Micro math educator here.. I'd like to defend the bottom pattern, at least for one circumstance. The fact is, memorizing these values is almost always needed for only one thing.. exams. The bottom pattern, despite all its mathematical taboos, is a very strong compression and therefore, easier to memorize. You get 5 values for the cost of memorizing the denominator, the counting sequence and a square root. The other approaches, albeit mathematically honest, are just harder to memorize. And when students are pressed for exams, they care about cheap memorization.
I'll also add that in my student/teaching history in the US (maybe different in the UK?), we use the rationalized values for pi/4, meaning three of our values are already common denominator, so why not the other two values as well?
When I was a student, I always found the unit triangles very easy to remember. It relies much less on rote memorisation of facts (which is hard), and instead on simply developing understanding and applying known rules (in this case, basic SOHCAHTOA trig) to a simple situation.
So, the solution is that education should change, and do not force students to memorize something they do not understand, do not ask memorized values in exams. Memorizing number multiplication results under 10 are useful, but I neved knew why sine values are such overrated. Understanding something is useful, no matter what that thing is. Sine results are used in common exam tasks, but hey, why doesn't the exam paper provide a table with those values, and let the student focus on the calculation process instead?
The more i had to memorize, the less i liked math. No issue with looking things up, but come on in what reality is memorization for things like this a key asset other than that the general teaching world. "If you find yourself in a problematic situation you can't look up or use a calculator, most likely doing the math is the least of your problems."
As an engineer, I feel obligated to submit a true engineer approach. For any x, sin x = 0, with +/- 1.1 error tolerance. (experiments show the error is between -1 and +1, but let's add 10% safety margin).
THIS is precisely why unexpected oscillations appear in everything from bridges to electronics! If only mathematicians could devise a more accurate way of determining the value of sin x.
@@londonalicante Why calculate a more accurate value if you can just pour twice as much concrete? It's when the engineers start skimping on materials and try to be clever with maths to make up for the saved construction costs, that's when things collapse. Don't try to make it never to explode. Just make sure when it inevitably explodes (as it will no matter how hard you try to prevent it), it won't kill anyone and can be rebuilt in a couple days.
This is a memorization tool, not an understanding tool. And it will continue to be useful as long as math education prioritizes memorization over understanding.
Understanding is harder to test than memorization - or - it is easier to scale up memorization testing than understanding/functional testing. As long as we want to score learning, we will value memorization.
I don't often disagree with Matt but I've got to be honest, this seems like an unnecessary takedown of a harmless mnemonic. Plus, too much time was spent talking about the "arbitrary" values, but the mnemonic only exists because those values were *already* chosen as the values for which sin and cos exact values are considered "known" -- usable without calculation or justification. In the end, I like the triangles, but I also think this pattern is really helpful. And after all, there's no reason to just have one! When I was in trig and calculus in high school, I used a combination of the sqrt(n)/2 and the circle + wave picture to quickly recall the values when I needed them (which was a lot, of course). Eventually, the bigger ideas soaked in (as did these values) and so I stopped needing it, but that took time. Waiting until everything is fully absorbed and internalized before moving on is not really the most efficient way to learn, especially when a cheap memorization trick can carry you forward until that happens. Overall, I think it's fine when presented as a mnemonic, emphasizing that the pattern doesn't reveal any deeper truths, but can be used to remember a few distinguished values in a pinch.
That's definitely better than using your calculator for everything, because this way, you actually engage with it and *will* at some point remember it.
the importance of those angles is in fact that they have a "nice" number as their solution, and thus are used in exams where no calculators are allowed. Even as a student in university you encounter this in beginning mechanics classes, where the angle of two bars are usually chosen as 30° 45° or 60° (or the radiant counterpart) as those are first, simple to see by eye, and second, have these simple "clean" resulting numbers.
I always use the triangles so I can derived the values quickly. The triangles give you the sine, cosine and tangent values. Also since the triangles are more visual it is easier to remember.
Thank you for pointing this out, I was thinking the same thing: Had I seen this "non-pattern" during my early physics studies, it would've been quite helpful on numerous exercise sheets. Remembering some example triangles would've been fine too, but I think I'd need to actually draw them, which wouldn't be necessary with the meme-pattern
it's also a fact of physics that those angles maximize something (projectile motion) maximum vertical height then 90 degrees with the horizontal. maximum horizontal range, then 45 degrees with horizontal. maximum time and range simultaneously is 60 degrees.
i remember discovering this pattern when i was learning trig, and it did help a lot. although i remembered it as sqrt(0/4), sqrt(1/4), sqrt(2/4),sqrt(3/4),sqrt(4/4). and then i remembered that cosine would have the corresponding values to make it add to 1 inside the square root. since sin^2 + cos^2 = 1
THANK YOU. I thought for 8 minutes that that would be what the video would end up at. But no, he kept iterating his "your meme is bad and you should feel bad" pitch. I thought, "just square the sin values, and BAM, not a pattern of sin itself but of sin². And the theta values aren't THAT random, use two thetas that add up to 90° and the sin² values will add up to one. Because sin² (theta) and cos² (90° - theta) are the same." He came close with his "quarter wave of sin leads to semiwave of cos²" but never stated it explicitly. What a math-boomer...
I don't know if you're aware Matt, but at GCSE 0, 30, 45 60, & 90 are the sine values you're required to memorise for the non-calculator exam. That's why those numbers were 'arbitrarily' chosen, and why people are so eager for an easy way to remember them.
As everyone else has already pointed out, the angles aren't arbitrary. The angles are what you are required to memorize for non-calculator portions of most modern math tests, which is why the pattern is helpful. It means that you have to do less work to memorize it.
Also, the angles aren't arbitrary because they are the limit-2 and limit-3 harmonics of the unit circle (in the complex plane). When you square those values, you still end up on a limit-2 or limit-3 root. 30° is a nice 1/2i multiple, 45° maps to 90° (1i) and 60° maps to 120°, which is also 1/2i. The reason pi/5 (36) is less interesting is when you square any of the 5th roots of the unit, you still are on a 5-limit. You have to raise them to the 5th power to get any rationals. But since euclidean metrics are always degree-2, there's no geometry of 5-limit root angles which gives clean ratios the same way 2 and 3 limit roots do.
If you are used to memorizing these values without actually understanding how they came about then it feels like they are arbitrary. They are not arbitrary at all! And they are not selected by the curriculum!!
There is one reason people prefer the bottom version, which seems to have gone completely unnoticed here: It's easier to remember. The majority of people looking for some deeper insight are the people who likely already know these exact values.
In defense of the meme: The angles 30, 45, 60 and 90° come up far more frequently than most other angles. 36° or π/5 just isn't needed nearly as often. So being able to put the most common angles into such a "pattern" is still helpful to memorize these values. Most other values are more complicated to memorize and not needed enough to be granted precious memory capacity.
This has nothing to do with "finding interesting mathematical patterns". This is about memorizing a bunch of values to your exam using as little brain power as necessary. And I think that the bottom pattern does a very fine job at that. You write that line (which is easy enough to memorize), simplify, invert the order to have the cosine line. Divide to have the tangent line and you are done, at least in the first quadrant. I think it is a good practice to students that are starting to deal with trigonometry until it gets "in their blood" over repetition in many years.
I am a precalculus trigonometry student. I do find it useful, but I would really like to understand how to approximate sin(theta), cos(theta), and tan(theta), etc., to like 3 or 4 decimal points, instead of just letting the computer do it. I understand that it isn't something that can be put into a non-approximating equation, but computers can do it to a few decimal places quickly WITHOUT lookup tables (only a cache for the most used results), so it must be possible to learn that. Is it too advanced for my level? I am not a massive math guy but I am very good with computers and understand the low level stuff (ALU, etc.)
I'm an electrician and when dealing with alternating current, we deal with sin waves. But this video made me realize that when we bend conduit, especially to make offsets, we are basically physically creating sin waves. For example, when calculating the distance between the two bends that make up an offset, we use a ratio that depends on the distance we need to offset and the angle we want to use. The easiest to remember is 30 degree angles, as the ratio is 2. So, if I want a 37 centimeter offset using 30 degree angles, I need to keep 74 centimeters between bends. For a 45 degree angle, the ratio is 1.4 and for a 60 degree angle we need a 1.2 ratio. We also use 22.5 degrees, with a ratio of 2.6 and a 10 degree angle with a ratio of 6. This is for electrical conduit in buildings, we don't really need these to be more precise but it was nice to see where these ratios come from :)
I used to run a cutting table which cut shapes out of plate metal. I had to draw the shapes myself on paper using a marker, protractor, straight edge, etc. It helped a lot to know trigonometry to figure out measurements that weren't given on the prints. I think I used more advanced math for that job than I do as an engineer.
The real problem you should have discussed is the reason why that meme was necessary. Because students are taught that they must memorise these 5 magic numbers, 3 of which are fractions with square roots in them, completely out of context with no logic behind it, for the sake of exams. Students see no pattern in them and therefore it is hard to memorise.
Yep, my teacher taught the bottom method. Later teachers tried to teach the triangle method but it's just nowhere near as easy to remember a shape than a table of numbers.
That's so stupid, even before calculators existed, people used sin tables, cos tables, log tables and so on, nobody remembered all of that stuff back then (and calculating it on the fly every time would have been very tedious) and now we do have calculators, why do students still have to remember those values? I mean, yeah, if you know what a unit circle is or just how the sin graph looks, you should know where the 0s and 1s are, but anything with a square root is not something you should have to remember. I don't get it.
I just ran into this. My ALEKS learning software was asking me to calculate sin() and it just told me to basically memorize the unit circle coordinates for the basic angles 30, 45, 60, 90 and so on. I was like... wtf, surely there is a way to not have to memorize all those weird coordinates, I literally cant memorize that. Turns out, yeah, the only way is using something called a "taylor series" which is above my level but appears to be an approximation. Idk what to do. The taylor series has this weird "E" symbol though. It looks kinda like a for loop. I know programming very well so, maybe I will just try to find the computer algorithm and memorize that lol.
