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Tangent is very useful for converting angles to slopes. Tangent of an angle is literally precisely the slope of any line with that angle from +x.

Yeah, a circle was what I taught myself what sin and cos were about when programming in Scratch as a child. And when school started explaining sin and cos with triangles I was like "huh?"

As someone learning game programming with almost zero useful formal mathematical education (through no ones fault other than my own - I was just a terrible math student), this was really accessible. Thank you for taking the time to put this stuff together in such a nice format.

In defense of the triangle, everything with the unit circle is literally just an application of the triangle! And in my opinion the most intuitive explanation of the weird tan graph actually comes from looking at the triangle: since tan is opp/adj, what happens when the adjacent side gets shorter and shorter? The output is going to get bigger and bigger until you get ridiculously big numbers (ie: 10/1=10, 10/0.00001=100000, 10/0.00000000001=100000000000, etc.), then when the length adjacent side is 0... (Also, quick correction at 4:47, the visuals for the Bhaskara I's Approximation show it matching for [0,2π], though it actually breaks off pretty soon after π.)

I have been wondering what Sin Cos and Tan were for ages. I could never find any halfway descent explanation of what they were only that they were. Thank you immensely.

5:35, Recently seen this concept again in the Kaze Emanuar M64 optimization videos. Using a folded polynomial solution, with 1/8th of the sine wave to reconstruct the whole sin/cos graph.

Very good video. I appreciated the part where you show how the symmetries of the sine function can lead to very efficient and accurate Taylor expansion computations. A note I will make is while a circle can be divided into any number of parts, the choice of 360 wasn't completely arbitrary historically. 360 is a highly composite number, that is it has more factors than any number before it. You can easily divide it by 2, 3, 4, 5, 6, 8, ,12 ,30, 60, 120... (there are several more). It's easy to imagine why this was useful for say, cartographers and navigators trying to make easy shortcuts for computing angles.

I can't wait for your math course! I've wanted a video exactly like this on trig functions for a while. It's super helpful to develop an intuitive understanding of these things

Thanks! Solid video as always! Can't wait for the new course

I hated trigonometry in high school because any time I asked what the trig functions were I would just get a vague hand-wavy "ratio of triangle sides" explanation. Even though I eventually learned all of this on my own eventually I still appreciate you giving a more intuitive explanation.

That was the most useful explanation on the topic i've seen in a while. Congrats!!!

Digital logic designer here. I use the CORDIC algorithm to calculate sine and cosines because it is simple, fast, and small (in terms of chip area) and involves only bitshifts additions and subtraction. You can also use it to calculate absolute magnitude of a vector, or atan2() of an x/y pair.

this video answers one of my questions i’ve been wondering for years! beautiful video

Love the video. Unlike many commenters here I learned trigonometry exactly as explained in the video with a circle of radius 1. In fact trigonometry was the first time that math "clicked" for me. I guess having a good way of visualizing what is being explained helps me a ton.

This was so helpful. I don't know if I missed it, but I don't think I've ever seen tan explained like that, and I certainly had no idea why it made such weird shapes on the graph, but now it makes simple and intuitive sense, thank you!

all my life ive never needed this video until literally last week this is perfect amazing video my friend

Thank you! Looking forward to the math course from you.

This is one of the best videos I've seen on this topic! As an oldskool gamedev, the dot product, lookuptables, and quadrantifying/octantifying periodic values were very useful. This video just makes understanding them easier even for us retired users. Have you made videos on splines? I have been using both the Bezier and Catmull-Rom to do enemy patterns in SHMUPS. They're way easier to predict than using a combination of trig functions. A piece on how to derive an angle in regard to "t" as well as reparameterizing them so that points across splines are uniform in distance would be swell. Looking forward to your next vid!

this, this is exactly what I needed, I've had this question for a while now so thanks so much for this honestly awesome video

Can't thank you enough I've been searching for this since forever

Well this video is really intuitive and useful, people use these built in libraries directly in their code and they don't know how the background process goes, thanks a lot for this video

this is a really well made video, and I actually liked it....I honestly, like a lot of people, copy a lot of math functions from other code when I am less familiar with it...then I store it away in a kind of cheat sheet until I get comfortable with the function....

