream. i dont want any argument to come out of this though, it was probably just made as a joke.

[@tolep]

how do you engage "slow mode" in clasroom?

[@shishenliart]

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.

[@ybc8495]

my 8 year old son is your fan, I suspicious whether he really understands, but seems he get the most multiple choice correct.

[@maxwellsequation4887]

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.

[@connorkearley7381]

he do 😼

[@2snipe1]

Omg, I literally got chills at the explanation near 1:07:00. That makes perfect sense!!

[@thechiliman500]

One issue (that may have been noticed later) is often times when swapping between questions that have an answer that is correct and not an opinion, if the following question also has a single correct answer then it shows up as green before it resets. (Example 45:39) Hopefully that can be fixed since it really takes quite a bit a way from the unknown aspect of the live questions. Otherwise I'm loving these lectures. Keep them up :)

[@3blue1brown]

Thanks for letting us know!

[@NoriMori1992]

How is 45:39 an example of that?

[@joshinils]

@@NoriMori1992 because the beginning has since been choppen off and all timestamps created before that now reference something else

[@NoriMori1992]

joshinils Yeah, I realized that literally a minute later.

[@davidgould9431]

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.

[@squibble311]

noice

[@afonsoferreira5171]

Dude this was great, I really think people like you are the pioneers in the must needed change to our education system. Keep up the good work

[@varunraju1569]

Thank you so much, Grant. Really, watching your videos is an absolute treat. What you are doing is incredible and you have certainly helped me appreciate the beauty of math.

[@varunraju1569]

The unit circle is probably the most profound idea I have learnt about in high-school

[@stylis666]

I've learned to first read through the question and the answers if it's a multiple choice and then figure out what is and isn't relevant. It took me ten minutes to get this leaning tower. I first drew a tower, then I swirled it around to simulate years, then I had to adjust the ground because the tower wasn't upright anymore according to the story, but I had already drawn it, in detail. Then I wondered how years made the tower lean and why not everything that is several years into existence isn't leaning, which did of course totally explained my slanted up ground. Then I figured out that the time wasn't relevant to that fact. Then I read the question and figured out that the tower perhaps having been built upright wasn't relevant to the question either. Then I calculated the answer and found out that the correct answer wasn't in metres but in maths, so I got out my calculator for nothing. Some confusing stuff. I did get the correct -answe- calculation though. And my tower and angle thingy totally look like a dick and balls and the sun is directly overhead and smiling, so that's a win.

[@gurpchirp]

​@@stylis666 brilliant.

[@theleviathan3902]

Learnt

[@mr.morgans2305]

27:56 i saw a glimpse of elegance in mathematics at 27:56 , as you scale each side by h , also take different cases of scaling down by o and a. in the case of scaling down by o , you get hypotenuse = h/o adjacent = a/o opposite = o/o = 1 apply pythagoras theorem now ^^

[@shitalsavekar7757]

I missed initial lectures of Trigonometry in my 9th standard. And then it was roller coaster ride till I finished my engineering.

[@danielm.1441]

"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*

[@csanadballa8561]

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.

[@I_like_pi_]

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_like_pi_]

@Hassan Akhtar And also in economics for profit. The important part is the value, not the name, since we are discussing which value aliens would use.

[@ryanfuxa6738]

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_like_pi_]

@@ryanfuxa6738 Smaller is not always simpler. If that were true, then they would choose 1/π instead of π. Or they would choose √π/42357882214578995322. That number is a lot smaller, but it definitely isn't simpler. If they used that value as the circle constant (let's call it x), then the area of a circle would be 42357882214578995322(xr)^2, and that definitely is more complicated than πr^2

[@javierbg1995]

48:23 you can see the next question's answer for a sec right there, be careful with the UI! Otherwise a really nice and simple interface, loved it

[@ashwinjain5566]

dude its all for free cut him some slack

[@javierbg1995]

@@ashwinjain5566 of course, I wasn't criticizing or anything, just pointing it out in case it can be fixed. We are lucky we have people like Grant & his crew!

[@ashwinjain5566]

@@javierbg1995 well, you are right that we've people like him

[@DavidLindes]

@@javierbg1995 or perhaps "criticizing" isn't always a bad thing. A gentle constructive critique can help make things better! And that's the spirit I read your original comment in. :)

[@BillionPlusOne]

I was moderately smart but very lazy in high school. Thirty years later I'm finally starting to understand the beauty and elegance of mathematics, thanks to your videos. It's like seeing a glimpse of some really deep beauty, like getting out of Plato's cave. Thanks so much, you're the best maths teacher in the world!

[@akshitsharma187]

"You think it's about triangles but it's really about circles" I remember this quote. It was said by SAL KHAN from khan academy.

[@carterswafford2222]

it's not that origninal, so eh

[@ameenh4122]

@@carterswafford2222 lol

[@DoctorHomunculous]

This lecture reminds me of the moments i was blown away by the best math teachers ive had. Ive never seen such good fundamentals taught on youtube. Exactly what i was looking for to refresh my trig

[@sermuns]

This is absolutely incredible. I love your approach to math, Grant.

[@useruserov8668]

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.

[@GravyTraining]

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!