@@GGysar It can be super useful to know these values, not having to use a calculator, and also being able to get an exact answer (if your calculator only provides decimal approximations). Conceptually it's also super useful to understand the unit circle, and with knowledge of these values and the unit circle you can calculate the values of so many other angles in your head (e.g. sin(pi/6) = sin(5pi/6) and you will also know which ones will be the negative versions.
Considering that I have never had any problem at all seeing sqrt(0)/2 as 0, sqrt(1)/2 as 1/2 and sqrt(4)/2 as 1, the bottom part has NEVER caused me to "lose" the insight of 0, 1/2, 1. However, when I noticed the bottom pattern at school, I remember being the only one never needing to memorise the values or look them up in a table, meanwhile, I saw all the other kids struggling with those values. I think the mathematician outrage over the bottom one is unjustified.
the outrage is warranted these people are very passionate when it comes to math so tricks like this would be considered blasphemy as it ignores the "mathematical beauty" in favor of "aesthetic" unfortunately, not every student (as trigonometry is mandatory in school) is as passionate and I don't blame them (because me neither lol)
@@fernando4959 I appreciate mathematical beauty but even as a kid, I marvelled at the fact that 4, 2(2), 2^2, sqrt(16), etc. are literally the same number expressed in different notations. This fact alone was beautiful to me as a kid. It also allowed me to separate the logical object itself from the notation representing that object. This separation of objects from notation also allowed me to quickly understand the debates about pi vs tau, the poor representation of exponents, roots and logarithms (as explained by 3blue1brown) and many other issues. Because I familiarised with this concept as a kid, I see no difference between base-10, base-12 and other base numbers, whereas others complain about how base-12 gives easier division with less fractions but we're still using base-10. You're not losing any mathematical beauty. You're changing notation. No mathematical structure suffers, only mathematical notation.
But if you understood how it relates to triangles and circles, you would have the benefit of easy memorization AND you'd actually know what it is you're computing. The "trick" only saves you any effort if you have no idea how sin and cos are defined.
Yeah, this was how I remembered it for the GCSE maths non-calculator paper. We were required to “learn” (read, memorise) the values for sin, cos and tan for the angles 0, 30, 45, 60 and 90 degrees. Once you remember the sine values, you can easily just shift it for cosine, and then calculate tangent.
This *absolutely* reveals a real and illuminating mathematical pattern, it's just it's a pattern in the angles when you linearly increase the sine-squared function. In the first quadrant as we increase the sine-squared function through increments of 1/4 starting at 0 we see 0, 30, 45, 60, 90 and we can examine what happens to the angles when we use different increments with the same function - when are they "nice" angles? Or we can observe that since sine-squared is a rational function of the coordinates of a triangle we can calculate it over finite fields where the logarithmic nature of an angle causes difficulties.
I think Matt's point is that 0 30 45 60 90 isn't a pattern. It goes up by 30 15 15 30, and then it stops. But now I write the values out, I'm not so sure.
0, 30, 45, 60, 90 is literally a perfect pattern though. It cuts a right angle in halves and thirds. It is two extremely basic patterns on top of each other.
@@eekee6034 but that's why it is useful to me, as a precalculus trig student, because the test wants me to memorize 0, 30, 45, 60, 90 theta as coordinates. It isn't a pattern forever, its more like a weird way to mentally compress the data. Its like if you gave a computer a sequence of unrelated equations, and it found a simple rule that let you derive the other patterns by performing one equation in a loop, so you only have to memorize that and the first one.
Nice. As mentioned in other comments, the formulas for 18, 36, 72 degrees can be written using sqrt(5) or the golden ratio (sqrt(5)+1)/2. See also en.wikipedia.org/wiki/Exact_trigonometric_values. There are some whoppers in that list.
@@dranorter a problem exists that these are all values that result from solutions to complex-number polynomial equations for integer roots of a square-root of -1; values that originate from powers of the same root _will_ be related by a sort of pattern that eventually ends on ±1 (and then oscillates), but smaller roots involve ever more complicated equations to solve, with ever more complicated solutions, so there's no guarantee or even likelihood of a simple (finite) rule
So, I actually did this back in high school when I had to memorize the x,y coordinates of the unit circle. I would just put root over two on everything, then fill it out 1,2,3,4. Counting up for Sine and counting down for cosine all the way around. and then go back and throw the appropriate negatives on them. It made it easier to remember because there was less to remember. Not the most mathematically elegant, but definitely the most test-takerly elegant.
I remember my calc 2 professor in college got mad that people didn't know their sine and cosine values, so he wrote out this trick on the white board and everyone was shocked at how genius this approach was. Apparently this is how everyone learned it in 1980s soviet russia
This is the first video where I think you missed the point. In all my math classes (pre-algebra up to Calc III) the only trig values I was ever tested on were 30deg, 45deg, 60deg, and 90deg with the expectation to know how to translate those around the unit circle. To a new trig student, the angles and values are completely arbitrary so if you need to know them for a test, you need a trick to memorize them. That's what the second image accomplishes. Like it or not 90+% of all math students only need to know this stuff for the test where they usually can't use a calculator.
It's not just a meme. It's an easy way to memorise those values. And the cosine values are just the reverse order of the sine values. They're neat mnemonics, providing an effective way for (at least trigonometry beginners) to acquire them. I teach this to all my maths students, and of course also explaining how those values are calculated.
i disagree with this take, as no-one is claiming this rule has any meaning other than just being a coincidence, or using it to extrapolate to find other values. It is simply a way to remember the some of the values if you don't have a calculator, as this way is easier to remember and recall in exams than the conventional notation which has no pattern to aid memorisation.
@@wolfie54321 no i totally agree in teaching the understanding like that, but once a student understands why and the method your talking about, if they want to chose a different method they know for memorisation then i think this is fine. like i understand trig but i may still use this method if you asked me to recall these values.
You can memorise traditional values, memorise specific triangles, do it in radians, etc. But if want to pass the test, meme is easiest to remeber, just don't forget to simplify.
There is a "Nice" way to simplify the values for sine and cosine of 18°,36°,54°, and 72°; but it involves the use of the Golden ratios: φ = 1.618... and Φ = -0.618... It goes like this. For sine: Sin(18°) = ½*√(0+Φ²) Sin(36°) = ½*√(1+Φ²) Sin(54°) = ½*√(0+φ²) Sin(72°) = ½*√(1+φ²) Notice How the first numbers have Φ while the last use φ, and that the 0 and 1 alternates. For cosine the opposite is used: Cos(18°) = ½*√(1+φ²) Cos(36°) = ½*√(0+φ²) Cos(54°) = ½*√(1+Φ²) Cos(72°) = ½*√(0+Φ²) It's weird but I prefer it than using nested square roots.
When he showed the nested square root I immediately called out "What's a phi doing here‽" So ironically, the mnemonic after _did_ help reveal a deeper pattern.
The production values of this video are so good that they’re invisible. Kudos to Keyboard Cat throwaway animations, and the excellent Meme-screenshot video windows at the end.
I learned the 5-finger trick back when I was a tutor in college. One of my students taught me it, and my mind was kind of blown. I get why educators don’t teach it, since it glosses over the underlying math and suggests a pattern where there is none, but it was still a fun trick.
I think there might be a real pattern! A different mnemonic can be formed that's similarly linear and includes all multiples of pi/6 and pi/4. I don't see how to include pi/10, but there's probably a way; I'd expect there's some sequence of increasingly complicated mnemonics, connected with some rigorous scheme for approximating sine.
I like it, it helped me memorize the values of the common angles in the other quarters (180-270 for example) way more easily. Definitely a better way to learn them (for exams), I think you're being a bit too harsh here.
I think he recognizes that the bottom is easier to memorize, but it just hurts too much from a mathematical perspective. The reason why all these sin values are what they are and are so nice is because they’re all special cases, and so it’s probably better to store them all as special cases, with the source of these special cases being the triangles mentioned at the end of the video. Making a pattern to memorize values feels gross as a result because they’re all special cases, there is no true pattern. (Also, after some time of working through the triangles you will remember that 0,1/2, sqrt2/2, sqrt3/2, 1 are the relevant values, in which sketching out the unit circle definition will make finding any value pretty fast, provided you can draw estimated angles)
@@rli1618 It's a great learning tool. I learnt in an instant all the values I needed to know which I would have struggled with without that tool. I struggle with arbitrary numbers and the actual maths behind it is too complicated.
@@rli1618 As someone doing a master in mathematics, it's a great learning tool. I have a lot of trouble with memorisation and this tool is the formost reason I know my trig as well as I do. Special cases is fun and all that but the only time you need the conversions is on a test and on a test the only thing that matters is ease of memorisation. In my entire study of mathematics I have never seen any use of these nice special cases and have only ever known these values as some stuff to memorise.
I think most commenters here are missing the point, since the video mentions 2 other mnemonics that are arguably better. The thing is that this mnemonic in particular is kinda misleading, because it appears to be a mathematical pattern when it's actually just an useful coincidence.
I actually noticed the whole sin30 = √1/2, sin45 = √2/2, and sin60 = √3/2 thing as a kid when I was taking trig, definitely made it a bit easier to memorize!
10:45 Ben Sparks' drawing is set to become an icon of educational maths for its simplicity yet meaningfulness. It could be named "maths in a rush". T-shirts NOW!!!
I independently noticed this pattern about 25 years ago in high school. It is a GREAT way to memorize this. That doesn't mean there is any deeper connection. It is a pneumonic, not a pattern. Like SOHCAHTOA.
In highschool, I noticed this too, but I noticed it as sin30=sqrt(1/4) sin45=sqrt(2/4) and sin60=sqrt(3/4) Which also made it easy to remember the cosines as well, because sin^2+cos^2=1
@@ethanbottomley-mason8447 For me it depends. If I have calculations that have a sqrt(2) * sin(60) or sqrt(2) * cos(30) then I would use 1/sqrt(2) Else I would use sqrt(2)/2, depends on the calculations for me
These values are commonly expected to be known by heart. To be frank, i think it's really unfair for teachers to demand a pattern be memorized and then disavow the easiest way to memorize it.