This 5 minute of trig taught me more than my math teacher could in 6 months

Brilliant video! You've answered the question in an interesting, complete and easy to follow way 😁

5:27 lovely reference! anyway, i do loved the whole explanation and you showing the actual code / mathematical definition

Love seeing the visualisation of this stuff!

Great video! Would love to see an explanation of how logs are computed!

Its funny how when you need to learn something you learn it alot easier than when youre forced to learn it. I willing learnt about the different trig functions a while back. Nice to see the OG making a vid on it😆

I never understood trigonometry before and even in coding all I needed to know that they make waves that I can exploit to make something move smoothly back and forth. But now I finally understand... until I forget again because I still won't use them to their fullest.

Much love Simon. I love your videos man. You’re a smart guy and elite programmer.

I've fiddled around with approximating sines and cosines. I like the idea of reducing things to the range from 0 to pi/2. One more step, I've found, is to care only about the range from 0 to pi/4. What happens from pi/4 to pi/2? Well, that'd be the complementary function of the complementary angle. Like, the sine of 80 degrees would be the same as the cosine of 10 degrees. Polynomial expansions close to zero tend to get very accurately very quickly. So that region around pi/4 (45 degrees) is where things would converge the slowest, but even then, I think we've got an approach for that. Remember that sin(a - b)= sina*cosb + cosa*sinb, so if you wanted to calculate the sine of 40 degrees, that's the sine of (45 - 5) degrees. Okay then, so we'd have to calculate sin(45)*cos(5) - cos(45)*sin(5), and sin(45) = cos(45) = 0.7071etc, so we've got very little to calculate. The answer will be 0.7071etc*(cos(5) - sin(5)), and those will converge quickly. At this point we're approximating just from 0 to pi/8.

Exactly what I needed! Thank you!

Thank you for this wonderful explanation!

I think I really learned to appreciate trigonometry after learning it in school. It's just a beautiful concept that just is so satisfying to watch. (And it has some cool ways to apply it in engineering to make fun stuff)

i always found it easier to view tan as the slope of the hypotenuse. it has asymptotes at pi/2 and 3pi/2 because the hypotenuse is vertical there. the reason the slope is equal to that length you showed is because slope is change in y over change in x, but when the radius is 1, the change in x becomes 1, so it’s just the change in y that matters. this is the length of that vertical line it sound strange but it you draw it all out and try to find the slope of a triangle in a circle manually and use 1 radius to be 1 unit, you’ll see it really easily. just remember to extend the hypotenuse

I look forward to picking up the math course!

this was amazing and very helpful

I spent a whole day trying to figure out how to convert my X,Y speed parameters into a simple vector that just described the angle in 0-360 degrees and the amplitude. I then came across and remembered the old trig formulas and bodged my way through. I laughed so hard when I saw the circle diagram here.

Well. Damn. That was a very educational visualization of SinCosTan. Especially the movement to graph. Thank you.

Looking forward to this course

Man, that was excellent! I always wondered how this was done, honestly a little disappointed I didn't figure it out myself.

In fact you can reduce the range even further, by noticing that the sin function, for x ranging from pi/4 to pi/2 is a mirror of the cos function between 0 and pi/4. And the Taylor series in the 0 to pi/4 range converges to an optimal precision after very few terms, both for sin and cos. This is what I did, without suggestions from anyone, on my Z80, in assembly code, back in the early eighties.

This method works pretty well for sine, cosine, and tangent. There’s some other optimizations you can do, such as representing the Taylor polynomial with nested multiplication. Also, for arcsine, arccosine, arctangent, and log, some basic identities come in handy.

Thanks alot Simon!!! Sent to my friend who's stuck on same subject what a neat synchronicity 😃👍. Happy Coding

Congrats on 100k subscribers :) Have been following your channel for a while now and absolutely love your content. Even the video about NFT minting, people were just bashing everything NFT related. The tech behind it is still interesting so please don't stop making videos about topics you like to learn about even if they are not graphics/game-dev related

Great video as I’m going into calculus and hadn’t quite grasped the relationships and just been working out their inverses & antiderivatives & derivatives. I love a similar videos on csc, sec, and cot if your up for it

Thank you. This remind me that I could implement various versions of approximate sin and cos functions depending on the accuracy I need for my application, and thus get close enough results faster than when using standard sin and cos function defined by the language or the default Math library. Standard sin and cos function defined by the language, its standard library or the ALU are in most cases compromises that are at the same time way to slow for real time artistic applications and way to inaccurate for science research. It's good to master this so we know what we do, always do enough without overdoing.