[@cbbuntz]

7:34 Although there are explicit formulae for powers of sine and cosine, you can use the definitions of chebyshev polynomials to reverse engineer what cosines make cosine raised to a power by taking the inverse of the matrix (because they already answer "what combinations of cos^k make up cos(n)"). Chebyshev type II gives the answer to "what combinations or cos^k * sin give you cos(n)*sin". Generalizing this to all jacobi polynomials, you can get the answer to "what combination cos(n)*sin^m make cos^k*sin^m?" or the reverse, or all sorts of variants that are phase shifted, or whatever other problems you want to solve about sinusoids raised to powers. It's pretty easy in matlab or python or something, but a little complicated to fully explain in a youtube comment

[@nathanjunker2523]

I must say, you had me get up out of bed at 2:00 am to play with my graphing calculator

[@kfactor99]

casual math talk, gaining interest and insight into various topics. love it

[@zdlax]

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.

[@sourabhjain1088]

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.

[@martinepstein9826]

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

[@okuno54]

@wise ol' man b-but Algebra ≅Geometry!

[@KKKVVV-ox6sm]

your channel is a goldmine

[@killymxi]

- Has an ideal ruler to draw perpendiculars; - Totally ignores it.

[@farhansadik5423]

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!

[@sanatanmeaning]

44:02 thats a pure smile,from the bottom of your heart .

[@hannamuller1805]

I bloody LOVE the way he teaches!!! I wish schools taught like this I would be so smart ):

[@raihanakram530]

44:54 Ramanujan's Constant - Reaction

[@colinthomasson3948]

I cannot thank you enough for theses Lockdown Lectures. They are quite unrivaled for intellectual lucidity, selflessly providing such a treasure of knowledge of inestimable worth

[@ASLUHLUHC3]

This video was so good I went from knowing almost nothing about the fundamentals of trigonometry to deriving sinθ=opp/hyp and cosθ=adj/hyp on my own. It would've been good if you went more into tan, but after further research, I learned geometrically why tanθ=sinθ/cosθ , and was able to derive tanθ=opp/adj

[@Basaltq]

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.

[@jeffspaulding9834]

No wonder every time I've gone to the store they're out of unit circles. Grant's been hoarding them!

[@InoceramusGigas]

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 👍

[@strat5520]

Here I sit, at 33yrs old, watching 3b1b like Im a student again.

[@sirpikapika1129]

Thanks to this video I was able to answer @ on every multiple choice question on my final, thanks Grant!

[@marcusyarema7564]

Grant looks like a young Conan O’Brien

[@ka7al958]

Mixed with Matthew mcconaughey

[@paradoxicallyexcellent5138]

He's definitely Irish, I believe he said as much in the Ben Ben Blue podcast a couple years ago.

[@avneeshkhanna]

Grant looks like if Conan O'Brien went to Harvard....oh wait

[@SupratimSamanta]

You all are wrong, Grant looks like Eddie Redmayne. 😄

[@bbbbbbb51]

More like a young Norm Macdonald

[@thigtsquare950]

Paradoxically this lecture was so simple and at the same time so complex! I wish I had lectures like these when I was in university.

[@NonTwinBrothers]

Lmao the JSON comment at 1:14:37

[@bananastuff2840]

Can you explain what he meant?

[@RG001100]

@@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).

[@amirmohammadjalili2676]

Man, we can never thank you enough. please never stop this

[@gheorghegeorgescu7846]

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.

[@cfranc53]

So hard not to look at it once you notice it...

[@channelsixtyseven067]

Grant your videos have a therapeutic effect on me. Not just because of the mathematics but the graphics. I know this subject matter, but enjoy watching the programme all the same.

[@PatricioINTP]

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.

[@sailor5853]

I bet you know the size of all your fingers

[@sqrt-1646]

It is still taught in our school.

[@muhilan8540]

In Geometry we still learned how to construct a 30 degree angle, 45 degree angle, perpendicular bisector, etc.

[@RG-pr5xx]

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!

[@MooseBoost]

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)?

[@theflaggeddragon9472]

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!

[@darrinrasberryph.d.1458]

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!

[@pufthemajicdragon]

Yeah I've been out of high school for almost 20 years and you're making me as excited about math as I was when I was a freshman. (Which was very excited, btw.)

[@l3nn4rt24]

Timestamps: 9:15 start of stream / Introduction 12:16 contents 12:51 understanding cos(x) using Desmos 25:30 Trigonometry basics 59:50 cos^2(x) By the way: I noticed that the lowest labels of the Live Quiz tend to be shifted down by a bit... Usually the fourth an fourther, but sometimes also the third.

[@prateekagarwal7491]

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!

[@squibble311]

tanθ is also the slope of the radius from (0, 0) to (cosθ, sinθ); its sinθ/cosθ, the rise/run

[@tuatarian6591]

I really love the encouragement to pause & ponder, It really did give me a better grasp over the concepts

[@r75shell]

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.

[@drpkmath12345]

r75shell yes right. Good point!

[@AniseedBalls]

This was how I was taught it. I think it's better because the measure is on a line that doesn't move.

[@ZoraZoldier]

Makes sense. However I feel it might get more confusing once you get to angles outside of the first quadrant of the circle.

[@tachikoma-gg]

The treble levels of your mic are kinda low so it's hard to hear you articulate words.

[@donegal79]

i suspect its your hearing.thats the problem