Similarly to what Mutual Information said, I think the value in this is just that it's really easy to memorize, though I'd say the same is true outside of exams too - because those are very common values. Knowing how to derive them is of course better from a mathematical standpoint, and that's how I'd do it as well; however if you just want a quick and easy way to remember these values and don't care it doesn't generalize, this is a good way to do it imho. (Also regarding the finger technique, I wouldn't say it's that much better, considering that you can get the exact same thing by knowing that sin and cos are symmetric within that quarter circle)
That final method of drawing the two triangles is what my math teacher taught me and what I use every time! I use my thumb and finger to remember which corner is 60 and which is 30 :P
Wow I just noticed the beard growing when I was jumping around the video. What a subtle feature... I admire the determination that must have been involved in pulling it off. (Or shaving it off, perhaps.)
Actually, there *is* at least one proper pattern: If the sine of an angle alpha equals sqrt(x)/2, then the sine of the complementary angle pi/2-alpha equals sqrt(4-x)/2. Same for cos, by the way. So, the left-right symmetry in the table is not just coincidence, it is real. That also holds for other angles, say sin(pi/8)=sqrt(2-sqrt(2))/2 so sin(3pi/8)=sqrt(2+sqrt(2))/2 Or sin(pi/5)=sqrt((5-sqrt(5))/2)/2 so sin(3pi/10)=sqrt(4-(5-sqrt(5))/2)/2=sqrt((3+sqrt(5))/2)/2 which simplifies to (1+sqrt(5))/4. And another similar pair for sin(4pi/5) and sin(pi/10). Not to mention sin(pi/12)=sqrt(2-sqrt(3))/2 and sin(5pi/12)=sqrt(2+sqrt(3))/2 Some of these pairs are quite elegant. (Addendum: watching the 2nd half, this is the nice symmetry Matt talks about, in a slightly different and completer way.)
Well if one goes further and writes it using a reverse order 4 to 0 sin 0⁰: √(2²-4)/2 sin 30⁰: √(2²-3)/2 sin 45⁰: √(2²-2)/2 sin 60⁰: √(2²-1)/2 sin 90⁰: √(2²-0)/2 then one can see the pattern for the sides of the triangle they come from since sin x = O/H = √(H²-A²)/H, where H=2 so that all these values come from right angled triangles with hypotenuse 2 or alternatively from a circle of radius 2. But that's probably not how mnemonics work.
worth noting that the pattern is like this in part because: sin²(θ) + sin²(τ/4 - θ) = 1 so we are seeing three special cases of that: (√0/2)² + (√4/2)² = 4/4 = 1 (√1/2)² + (√3/2)² = 4/4 = 1 (√2/2)² + (√2/2)² = 4/4 = 1
In a way it’s quite arrogant and condescending for a bunch of maths educators to have sat around laughing at the masses going through the education system for finding a way to help themselves quickly recall some information that will help them get through life more easily. The vast majority are not going to derive any meaning from a wavy line or some right-angle triangles, they’re going to respond to the sequence the same way they respond to putting a finger down to find a 9 times table, or counting knuckles to work out which months have 31 days, or holding out hands to work out which one “says L for left”. Maybe it’s not a real sequence or pattern, but it works. That’s all that matters. Kinda disappointed to hear Matt and Katie and others stoop to this kind of mockery.
I was actually a little proud of myself in school when I noticed this pattern myself to remember the values easier. I guessed that there wasn't some special reason/pattern behind it since 0, 30, 45, 60, 90 isn't equally spaced, but I never looked into it further because I figured there wasn't anything interesting underlying it. It does make it easier to remember, though.
As a Mathematician, I kept waiting for the part where I’m supposed to be “angry” about the meme Honestly, you talked for 12 minutes without actually saying anything. The meme is literally just a useful mnemonic. You’re making this way deeper than it needs to be
It's a pretty good approximation for other angles if you take a weighted average. For example, 40 is two thirds of the way from 30 to 45, so its sine is around √(1+2/3) / 2. It's only 0.002 off !
@@hareecionelson5875 Nope, not an engineer, I'm studying pure math. It's just that where there's smoke, there's fire. In this case, the fire is the approximation we get for the sine. It's not an infinitely precise approximation like Taylor series, but it's a lot better than the first few terms.
Yeah, the pattern is nice because it's easier to memorize (for people with math-inclined minds that can easily work out sqr(2)/2 simplifies to 1/sqr(2), the rest should be trivial to simply for anyone who can do that.) not because it reveals any underlying meaning.
You mean non math inclined right? I mean people will memorize things differently but i think if you're more math intuitive the logic is what you remember not the sequence. And you find the sequence back that way. And there's no arguing that a series in a table is much worse logical reason than the way the unit circle works. Because the unit circle links everything in trigonometry.
@@SquintyGears but how do you quickly calculate the values, especially sqrt(3/4) from "the logic"? Do you calculate the infinite series? This is just a good way to save time in trigonometric calculations, by knowing some values. Maybe mathematicians do not like this, but most people who regularly use maths are not mathematicians. Compare for example Feynman, who considered knowing some values out of your head, to be really useful.
@@Henriiyy well outside of very specific tests that at the same time force you to know the values AND not have a calculator it's not very useful. and that kind of exam isn't the norm because wtf are you going to do with sqrt(3)/2 in any other following calculation.
@@SquintyGears sqrt(3)/2 is way better than sin(pi/3) because you can use it further and combine it with other expressions. I use maths every day and haven't taken a trig exam in many years, and it's still useful. 30,45 and 60 degrees are just very common angles. Edit: yes, I could use a calculator, but when you use this all the time, it would be tremendously tedious to type it in every time, just like it's good to know the times tables instead of using a calculator for 8*6 every time.
This memorization pattern has led me to an interesting approximation for sin(x). The formula suggests that sin(x) = √(some f(x))/2, therefore we must find some f(x) = 4sin²(x). Using a polynomial interpolation formula for the values ((0,0),(30,1),(45,2),(60,3),(90,4)), we find f(x) = -x³/81000 + x²/600 - x/180, which is a nice enough expression. So, sin(x) ≈ √(-x³/81000 + x²/600 - x/180)/2 gives exact values for the listed above points, and gives nice approximations for the intermediate values. The maximum error is close to +2% between 5º and 15º and -2% between 75º and 85º and close to +-0.2% between 27º and 63º. BTW, if we use the angle in pi radians, that is, z = x/180, the expression gets even simpler: sin(z) ≈ √(-z(72z²-54z+1))/2
I don't think I can trust the aesthetic sensibilities of a pi apologist when it comes to anything having to do with the unit circle. This is a man who wholeheartedly believes that representing a 3/4 rotation around a unit circle as 3/2*pi radians is more mathematically illuminating than 3/4*tau radians
Yes, exactly. This is *the* scenario for which the case for tau is the strongest. Pi simply should not be used for radians, it completely obscures the obvious logic of the unit to stick a diameter-based constant in there.
the only thing in my mind is Matt doing the video almost backwards and slowly trimming his beard, it way too perfect I didn't realize until he put the jacket
Careful Matt you explained what ‘sin’ means in an easily understood way. Maths teachers in schools have being trying to hide that information from students for decades.
I think it’s always worth mentioning _why_ students find these mnemonics attractive: bad maths education. For this one, specifically, the “bad maths education” is tests and quizzes that make you do algebra with memorized formulae and values, where the formulae are the trig identities and the memorized values are trig functions at these four values, so you can calculate an answer, which the quizzes require, for some reason.
Nice transition around 6:17 - beard transition was smooth, kept camera in the same location, and very slight difference in environmental lighting. Also, that's a rad shirt - bit more obvious transition when it switches from white to black print.
First of all, who writes 1/sqrt(2) this is an awful non-standard notation, it's always sqrt(2)/2 Second, it's a very useful memorization tool. I personally draw a quarter circle when I fail to remember which one has which value, vertical is sine, pick the middle, draw the angle, it's short, so it's 30°, right, sin(30)=1/2, thus cos(60)=1/2, and the opposites are sqrt(3)/2 As to explain I would actually use a 3d graph that looks like a spring, show from the front, and from the sides, and it's so good at explaining how the graphs relate to each other
I love the subtle commitment to growing the beard over the coarse of filming, to tie in with the meme at the end. Well done Matt, that gave me a good laugh when I noticed. Wish I could give me 'likes' for this effort. Would Steve show the same commitment and shave his beard at the start of a video... I think not.
How I wish I did exams: Draw the trig circle. Simplify √0/2 √1/2 √2/2 √3/2 Plot them on the circle. Use the circle for reference. Write the exam answers below it. But I didn't. I should have... I do recall imagining a trig circle and rotating my pen in exams, but this mnemonic would have helped extra. At least to get me started.
I disagree. When I saw this "pattern" years ago I loved it. The reason being: it helps. A lot. I am not against memorization in math as long as you know what is going on on the back stage so to speak. Once you do, whatever method you use to remember sine and cosine of the notable angles is valid. And this is a great one 'cuz it's easy. So easy even some more elderly people I help learning basic math have no trouble with it.
To be honest I don't even understand why people did to memorize sine values to begin with it's simple enough to draw the triangles these angles are derived from. I got through all of biophysics undergrad with this (can never remember which sin(x) value is 1/2 and which cos(x) value is 1/2)
can we just acknowledge that Matt had to grow that beard and record bits for a meme video along the way :) these details are a big part of why i like this chanel so much! really entertaining and informative as usual. thanks Matt :)
Even though it may be a taboo I remembered these values because one day I saw the values and in maths class for some reason square root of 1/4 is 1/2 came up and I realised the the pattern I used was root of 0/4, 1/4 … 4/4 so yea I would not have had my appreciation of trigonometry if I didn’t learn it this way
A mnemonic I don't need to memorize is more useful to one I need another mnemonic to remember. Chances are anything a mnemonic is going to help with is going to be 0 30 45 60 or 90 degrees anyway, and if none of those angles are the ones you're using, you're going to dig out a calculator and trig cheatsheet anyway. This is a practical mnemonic for people who actually need to use their angles and be done with them 20 seconds later, not just appreciate them in abstract.