Looking forward for the "Game Math Explained Simply" course. Your explain thing really well :D

yeah. this makes much clear sense. thanks for sharing.

fantastic explination!

this is a great video and a wonderful presendation not just due to my lack of intuitive mathematical knowledge

Very nice visualizations!

The title of this video is exactly the question that's been annoying me for a really long time. Thanks so much for making this.

Thank you, short and concise, as I like it. Me being older, I try to pick up geometric algebra from UKposts videos, at high school level. It took me a while to figure out why if u = ae1+be2 and v = ce1+de2, then u.v is both ac+bd and |u||v|cos(th). I did, and your video shows it clearly. From a stupid looking formula, the inner product proves to be very useful, I really start to like geometric algebra!

I remember seeing a discussion on a game dev forum where the approach of 6:38 was used to fit a quadratic to that part of the graph, and the fit was quite good despite the simplicity. You could then use Newton's method to refine the result if you wanted more accuracy. OTOH if you're working at the hardware level, CORDIC is the usual algorithm of choice since it avoids any multiplies.

As a math teacher, this is a spectacular video. Awesome work.

I heard my maths professor mention this YESTERDAY. I did zero research, didn't look anything about it yet youtube still recommended this to me. Jeez... how am I supposed to believe I don't live in simulation?

Just found this channel - and while I do a lot of math for a living, this was such a nice presentation that you earned a subscriber

Wow, very well done!

School does everything wrong. This is a much better explanation than anything school has ever taught me. I just luckily happened to figure out these circle rules on my own when playing with code, but the school system doesn't provide the intuitive or useful explanations of math 😔

Being an old time dev, I have always worried about how sin & cos are being computed in case they are expensive, as I'd heard how early 3D engines used the lookup table approach as a speed hack. But seeing how it's done these days has set my mind at ease. Thanks! :)

THANK YOU IVE ALWAYS WANTED TO KNOW

Nice video and well explained. I really like that you also showed some code. One thing I noticed is that in the code at 6:50 it should be “switch (quadrant)”, but I guess you added this to check whether or not we’re paying attention? 🤓

This is how sin and cos routines in various math libraries work, and it is often very possible to write an algorithm for trig operations which is lower latency than using a single CPU instruction with the trade off being less precision. Though latency can also be a little misleading at times since those operators are highly parallelized. If the FSIN instruction takes 170 cycles to give a result that doesn't mean your computer spends all that time waiting for an answer. It will try to do other operations with that core while it waits and can even have several FSIN operations being calculated simultaneously. So which is faster in practice can be more nuanced, as always seems to be the case with modern super scalar architectures. In hardware these operators are usually implemented using the CORDIC algorithm. The resources you linked discuss that a little. Essentially you implement a more general function that rotates a vector (x, y) by an arbitrary angle. For anyone with textheworld installed: [;\begin{bmatrix}x_{out} \\y_{out}\end{bmatrix}=ROT(\theta)\begin{bmatrix}x_{in}\\y_{in}\end{bmatrix}=\begin{bmatrix}x_{in} \\y_{in}\end{bmatrix}\begin{bmatrix}cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta)\end{bmatrix};] Sine and cosine in that situation are just the x and y coordinates that result from rotating the vector (1,0). Using only rotation in various ways it's possible to implement several other tricky functions as well including inverse trig functions, logarithms, hyperbolic trig functions, square roots, and even standard multiplication and division. The documentation for the Xilinx CORDIC IP core is pretty good for seeing how all that works in practice. The CORDIC algorithm takes advantage of the fact that rotation by an arbitrary angle can be expressed using a series of rotations by different angles which sum to that angle. For example to rotate by 67 degrees you could rotate a vector by 45 degrees, then by 30 degrees, then by -8 degrees. These smaller angles are chosen using powers of 2 so that the algorithm can use a bit shift operation rather than a more expensive multiplication. With this structure CORDIC calculates an answer to N bits of precision with N iterations of shift-addsub. The output is still an approximation, but its as precise an implementation of the trigonometric operations as floating point numbers are able to express.

super pumped for your math course!

im so glad bob burger taught me how to compute sine and cosine

I'm so glad I watched this before my kids. I can't have them hear such an explicit description of how radians are made.

I've been working on low level graphics functions for some ten years now and it blows me away how bad people are at explaining trig functions. You, my dude, have nailed it. The unit circle is easily the best way to visualize how they work. Looking forward to see how you simplify matrices and quaternions, assuming you'll be doing those. The best way I've found *personally* to represent the former is using basis vectors and showing how it's like a box within a box within a box, if that makes sense. The latter's a bit trickier, but I usually imagine it as a ray emitted from a given point, with everything rotated a given number of radians around that ray. Hope this channel grows. I'll do my part. :p

One of the neatest tricks I ever learned was called CORDIC, which is something like Coordinate Rotation Digital Computation. It expresses an angle as sums and differences of smaller angles which are chosen so that rotation by those smaller operations are trivially simple, operations on bits. Since a rotation of the vector (1,0) can give you sine and cosine, those are easy to compute. But suppose you are given x and y and you want to compute the angle that rotates it onto sqrt(x^2+y^2) - e.g. find the angle. It's all additions and subtractions. You can also get any of the trig functions with similar tricks. If you have to build the hardware implementation of trig computations, it's really easy.

Very interesting. You make it very easy to understand. Thank you. You forgot to mention the software used to make those trigonometric graphics.

Great video! What programs did you use to make this video?

Great video, easy to learn from. I personally, find it much easier to see tanθ as the slope of the terminal arm. That is, once you place your θ and extend the terminal arm to the circle, the arm's rise/run is tanθ.

Very interesting. Do you expect to make a similar video about logarithms? I'm quite curious as to how a computer computes log(x) efficiently.

I was diving into this recently, particularly Robin Green's writing on the subject, as I am - out of curiosity and for fun - building a simple game engine in Rust, and have found that Rust - as of writing - seems to lack bindings for SIMD-accelerated sine, cosine, and sincos functions, which I seem to remember having seen in C++, so I decided to try and see if I could create an efficient implementation using SSE or AVX intrinsics. In the document Robin Green lists at the end of his blog I saw code that superficially resembles the code shown at 6:46 though the switch statement seems different. Green uses 'switch(k & 3)' and then three case statements (with constants 0, 1 and 2) and then a 'default' fallback case. I think this works because it is guaranteed to be exhaustive, the AND with 3 means that all but the lowermost 2 bits are zeroed, effectively leaving a 2-bit integer, which of course can only encode 0, 1, 2 and 3. There's another difference though that confuses me. In the code shown in the video, integer 'quadrant' is defined as k mod 4, presumably to wrap k to range [0, 3], but then the switch statement refers to k and not quadrant. As far as I can tell, this means that k is not wrapped to the range of [0, 3] and 'quadrant' is not used. Am I correct in this, or am I misreading something?

lol, wow... That paper you showed at 7:05... I worked as a graduate intern on that exact team (especially, Shane and Ted), and I personally rewrote the entire test suite software for the extremely high performance code they were developing for the original Itanium (all of the "math.h functions", not just sine). Work included testing on beta versions of Win64 (and Linux), emulated IA-64 hardware, early IA-64 hardware revs, prototype compilers, and more. The team literally counted clock cycles for code paths through functions, maximally shaving them off.

Excellent video!

I enjoyed this video, would you consider doing one for other math concepts like matrices?

Yamaha's FM implementation used a quarter of a sine wave (premultiplied to ease of certain other calculations) as a lookup table.

Thats really great!

After reading many comments here by different persons, i collected a brief summary about the algorithm CORDIC (not the algorithm itself): X The CORDIC algorithm was used to compute Trig functions in hand-held calculators (HP and others), and computers with slow processors and small memory, but is specially useful for implementation in hardware as bit-shift operations and additions and substractions, which is very easy. You can also use CORDIC to compute the absolute value of a vector, and the function atan2() of any pair of x/y . The Baskhara Approximation formula can be used to compute sin(x), too (but maybe with less precision). A more modern algorithm than CORDIC ( and vastly superior, they say) uses Chevysev Series (or Approximation). Taylor series are very slow to converge, so they are not used to compute Trig functions.

I really enjoy watching your videos, what animation software do you use?