When I was a trig student I was greatly frustrated with how much of the math class was actually blind memorization of identities, but I was so grateful when my calculus professor showed me that pattern because that was a few less things I had to brute force memorize. So I have to disagree with all the hate towards it because half the "better" examples were exactly the ones I was happy to never use again. Great video though, just wanted to put in my experience as a student learning it.
I discovered a cool way to memorise the table back in highschool(it is an interesting fact on its own though). It was during Physics period that I noticed that I could use Malus' Law which is used in calculating intensity of light through a polarizer(I = I_0 times cos squared theta). If you square the sines of the angles 0°, 30°, 45°, 60° and 90°, you get 0, 1/4, 1/2, 3/4 and 1, respectively. Then I take their square roots to get what I want. Pretty neat!
5:21 "It makes sense looking at it geometrically why 1/3 gives you 1/2." Does it though? I mean you showed us that it sorta kinda looks like that but that didn't make me understand _why_ that's the case.
Sure, the geogebra file does leave some room for suspicion, but if you look at the sides of a 30 60 90 triangle you can see clearly why the sines of those anles are what they are, using only the pythagorean theorem.
@@unvergebeneid Whatever you purport the word “didactics” to mean, I’m not talking anything over Grade 3 geometry when I say “When there is a triangle with a right angle and a 60° angle, the side between the 30° angle and the 60° angle is exactly twice as long as the other side of the 60° angle.” I mean, you can even derive it yourself by folding an equilateral triangle in half, for crying out loud.
Although I agree with the comments that this is a useful pattern for memorization, even if it's an artificial one, I also think teachers should make clear it's nothing more than a memorization tool, not some hidden mathematical truth in plain sight.
the discrepancy is due to use case and demographic. mathematicians hate it for lack of simplicity, but plebs love it for the sudden insight of the pattern. it's not for YOU, it's for those who never realized there was a pattern
That's the mental model I figured out in high school to memorize the pattern and reproduce them on demand. While it might not exist in a technical sense, it's a useful mental tool for remembering something years or even decades after you last used it. I also used to teach a fluid flow model for the way electricity works for linemen apprentices. Voltage is pressure, resistance is friction, current is flow, and so on. People in physics forums used to rail against it because it's not exactly how it works, but they missed the point. The fluid flow module for electricity is a close enough approximation that it could be used for troubleshooting a downed power line after a storm, and that was all that was necessary. So the people who knew the exact physics of electricity could complain, but they were doing so from computers powered by a grid that was restored after a storm VERY QUICKLY because the linemen were able to troubleshoot using that model. It's the mathematical equivalent of using an acronym to remember a series of words. Nothing more.
I used to draw a little right triangle with 1, 2 and √3 sides. I'm sure everyone knows that triangle. All I had to remember was that sin(30)=1/2 and I could figure out the rest from that.
You don't even need to remember that since if you split an equilateral triangle (x=2) in half you get the 1 and 2 naturally. That's how it's illustrated in the video. You can also use the Law of Sines but that's assuming some further knowledge
Yeah, the math behind it is nice, but at no point did I get any points for that on exams. All I needed from this were the values and regardless of how weird the values were written, they were extremely easy to remember.
Not going to lie, I discovered this on my own back in high school and I still use it as my go-to mnemonic for values of sine at angles of 0, pi/6, pi/4/ pi/3, and pi/2. From there I can use basic trig properties to get any of the trig functions at any of the special values on the circle. Is it accurate to the heart of how trig works? Of course not, just like similar tricks like log(1) + log(2) + log(3) = log(1 + 2 + 3). But when I need to guesstimate something or simplify an expression (read: exam question), it's a lot quicker than re-deriving values from first principles and more reliable than pure memorization, at least for me. Every model is wrong and some are useful. I would hesitate before teaching it as an educator for precisely the reasons you point out though, and if I did it would be with great caution to point out it's just a mnemonic. (Yes, I did read the pinned comment and your reply, and I agree with you both. I'm just pointing out my experience with the pattern since this is one I've used for a while.)
Mathematicians looking at a familiar set of values which have been expressed using their least common denominator: "There's no rational basis for writing it this way!" Ok, ok, they also used the exponential analog of a least common denominator. You've caught them generalizing an existing mathematical tool to reveal a relationship which was not previously obvious. Better tell them to knock it off double-quick!
Wish I had known about this, I could've sent in the perfect version. Alas, now it is too late and nobody will ever know. Jk for me it's jotting down a quarter circle and radians going up from 0/12 times pi to 6/12 times pi with 1/12 pi intervals. That gives the pattern 0-6/12ths pi for the angle. THEN, on the outside of the circle, come the sqrt(n)/2 ones, skipping the 1/12th and 5/12ths of pi. It's not about understanding from whence the values cometh (although I do use the circle just for that), it's a tool for memorizing them. You don't draw a triangle for every piece of pythagoras so there's no need for the hard-to-remember square roots and integers. Look at it from an information perspective. I can either remember 10 separate pieces of information per sine, cos and tan, OR I can remember 1 shape, 2 generating patterns and 2 exceptions. That's 30 pieces of information vs 5. It _is_ a superior way of remembering it, particularly with the quarter circle shape. It _isn't_ a superior way of teaching the origin of the values. But then, that's not what it's for.
I have tutored maths and futher maths GCSE & since the trig exact values need to be memorised there is an easy way of generating them from a table: 1. write the angles 0,30,45,60,90 along the top row 2. write sin in a separate column to the left of the angles, then the numbers 0-4 (inclusive) underneath the angles 3. write cos under the sin and then write the numbers 0-4 but backwards underneath the numbers in the top row 4. draw a big square root sign over the grid of numbers you have just drawn 5. Finally, draw a horizontal fraction line underneath the grid of numbers and put all of that over 2 Getting the values once you have this table is as easy as choosing the trig function and angle you want from the headings, then checking the corresponding number in the table, square rooting it, and writing it over 2. (eg for sin 60, look at sign and 60, check the cell they share, that being the number 3, then you have sin60=sqrt(3)/2 without a calculator) It sounds long writing it out like this, but in actual fact once you know how it is constructed, you can picture the table in your head and get the values from there, which is a huge time saver in the exam. And in case you forget it you can always draw it out again. Note: the way to get the Tan values from here (since they also have to be memorized) is using the identity tan = sin/cos
Totally agree it shouldn’t be used as learning tool or by anyone who hasn’t understood trigonometry to a good degree, but I think this is a good mnemonic for the people who know the meaning well and have to just remember the value quickly. I still have to draw on paper (or in my mind, not my fastest skill) the 30-60-90 or 45-90 triangles to infer the right values, and from now on I’ll begin using this method, because it is faster in terms of recalling time. I won’t forget the mathematical concept, but it’ll be faster for me to get the right one
Micro math educator here.. I'd like to defend the bottom pattern, at least for one circumstance. The fact is, memorizing these values is almost always needed for only one thing.. exams. The bottom pattern, despite all its mathematical taboos, is a very strong compression and therefore, easier to memorize. You get 5 values for the cost of memorizing the denominator, the counting sequence and a square root. The other approaches, albeit mathematically honest, are just harder to memorize. And when students are pressed for exams, they care about cheap memorization.
*Computer scientists loved this answer*
I'll also add that in my student/teaching history in the US (maybe different in the UK?), we use the rationalized values for pi/4, meaning three of our values are already common denominator, so why not the other two values as well?
When I was a student, I always found the unit triangles very easy to remember. It relies much less on rote memorisation of facts (which is hard), and instead on simply developing understanding and applying known rules (in this case, basic SOHCAHTOA trig) to a simple situation.
So, the solution is that education should change, and do not force students to memorize something they do not understand, do not ask memorized values in exams. Memorizing number multiplication results under 10 are useful, but I neved knew why sine values are such overrated. Understanding something is useful, no matter what that thing is. Sine results are used in common exam tasks, but hey, why doesn't the exam paper provide a table with those values, and let the student focus on the calculation process instead?
The more i had to memorize, the less i liked math. No issue with looking things up, but come on in what reality is memorization for things like this a key asset other than that the general teaching world. "If you find yourself in a problematic situation you can't look up or use a calculator, most likely doing the math is the least of your problems."
As an engineer, I feel obligated to submit a true engineer approach. For any x, sin x = 0, with +/- 1.1 error tolerance. (experiments show the error is between -1 and +1, but let's add 10% safety margin).
Add a fellow engineer, I like this comment.
Sin can reach 3 or even 4 in time of war.
@@Crazmuss 😂😂😂😂
THIS is precisely why unexpected oscillations appear in everything from bridges to electronics!
If only mathematicians could devise a more accurate way of determining the value of sin x.
@@londonalicante Why calculate a more accurate value if you can just pour twice as much concrete? It's when the engineers start skimping on materials and try to be clever with maths to make up for the saved construction costs, that's when things collapse.
Don't try to make it never to explode. Just make sure when it inevitably explodes (as it will no matter how hard you try to prevent it), it won't kill anyone and can be rebuilt in a couple days.
This is a memorization tool, not an understanding tool. And it will continue to be useful as long as math education prioritizes memorization over understanding.
Sometimes you just have to memorize stuff for quality of life.
Well said
Yeah! Beside basic arifmetis we need to think, not to remember that's the reason we have only do much actual mathematicians.
Great comment.
Understanding is harder to test than memorization - or - it is easier to scale up memorization testing than understanding/functional testing. As long as we want to score learning, we will value memorization.
I don't often disagree with Matt but I've got to be honest, this seems like an unnecessary takedown of a harmless mnemonic. Plus, too much time was spent talking about the "arbitrary" values, but the mnemonic only exists because those values were *already* chosen as the values for which sin and cos exact values are considered "known" -- usable without calculation or justification. In the end, I like the triangles, but I also think this pattern is really helpful.
And after all, there's no reason to just have one! When I was in trig and calculus in high school, I used a combination of the sqrt(n)/2 and the circle + wave picture to quickly recall the values when I needed them (which was a lot, of course). Eventually, the bigger ideas soaked in (as did these values) and so I stopped needing it, but that took time. Waiting until everything is fully absorbed and internalized before moving on is not really the most efficient way to learn, especially when a cheap memorization trick can carry you forward until that happens.