In the 70's I was slinging Fortran for Space Shuttle Simulators. When I started running the guidance software to support landing, everything I was monitoring was going along fine until just before landing when the simulated shuttle did a maneuver to transition from the steep glide slope to the shallow inner glide slope (-1.5 degrees) just before touchdown. The shuttle would suddenly climb , slow down, stall, and crash. I got with the vendor of the computer we were using and he pulled out the scientific subroutine library code-we were looking at hard copies here, no CRTs yet. It turned out that the sin function would start out by testing the value of the argument you were handing it then go to a unique Taylor Series expansion for that sized angle-this was partly necessary to get everything to execute in real-time. Sure enough, for certain small negative angles, the particular Taylor series it used was wrong and would return a positive sin instead of a negative one which is what it should be for a -1.5 degree angle. The guidance algorithm would then generate a positive (upward) altitude rate reference and issue commands to the control effectors to try to fly to it, which it did.

In first year university calculus we imagined that there are two functions s(x) and c(x) such that the first derivative (slope) of s is given by c and the 1st derivative of c is -s. It is demonstrated that s is sine and c is cosine. In Intel/AMD processors the microprocessor provides all of the primary mathematical functions. This is because it is more efficient to perform the calculations using the underlying hardware directly rather than exploiting the complex instruction set implementations of multiplication and division. This is not an advantage on reduced instruction set computers (RISC) such as ARM or RISC5 so they provide only the basic floating point operations. When you look at the x86 "hardware" instruction that provides sine you will observe that it returns *both* the sine and cosine values for the argument. Although this does permit obtaining the values of tangent and cotangent that is not why this is done. It is an artifact of the use of the Newton-Rapson algorithm to refine an initial approximation together with permitting further restricting the range of values that must be supported to just 0 through pi/4 by observing that sin(x) is cos(pi/2 - x) and cos(x) is sin(pi/2 - x). With modern inexpensive memory it is cheaper to use a table of, for example 256, values of sin and cos or any other transcendental function and then use the Newton-Rapson method to refine the value, especially to get 64 or 128 bit extended floating point values, than to use a polynomial. Each loop through the Newton-Rapson method increases the precision by a number of digits. All arithmetic operations more complex than multiplication depend intrinsically on gradually refining an approximation, usually by a variant of Newton-Rapson. For example look up videos on how to extract a square root by hand. Why do you divide the remainder by 2 times the current estimate? Because that is the 1st derivative of x^2! Also recall that you all learned how to do arithmetic by a table lookup. 7 times 8 is 56.

I would like to see going over modulo, since there are 2 definitions of it and different engines uses one of the two definintions.

You can also use half angle formula to get it to small angles for the Taylor series

Fantastic video

thank you for your knowledge sharing and useful explanation. Really thank you. I wish I was teach / learn this in school, not from youtube. :(

I think with the unit circle and using more lines you can also visualize COT, SEC, and CSC. If I am not mistaken, the unit hyperbola is used to visualize the HYPERBOLIC TRIG FUNCTIONS, Sinh, Cosh, Tanh, etc

This is the exact question I've been trying to figure out for weeks. Like what actually is in the black box of sin and cos.

Could you please make a video about application of calculus in game development?

I always wondered how these are computed Thank you so much I can almost sleep peacefully... though, how are then log and exp functions computed? That's the real question I couldn't find answer to

please make a video on quaternions!

holy shit this just explained everything that my pre calc teacher isn’t explaining

@4:07

I used scaling and symmetry with lookup tables and interpolation to do trig on my Commodore 64 in the 80s. There were, of course, no floating point instructions for the 6510 so it was some fairly gnarly bit munging in assembler. Commodore BASIC had trig support but sometimes you needed the extra 26K of RAM.

Great vid.

Never for a millisecond even questioned the ability for a computer to calculate sin(x) with perfect accuracy until I saw this thumbnail Then you explained how some games and stuff approximate it for speed, but now I'm realizing even on a *supercomputer* we've *never* been able to perfectly calculate sin(x), because it's a function based on π, an irrational number of infinite length which can't be represented in a computer. This existential crisis quickly led to another, realizing that we *will never be able to perfectly represent sin or any trig function because pi goes on forever, and that in a sense ALL of the maths we do, both on the computer and off, is like this: only theoretical on nature and every actual calculated number we use is just an approximation of some unobtainable truth of the universe and...* I think I need a drink

Very beautiful animation..

iirc, implementations usually divide by 2pi first, so using a 0-1 range i think referred to as turns is faster