Overall, I think it's fine when presented as a mnemonic, emphasizing that the pattern doesn't reveal any deeper truths, but can be used to remember a few distinguished values in a pinch.
That's definitely better than using your calculator for everything, because this way, you actually engage with it and *will* at some point remember it.
As a former math educator, I wholeheartedly concur with all of this.
the importance of those angles is in fact that they have a "nice" number as their solution, and thus are used in exams where no calculators are allowed. Even as a student in university you encounter this in beginning mechanics classes, where the angle of two bars are usually chosen as 30° 45° or 60° (or the radiant counterpart) as those are first, simple to see by eye, and second, have these simple "clean" resulting numbers.
I always use the triangles so I can derived the values quickly. The triangles give you the sine, cosine and tangent values. Also since the triangles are more visual it is easier to remember.
Thank you for pointing this out, I was thinking the same thing: Had I seen this "non-pattern" during my early physics studies, it would've been quite helpful on numerous exercise sheets. Remembering some example triangles would've been fine too, but I think I'd need to actually draw them, which wouldn't be necessary with the meme-pattern
it's also a fact of physics that those angles maximize something (projectile motion)
maximum vertical height then 90 degrees with the horizontal.
maximum horizontal range, then 45 degrees with horizontal.
maximum time and range simultaneously is 60 degrees.
i remember discovering this pattern when i was learning trig, and it did help a lot. although i remembered it as sqrt(0/4), sqrt(1/4), sqrt(2/4),sqrt(3/4),sqrt(4/4). and then i remembered that cosine would have the corresponding values to make it add to 1 inside the square root. since sin^2 + cos^2 = 1
I like that one a lot better
This is what I used too. The connection to that identity / pythagorean theorem is nice and easy to visualize.
You could just reverse the sequence for cos - sqrt(4/4), sqrt(3/4), sqrt(2/4), sqrt(1/4), sqrt(0/4)
THANK YOU. I thought for 8 minutes that that would be what the video would end up at. But no, he kept iterating his "your meme is bad and you should feel bad" pitch.
I thought, "just square the sin values, and BAM, not a pattern of sin itself but of sin². And the theta values aren't THAT random, use two thetas that add up to 90° and the sin² values will add up to one. Because sin² (theta) and cos² (90° - theta) are the same." He came close with his "quarter wave of sin leads to semiwave of cos²" but never stated it explicitly. What a math-boomer...
I don't know if you're aware Matt, but at GCSE 0, 30, 45 60, & 90 are the sine values you're required to memorise for the non-calculator exam. That's why those numbers were 'arbitrarily' chosen, and why people are so eager for an easy way to remember them.
*45 instead of 35 but yes, this is exactly why
Quickly sketching two triangles gives you all of them like the video
They are not arbitrary, they are factors of pi in radiants
@@m3ducraft They're completely arbitrary in that you don't require factors of pi in radians for any specific task.
@@Gabu_ they are kiiiind of not arbitrary because they come out from a regular triangle and a square...but I get what you mean
As everyone else has already pointed out, the angles aren't arbitrary. The angles are what you are required to memorize for non-calculator portions of most modern math tests, which is why the pattern is helpful. It means that you have to do less work to memorize it.
Also, the angles aren't arbitrary because they are the limit-2 and limit-3 harmonics of the unit circle (in the complex plane). When you square those values, you still end up on a limit-2 or limit-3 root. 30° is a nice 1/2i multiple, 45° maps to 90° (1i) and 60° maps to 120°, which is also 1/2i. The reason pi/5 (36) is less interesting is when you square any of the 5th roots of the unit, you still are on a 5-limit. You have to raise them to the 5th power to get any rationals. But since euclidean metrics are always degree-2, there's no geometry of 5-limit root angles which gives clean ratios the same way 2 and 3 limit roots do.
They are arbitrary, but the arbitrary decision was made my the people making curriculums, not the meme maker
If you are used to memorizing these values without actually understanding how they came about then it feels like they are arbitrary. They are not arbitrary at all! And they are not selected by the curriculum!!
@@melk2950 these angles were not selected by the people who make curriculum!!
@@defenestrated23 no they're just π/2, π/3, π/4, π/6 (I don't know why they skipped π/5)
There is one reason people prefer the bottom version, which seems to have gone completely unnoticed here: It's easier to remember. The majority of people looking for some deeper insight are the people who likely already know these exact values.
I'd say the two triangles method is even easier to remember AND is grounded in proper understanding.
The two triangles is the one I use. It is visual so easier to remember.
Exact values don’t exist
Fact: all geogebra files are made by Ben sparks. No exceptions
They don't call them Sparks' Lines for nothing...
I recently made a GeoGebra file and whished I'd have a Ben Sparks to make it pretty for me :(
He's the only one smart enough to learn how to use that program.
@@YourMJK you ARE Ben Sparks you just don't know it yet.
In defense of the meme:
The angles 30, 45, 60 and 90° come up far more frequently than most other angles. 36° or π/5 just isn't needed nearly as often.
So being able to put the most common angles into such a "pattern" is still helpful to memorize these values.
Most other values are more complicated to memorize and not needed enough to be granted precious memory capacity.
This has nothing to do with "finding interesting mathematical patterns". This is about memorizing a bunch of values to your exam using as little brain power as necessary. And I think that the bottom pattern does a very fine job at that. You write that line (which is easy enough to memorize), simplify, invert the order to have the cosine line. Divide to have the tangent line and you are done, at least in the first quadrant. I think it is a good practice to students that are starting to deal with trigonometry until it gets "in their blood" over repetition in many years.
I am a precalculus trigonometry student. I do find it useful, but I would really like to understand how to approximate sin(theta), cos(theta), and tan(theta), etc., to like 3 or 4 decimal points, instead of just letting the computer do it. I understand that it isn't something that can be put into a non-approximating equation, but computers can do it to a few decimal places quickly WITHOUT lookup tables (only a cache for the most used results), so it must be possible to learn that. Is it too advanced for my level? I am not a massive math guy but I am very good with computers and understand the low level stuff (ALU, etc.)
I'm an electrician and when dealing with alternating current, we deal with sin waves. But this video made me realize that when we bend conduit, especially to make offsets, we are basically physically creating sin waves. For example, when calculating the distance between the two bends that make up an offset, we use a ratio that depends on the distance we need to offset and the angle we want to use. The easiest to remember is 30 degree angles, as the ratio is 2. So, if I want a 37 centimeter offset using 30 degree angles, I need to keep 74 centimeters between bends. For a 45 degree angle, the ratio is 1.4 and for a 60 degree angle we need a 1.2 ratio. We also use 22.5 degrees, with a ratio of 2.6 and a 10 degree angle with a ratio of 6. This is for electrical conduit in buildings, we don't really need these to be more precise but it was nice to see where these ratios come from :)
and people ask -- how will we ever use trigonometry in real life 🤮🤮🤮🤓🤓?
Ah neat! The pattern doesn't really aid memorization when decimal values are needed.
@@singh.ayushman and carpentry
I used to run a cutting table which cut shapes out of plate metal. I had to draw the shapes myself on paper using a marker, protractor, straight edge, etc. It helped a lot to know trigonometry to figure out measurements that weren't given on the prints. I think I used more advanced math for that job than I do as an engineer.
The real problem you should have discussed is the reason why that meme was necessary. Because students are taught that they must memorise these 5 magic numbers, 3 of which are fractions with square roots in them, completely out of context with no logic behind it, for the sake of exams. Students see no pattern in them and therefore it is hard to memorise.
Yep, my teacher taught the bottom method. Later teachers tried to teach the triangle method but it's just nowhere near as easy to remember a shape than a table of numbers.
That's so stupid, even before calculators existed, people used sin tables, cos tables, log tables and so on, nobody remembered all of that stuff back then (and calculating it on the fly every time would have been very tedious) and now we do have calculators, why do students still have to remember those values? I mean, yeah, if you know what a unit circle is or just how the sin graph looks, you should know where the 0s and 1s are, but anything with a square root is not something you should have to remember. I don't get it.
I just ran into this. My ALEKS learning software was asking me to calculate sin() and it just told me to basically memorize the unit circle coordinates for the basic angles 30, 45, 60, 90 and so on. I was like... wtf, surely there is a way to not have to memorize all those weird coordinates, I literally cant memorize that. Turns out, yeah, the only way is using something called a "taylor series" which is above my level but appears to be an approximation. Idk what to do. The taylor series has this weird "E" symbol though. It looks kinda like a for loop. I know programming very well so, maybe I will just try to find the computer algorithm and memorize that lol.
@@zachb1706 probably because you learned the numbers first?
@@GGysar It can be super useful to know these values, not having to use a calculator, and also being able to get an exact answer (if your calculator only provides decimal approximations). Conceptually it's also super useful to understand the unit circle, and with knowledge of these values and the unit circle you can calculate the values of so many other angles in your head (e.g. sin(pi/6) = sin(5pi/6) and you will also know which ones will be the negative versions.
Considering that I have never had any problem at all seeing sqrt(0)/2 as 0, sqrt(1)/2 as 1/2 and sqrt(4)/2 as 1, the bottom part has NEVER caused me to "lose" the insight of 0, 1/2, 1. However, when I noticed the bottom pattern at school, I remember being the only one never needing to memorise the values or look them up in a table, meanwhile, I saw all the other kids struggling with those values. I think the mathematician outrage over the bottom one is unjustified.
*applauds*
AAAAAUGHRULIRRRRUUUAAEEEAHHHGHGHGUEEE RAGEEE
the outrage is warranted
these people are very passionate when it comes to math so tricks like this would be considered blasphemy as it ignores the "mathematical beauty" in favor of "aesthetic"
unfortunately, not every student (as trigonometry is mandatory in school) is as passionate
and I don't blame them (because me neither lol)
@@fernando4959 I appreciate mathematical beauty but even as a kid, I marvelled at the fact that 4, 2(2), 2^2, sqrt(16), etc. are literally the same number expressed in different notations. This fact alone was beautiful to me as a kid. It also allowed me to separate the logical object itself from the notation representing that object. This separation of objects from notation also allowed me to quickly understand the debates about pi vs tau, the poor representation of exponents, roots and logarithms (as explained by 3blue1brown) and many other issues. Because I familiarised with this concept as a kid, I see no difference between base-10, base-12 and other base numbers, whereas others complain about how base-12 gives easier division with less fractions but we're still using base-10.
You're not losing any mathematical beauty. You're changing notation. No mathematical structure suffers, only mathematical notation.
But if you understood how it relates to triangles and circles, you would have the benefit of easy memorization AND you'd actually know what it is you're computing. The "trick" only saves you any effort if you have no idea how sin and cos are defined.
Yeah, this was how I remembered it for the GCSE maths non-calculator paper.
We were required to “learn” (read, memorise) the values for sin, cos and tan for the angles 0, 30, 45, 60 and 90 degrees.
Once you remember the sine values, you can easily just shift it for cosine, and then calculate tangent.
I think it actually is a nice pattern, and none of the quibbling makes it any less nice :)
The amount of time you put into these videos is beardgrowing!
I spent far too long rewinding the video to spot the edits 😆
Why nobody ever talks about that?!
I was looking for comments like this.. this is way better than whatever I would've come up with..
Also wonderful video.. I totally love your humor.
Matt Parker: Procrastinaire
"I should really finish that sine video.........nah."
Came here to make this comment.
This *absolutely* reveals a real and illuminating mathematical pattern, it's just it's a pattern in the angles when you linearly increase the sine-squared function. In the first quadrant as we increase the sine-squared function through increments of 1/4 starting at 0 we see 0, 30, 45, 60, 90 and we can examine what happens to the angles when we use different increments with the same function - when are they "nice" angles? Or we can observe that since sine-squared is a rational function of the coordinates of a triangle we can calculate it over finite fields where the logarithmic nature of an angle causes difficulties.
I think Matt's point is that 0 30 45 60 90 isn't a pattern. It goes up by 30 15 15 30, and then it stops. But now I write the values out, I'm not so sure.
0, 30, 45, 60, 90 is literally a perfect pattern though. It cuts a right angle in halves and thirds. It is two extremely basic patterns on top of each other.
@@eekee6034 but that's why it is useful to me, as a precalculus trig student, because the test wants me to memorize 0, 30, 45, 60, 90 theta as coordinates. It isn't a pattern forever, its more like a weird way to mentally compress the data. Its like if you gave a computer a sequence of unrelated equations, and it found a simple rule that let you derive the other patterns by performing one equation in a loop, so you only have to memorize that and the first one.
It's also worth noting, the mathematical pattern in the meme does generalize a bit. For example we can add sin(15) and sin(22.5). Sin(0) = √(2 - √4)/2, sin(15) = √(2 - √3)/2, sin(22.5) = √(2 - √2)/2, sin(30) = √(2 - √1)/2, sin(45) = √(2 - √0)/2, sin(60) = √(2 + √1)/2, sin(67.5) = √(2 + √2)/2, sin(75) = √(2 + √3)/2, sin(90) = √(2 + √4)/2.
Nice. As mentioned in other comments, the formulas for 18, 36, 72 degrees can be written using sqrt(5) or the golden ratio (sqrt(5)+1)/2. See also en.wikipedia.org/wiki/Exact_trigonometric_values. There are some whoppers in that list.
Nice
This could be a real patter rather than that Matt has criticized.
@@emekdulgeroglu3914 I keep staring at it and haven't found a 3rd, even more detailed version yet. But it's definitely possible.
@@dranorter a problem exists that these are all values that result from solutions to complex-number polynomial equations for integer roots of a square-root of -1; values that originate from powers of the same root _will_ be related by a sort of pattern that eventually ends on ±1 (and then oscillates), but smaller roots involve ever more complicated equations to solve, with ever more complicated solutions, so there's no guarantee or even likelihood of a simple (finite) rule
Matt : "a 1/3 of a 1/4 is a 1/2"
Me trying to process that: .......so this is what an aneurism feels like
classic Parker
So, I actually did this back in high school when I had to memorize the x,y coordinates of the unit circle.
I would just put root over two on everything, then fill it out 1,2,3,4. Counting up for Sine and counting down for cosine all the way around. and then go back and throw the appropriate negatives on them. It made it easier to remember because there was less to remember. Not the most mathematically elegant, but definitely the most test-takerly elegant.
Hey, same thing I did! High-five!
Every trig test I ended up drawing a little unit circle at the top and filling it in so I had easy reference.
I remember my calc 2 professor in college got mad that people didn't know their sine and cosine values, so he wrote out this trick on the white board and everyone was shocked at how genius this approach was. Apparently this is how everyone learned it in 1980s soviet russia
This is the first video where I think you missed the point.
In all my math classes (pre-algebra up to Calc III) the only trig values I was ever tested on were 30deg, 45deg, 60deg, and 90deg with the expectation to know how to translate those around the unit circle. To a new trig student, the angles and values are completely arbitrary so if you need to know them for a test, you need a trick to memorize them. That's what the second image accomplishes. Like it or not 90+% of all math students only need to know this stuff for the test where they usually can't use a calculator.
It's not just a meme. It's an easy way to memorise those values.
And the cosine values are just the reverse order of the sine values.
They're neat mnemonics, providing an effective way for (at least trigonometry beginners) to acquire them. I teach this to all my maths students, and of course also explaining how those values are calculated.
i disagree with this take, as no-one is claiming this rule has any meaning other than just being a coincidence, or using it to extrapolate to find other values. It is simply a way to remember the some of the values if you don't have a calculator, as this way is easier to remember and recall in exams than the conventional notation which has no pattern to aid memorisation.
@@wolfie54321 no i totally agree in teaching the understanding like that, but once a student understands why and the method your talking about, if they want to chose a different method they know for memorisation then i think this is fine. like i understand trig but i may still use this method if you asked me to recall these values.
Well done on finally hitting a million subs! I know you didn’t quite beat Steve Mould there but we can consider it a Parker victory
I love the beard growing and evolution of the tee-shirt, Matt, as the video progresses. Super sneaky.
Holy crap, I didn't notice this at all. Makes this video and meme feud even greater
I love the subtle nature of Matt slowly growing a beard throughout the video!
As well as the shirt's logo coming in over time for the meme format, didn't even catch that! So brilliant!
You can memorise traditional values, memorise specific triangles, do it in radians, etc.
But if want to pass the test, meme is easiest to remeber, just don't forget to simplify.
There is a "Nice" way to simplify the values for sine and cosine of 18°,36°,54°, and 72°; but it involves the use of the Golden ratios: φ = 1.618... and Φ = -0.618... It goes like this.
For sine:
Sin(18°) = ½*√(0+Φ²)
Sin(36°) = ½*√(1+Φ²)
Sin(54°) = ½*√(0+φ²)
Sin(72°) = ½*√(1+φ²)
Notice How the first numbers have Φ while the last use φ, and that the 0 and 1 alternates. For cosine the opposite is used:
Cos(18°) = ½*√(1+φ²)
Cos(36°) = ½*√(0+φ²)
Cos(54°) = ½*√(1+Φ²)
Cos(72°) = ½*√(0+Φ²)
It's weird but I prefer it than using nested square roots.
Oh wow! That's really cool!
This comment needs to be way higher up!
I'm left wondering whether there's an even nicer way though...
When he showed the nested square root I immediately called out "What's a phi doing here‽" So ironically, the mnemonic after _did_ help reveal a deeper pattern.
I like how his beard grows like a sine function over time
The production values of this video are so good that they’re invisible. Kudos to Keyboard Cat throwaway animations, and the excellent Meme-screenshot video windows at the end.
I learned the 5-finger trick back when I was a tutor in college. One of my students taught me it, and my mind was kind of blown. I get why educators don’t teach it, since it glosses over the underlying math and suggests a pattern where there is none, but it was still a fun trick.
I think there might be a real pattern! A different mnemonic can be formed that's similarly linear and includes all multiples of pi/6 and pi/4. I don't see how to include pi/10, but there's probably a way; I'd expect there's some sequence of increasingly complicated mnemonics, connected with some rigorous scheme for approximating sine.
I like it, it helped me memorize the values of the common angles in the other quarters (180-270 for example) way more easily. Definitely a better way to learn them (for exams), I think you're being a bit too harsh here.
It's ridiculous to pretend it's supposed to be a mathematical claim. It's a learning tool for the common angles.
I think he recognizes that the bottom is easier to memorize, but it just hurts too much from a mathematical perspective. The reason why all these sin values are what they are and are so nice is because they’re all special cases, and so it’s probably better to store them all as special cases, with the source of these special cases being the triangles mentioned at the end of the video. Making a pattern to memorize values feels gross as a result because they’re all special cases, there is no true pattern.
(Also, after some time of working through the triangles you will remember that 0,1/2, sqrt2/2, sqrt3/2, 1 are the relevant values, in which sketching out the unit circle definition will make finding any value pretty fast, provided you can draw estimated angles)
@Mn M It’s not a good learning tool.
@@rli1618 It's a great learning tool. I learnt in an instant all the values I needed to know which I would have struggled with without that tool.
I struggle with arbitrary numbers and the actual maths behind it is too complicated.
@@rli1618 As someone doing a master in mathematics, it's a great learning tool. I have a lot of trouble with memorisation and this tool is the formost reason I know my trig as well as I do. Special cases is fun and all that but the only time you need the conversions is on a test and on a test the only thing that matters is ease of memorisation.
In my entire study of mathematics I have never seen any use of these nice special cases and have only ever known these values as some stuff to memorise.
I love how well you coordinated the recordings for the two Drakes at the end!
Alternate title: WhatsApp Group of Mathematicians Baffled by the Concept of a Mnemonic
I think most commenters here are missing the point, since the video mentions 2 other mnemonics that are arguably better. The thing is that this mnemonic in particular is kinda misleading, because it appears to be a mathematical pattern when it's actually just an useful coincidence.
It's even worse: *Mathematics Communicators
That trick at 9:14 is literally the same trick as the meme says
I actually noticed the whole sin30 = √1/2, sin45 = √2/2, and sin60 = √3/2 thing as a kid when I was taking trig, definitely made it a bit easier to memorize!
10:45 Ben Sparks' drawing is set to become an icon of educational maths for its simplicity yet meaningfulness. It could be named "maths in a rush". T-shirts NOW!!!
I independently noticed this pattern about 25 years ago in high school. It is a GREAT way to memorize this. That doesn't mean there is any deeper connection. It is a pneumonic, not a pattern. Like SOHCAHTOA.
SOH CAH TOA actually is a pattern, though
In highschool, I noticed this too, but I noticed it as sin30=sqrt(1/4) sin45=sqrt(2/4) and sin60=sqrt(3/4)
Which also made it easy to remember the cosines as well, because sin^2+cos^2=1
sqrt(2) / 2 is a way better way of writing it than 1 / sqrt(2) tho, can we agree on that?
No. 1/sqrt(2) is better, less numbers that aren't 1. It looks cleaner as a final answer. It is useless for calculating, but it looks nicer.
@@ethanbottomley-mason8447 If you think having a root in the denominator looks nice, I can't help you.
@@ethanbottomley-mason8447 For me it depends.
If I have calculations that have a sqrt(2) * sin(60) or sqrt(2) * cos(30)
then I would use 1/sqrt(2)
Else I would use sqrt(2)/2, depends on the calculations for me
@@Marconius6 Agree 100%. Standard convention is to always take your roots out of the denominator
Huge appreciation to the upper meme Matt for holding still the entire time.
Would've been cool if once upper Matt started moving, lower Matt had frozen in the 2nd pose.
Congrats on 1 Million
9:55
At this point he no longer reviews the meme, he *becomes* the meme
These values are commonly expected to be known by heart. To be frank, i think it's really unfair for teachers to demand a pattern be memorized and then disavow the easiest way to memorize it.
Educator: I found an easier way for the average person to remember something!
Mathematicians: *You godforsaken Plebian*
Similarly to what Mutual Information said, I think the value in this is just that it's really easy to memorize, though I'd say the same is true outside of exams too - because those are very common values.
Knowing how to derive them is of course better from a mathematical standpoint, and that's how I'd do it as well; however if you just want a quick and easy way to remember these values and don't care it doesn't generalize, this is a good way to do it imho.
(Also regarding the finger technique, I wouldn't say it's that much better, considering that you can get the exact same thing by knowing that sin and cos are symmetric within that quarter circle)
"Hun, how many bananas do we have left?"
"The square root of four divided by two!"
"Ok thanks!"
petition to have "maths meme corner" a regular series on this channel.
That final method of drawing the two triangles is what my math teacher taught me and what I use every time! I use my thumb and finger to remember which corner is 60 and which is 30 :P
Wow I just noticed the beard growing when I was jumping around the video. What a subtle feature... I admire the determination that must have been involved in pulling it off. (Or shaving it off, perhaps.)
Actually, there *is* at least one proper pattern:
If the sine of an angle alpha equals sqrt(x)/2, then the sine of the complementary angle pi/2-alpha equals sqrt(4-x)/2. Same for cos, by the way.
So, the left-right symmetry in the table is not just coincidence, it is real. That also holds for other angles, say
sin(pi/8)=sqrt(2-sqrt(2))/2
so
sin(3pi/8)=sqrt(2+sqrt(2))/2
Or
sin(pi/5)=sqrt((5-sqrt(5))/2)/2
so
sin(3pi/10)=sqrt(4-(5-sqrt(5))/2)/2=sqrt((3+sqrt(5))/2)/2
which simplifies to (1+sqrt(5))/4.
And another similar pair for sin(4pi/5) and sin(pi/10).
Not to mention
sin(pi/12)=sqrt(2-sqrt(3))/2
and
sin(5pi/12)=sqrt(2+sqrt(3))/2
Some of these pairs are quite elegant.
(Addendum: watching the 2nd half, this is the nice symmetry Matt talks about, in a slightly different and completer way.)
Well if one goes further and writes it using a reverse order 4 to 0
sin 0⁰: √(2²-4)/2
sin 30⁰: √(2²-3)/2
sin 45⁰: √(2²-2)/2
sin 60⁰: √(2²-1)/2
sin 90⁰: √(2²-0)/2
then one can see the pattern for the sides of the triangle they come from since sin x = O/H = √(H²-A²)/H, where H=2 so that all these values come from right angled triangles with hypotenuse 2 or alternatively from a circle of radius 2. But that's probably not how mnemonics work.
worth noting that the pattern is like this in part because:
sin²(θ) + sin²(τ/4 - θ) = 1
so we are seeing three special cases of that:
(√0/2)² + (√4/2)² = 4/4 = 1
(√1/2)² + (√3/2)² = 4/4 = 1
(√2/2)² + (√2/2)² = 4/4 = 1
In a way it’s quite arrogant and condescending for a bunch of maths educators to have sat around laughing at the masses going through the education system for finding a way to help themselves quickly recall some information that will help them get through life more easily. The vast majority are not going to derive any meaning from a wavy line or some right-angle triangles, they’re going to respond to the sequence the same way they respond to putting a finger down to find a 9 times table, or counting knuckles to work out which months have 31 days, or holding out hands to work out which one “says L for left”. Maybe it’s not a real sequence or pattern, but it works. That’s all that matters. Kinda disappointed to hear Matt and Katie and others stoop to this kind of mockery.
I was actually a little proud of myself in school when I noticed this pattern myself to remember the values easier. I guessed that there wasn't some special reason/pattern behind it since 0, 30, 45, 60, 90 isn't equally spaced, but I never looked into it further because I figured there wasn't anything interesting underlying it. It does make it easier to remember, though.
As a Mathematician, I kept waiting for the part where I’m supposed to be “angry” about the meme
Honestly, you talked for 12 minutes without actually saying anything. The meme is literally just a useful mnemonic. You’re making this way deeper than it needs to be
It's a pretty good approximation for other angles if you take a weighted average. For example, 40 is two thirds of the way from 30 to 45, so its sine is around √(1+2/3) / 2. It's only 0.002 off !
This only works for angles less than 45 degrees because sin is almost linear for those angles.
In any case, sin(x) can actively be approximated as simply x if x is a small angle.
@@seav80 No, it also works for other angles between 45 and 60, or between 60 and 90 as well.
For sin(70°), the error is only around 0.02.
Engineer spotted
@@hareecionelson5875 Nope, not an engineer, I'm studying pure math. It's just that where there's smoke, there's fire. In this case, the fire is the approximation we get for the sine. It's not an infinitely precise approximation like Taylor series, but it's a lot better than the first few terms.
Yeah, the pattern is nice because it's easier to memorize (for people with math-inclined minds that can easily work out sqr(2)/2 simplifies to 1/sqr(2), the rest should be trivial to simply for anyone who can do that.) not because it reveals any underlying meaning.
You mean non math inclined right? I mean people will memorize things differently but i think if you're more math intuitive the logic is what you remember not the sequence. And you find the sequence back that way.
And there's no arguing that a series in a table is much worse logical reason than the way the unit circle works. Because the unit circle links everything in trigonometry.
What "simplifies"? I prefer (√2)/2 to 1/√2 any day; integer denominators are much more valuable than unit numerators.
@@SquintyGears but how do you quickly calculate the values, especially sqrt(3/4) from "the logic"? Do you calculate the infinite series? This is just a good way to save time in trigonometric calculations, by knowing some values. Maybe mathematicians do not like this, but most people who regularly use maths are not mathematicians. Compare for example Feynman, who considered knowing some values out of your head, to be really useful.
@@Henriiyy well outside of very specific tests that at the same time force you to know the values AND not have a calculator it's not very useful.
and that kind of exam isn't the norm because wtf are you going to do with sqrt(3)/2 in any other following calculation.
@@SquintyGears sqrt(3)/2 is way better than sin(pi/3) because you can use it further and combine it with other expressions.
I use maths every day and haven't taken a trig exam in many years, and it's still useful. 30,45 and 60 degrees are just very common angles.
Edit: yes, I could use a calculator, but when you use this all the time, it would be tremendously tedious to type it in every time, just like it's good to know the times tables instead of using a calculator for 8*6 every time.
This memorization pattern has led me to an interesting approximation for sin(x). The formula suggests that sin(x) = √(some f(x))/2, therefore we must find some f(x) = 4sin²(x). Using a polynomial interpolation formula for the values ((0,0),(30,1),(45,2),(60,3),(90,4)), we find f(x) = -x³/81000 + x²/600 - x/180, which is a nice enough expression. So, sin(x) ≈ √(-x³/81000 + x²/600 - x/180)/2 gives exact values for the listed above points, and gives nice approximations for the intermediate values. The maximum error is close to +2% between 5º and 15º and -2% between 75º and 85º and close to +-0.2% between 27º and 63º.
BTW, if we use the angle in pi radians, that is, z = x/180, the expression gets even simpler: sin(z) ≈ √(-z(72z²-54z+1))/2
I don't think I can trust the aesthetic sensibilities of a pi apologist when it comes to anything having to do with the unit circle. This is a man who wholeheartedly believes that representing a 3/4 rotation around a unit circle as 3/2*pi radians is more mathematically illuminating than 3/4*tau radians
Yes, exactly.
This is *the* scenario for which the case for tau is the strongest. Pi simply should not be used for radians, it completely obscures the obvious logic of the unit to stick a diameter-based constant in there.
@@brianb.6356 but i like pie :-)
You know, for a second I thought you were one of those people who denied the existence of pi and all irrational numbers.
yeah i know all those fancy tricks but i’ve never really gotten it down. but this is the one that really stuck with me, so i’d say it’s worthwhile.
You never fail to throw something out of left field Matt, love it.
the only thing in my mind is Matt doing the video almost backwards and slowly trimming his beard, it way too perfect I didn't realize until he put the jacket
Careful Matt you explained what ‘sin’ means in an easily understood way.
Maths teachers in schools have being trying to hide that information from students for decades.
Congratulations on your 1 million subs
I think it’s always worth mentioning _why_ students find these mnemonics attractive: bad maths education. For this one, specifically, the “bad maths education” is tests and quizzes that make you do algebra with memorized formulae and values, where the formulae are the trig identities and the memorized values are trig functions at these four values, so you can calculate an answer, which the quizzes require, for some reason.
Nice transition around 6:17 - beard transition was smooth, kept camera in the same location, and very slight difference in environmental lighting.
Also, that's a rad shirt - bit more obvious transition when it switches from white to black print.
First of all, who writes 1/sqrt(2) this is an awful non-standard notation, it's always sqrt(2)/2
Second, it's a very useful memorization tool.
I personally draw a quarter circle when I fail to remember which one has which value, vertical is sine, pick the middle, draw the angle, it's short, so it's 30°, right, sin(30)=1/2, thus cos(60)=1/2, and the opposites are sqrt(3)/2
As to explain I would actually use a 3d graph that looks like a spring, show from the front, and from the sides, and it's so good at explaining how the graphs relate to each other
I went down a rabbit hole yesterday because of an idea that popped into my head, and I now want to see your take on it. Complex prime numbers.
I love the subtle commitment to growing the beard over the coarse of filming, to tie in with the meme at the end. Well done Matt, that gave me a good laugh when I noticed. Wish I could give me 'likes' for this effort. Would Steve show the same commitment and shave his beard at the start of a video... I think not.
I assumed he grew the whole beard before making the video and then proceeded to film backwards.
Very clever production to get the point across.
I was so caught up in the math I didn't notice the beard later. Love it.
How I wish I did exams:
Draw the trig circle.
Simplify √0/2 √1/2 √2/2 √3/2
Plot them on the circle.
Use the circle for reference.
Write the exam answers below it.
But I didn't. I should have...
I do recall imagining a trig circle and rotating my pen in exams, but this mnemonic would have helped extra. At least to get me started.
You're not wrong. I did this every trig test, and I went through those parts _so_ fast.
I disagree. When I saw this "pattern" years ago I loved it. The reason being: it helps. A lot.
I am not against memorization in math as long as you know what is going on on the back stage so to speak. Once you do, whatever method you use to remember sine and cosine of the notable angles is valid. And this is a great one 'cuz it's easy. So easy even some more elderly people I help learning basic math have no trouble with it.
Alternate title: people way too serious about math freak out over a way for high schoolers to memorize sine values
More like remember, not memorize.
exactly
To be honest I don't even understand why people did to memorize sine values to begin with it's simple enough to draw the triangles these angles are derived from. I got through all of biophysics undergrad with this (can never remember which sin(x) value is 1/2 and which cos(x) value is 1/2)
can we just acknowledge that Matt had to grow that beard and record bits for a meme video along the way :) these details are a big part of why i like this chanel so much! really entertaining and informative as usual. thanks Matt :)
Or it was all recorded on the same day but in reverse order. Was it a reference to Steve Mould?
Even though it may be a taboo I remembered these values because one day I saw the values and in maths class for some reason square root of 1/4 is 1/2 came up and I realised the the pattern I used was root of 0/4, 1/4 … 4/4 so yea
I would not have had my appreciation of trigonometry if I didn’t learn it this way
A mnemonic I don't need to memorize is more useful to one I need another mnemonic to remember.
Chances are anything a mnemonic is going to help with is going to be 0 30 45 60 or 90 degrees anyway, and if none of those angles are the ones you're using, you're going to dig out a calculator and trig cheatsheet anyway.
This is a practical mnemonic for people who actually need to use their angles and be done with them 20 seconds later, not just appreciate them in abstract.
When I was a trig student I was greatly frustrated with how much of the math class was actually blind memorization of identities, but I was so grateful when my calculus professor showed me that pattern because that was a few less things I had to brute force memorize. So I have to disagree with all the hate towards it because half the "better" examples were exactly the ones I was happy to never use again. Great video though, just wanted to put in my experience as a student learning it.
I discovered a cool way to memorise the table back in highschool(it is an interesting fact on its own though). It was during Physics period that I noticed that I could use Malus' Law which is used in calculating intensity of light through a polarizer(I = I_0 times cos squared theta). If you square the sines of the angles 0°, 30°, 45°, 60° and 90°, you get 0, 1/4, 1/2, 3/4 and 1, respectively. Then I take their square roots to get what I want. Pretty neat!
5:21 "It makes sense looking at it geometrically why 1/3 gives you 1/2." Does it though? I mean you showed us that it sorta kinda looks like that but that didn't make me understand _why_ that's the case.
I know, right? If there's one thing my math schooling drilled into me, it's "figure not to scale". Is it really 1/2? Or is it 0.5001234?
Sure, the geogebra file does leave some room for suspicion, but if you look at the sides of a 30 60 90 triangle you can see clearly why the sines of those anles are what they are, using only the pythagorean theorem.
Someone didn’t learn about 1:2:√3.
@@cmyk8964 I'm not talking mathematics here, I'm talking didactics.
@@unvergebeneid Whatever you purport the word “didactics” to mean, I’m not talking anything over Grade 3 geometry when I say “When there is a triangle with a right angle and a 60° angle, the side between the 30° angle and the 60° angle is exactly twice as long as the other side of the 60° angle.” I mean, you can even derive it yourself by folding an equilateral triangle in half, for crying out loud.
„I love that! A third of a quarter is a half!“ 😂
Although I agree with the comments that this is a useful pattern for memorization, even if it's an artificial one, I also think teachers should make clear it's nothing more than a memorization tool, not some hidden mathematical truth in plain sight.
the discrepancy is due to use case and demographic. mathematicians hate it for lack of simplicity, but plebs love it for the sudden insight of the pattern. it's not for YOU, it's for those who never realized there was a pattern
Never have I had such strong disagreements with basically everything Matt has said in a video!
"A third of a quarter is a half" Oh boy, we're getting spicy today.
This video is created in reverse order from the way shown.
We can observe how the beard keeps increasing gradually.
Love your explanations.
That's the mental model I figured out in high school to memorize the pattern and reproduce them on demand. While it might not exist in a technical sense, it's a useful mental tool for remembering something years or even decades after you last used it.
I also used to teach a fluid flow model for the way electricity works for linemen apprentices. Voltage is pressure, resistance is friction, current is flow, and so on. People in physics forums used to rail against it because it's not exactly how it works, but they missed the point. The fluid flow module for electricity is a close enough approximation that it could be used for troubleshooting a downed power line after a storm, and that was all that was necessary. So the people who knew the exact physics of electricity could complain, but they were doing so from computers powered by a grid that was restored after a storm VERY QUICKLY because the linemen were able to troubleshoot using that model.
It's the mathematical equivalent of using an acronym to remember a series of words. Nothing more.
I used to draw a little right triangle with 1, 2 and √3 sides. I'm sure everyone knows that triangle. All I had to remember was that sin(30)=1/2 and I could figure out the rest from that.
You don't even need to remember that since if you split an equilateral triangle (x=2) in half you get the 1 and 2 naturally. That's how it's illustrated in the video.
You can also use the Law of Sines but that's assuming some further knowledge
Yeah, the math behind it is nice, but at no point did I get any points for that on exams. All I needed from this were the values and regardless of how weird the values were written, they were extremely easy to remember.
The lesson here is, it's really easy to anger maths communicators with silly memes.
π = 3
The subtle growing of beard with each cut just to recreate the Drake meme is just perfect
Not going to lie, I discovered this on my own back in high school and I still use it as my go-to mnemonic for values of sine at angles of 0, pi/6, pi/4/ pi/3, and pi/2. From there I can use basic trig properties to get any of the trig functions at any of the special values on the circle. Is it accurate to the heart of how trig works? Of course not, just like similar tricks like log(1) + log(2) + log(3) = log(1 + 2 + 3). But when I need to guesstimate something or simplify an expression (read: exam question), it's a lot quicker than re-deriving values from first principles and more reliable than pure memorization, at least for me. Every model is wrong and some are useful. I would hesitate before teaching it as an educator for precisely the reasons you point out though, and if I did it would be with great caution to point out it's just a mnemonic.
(Yes, I did read the pinned comment and your reply, and I agree with you both. I'm just pointing out my experience with the pattern since this is one I've used for a while.)
Mathematicians looking at a familiar set of values which have been expressed using their least common denominator: "There's no rational basis for writing it this way!"
Ok, ok, they also used the exponential analog of a least common denominator. You've caught them generalizing an existing mathematical tool to reveal a relationship which was not previously obvious. Better tell them to knock it off double-quick!
Wish I had known about this, I could've sent in the perfect version. Alas, now it is too late and nobody will ever know.
Jk for me it's jotting down a quarter circle and radians going up from 0/12 times pi to 6/12 times pi with 1/12 pi intervals. That gives the pattern 0-6/12ths pi for the angle. THEN, on the outside of the circle, come the sqrt(n)/2 ones, skipping the 1/12th and 5/12ths of pi. It's not about understanding from whence the values cometh (although I do use the circle just for that), it's a tool for memorizing them.
You don't draw a triangle for every piece of pythagoras so there's no need for the hard-to-remember square roots and integers.
Look at it from an information perspective. I can either remember 10 separate pieces of information per sine, cos and tan, OR I can remember 1 shape, 2 generating patterns and 2 exceptions. That's 30 pieces of information vs 5. It _is_ a superior way of remembering it, particularly with the quarter circle shape. It _isn't_ a superior way of teaching the origin of the values. But then, that's not what it's for.
I have tutored maths and futher maths GCSE & since the trig exact values need to be memorised there is an easy way of generating them from a table:
1. write the angles 0,30,45,60,90 along the top row
2. write sin in a separate column to the left of the angles, then the numbers 0-4 (inclusive) underneath the angles
3. write cos under the sin and then write the numbers 0-4 but backwards underneath the numbers in the top row
4. draw a big square root sign over the grid of numbers you have just drawn
5. Finally, draw a horizontal fraction line underneath the grid of numbers and put all of that over 2
Getting the values once you have this table is as easy as choosing the trig function and angle you want from the headings, then checking the corresponding number in the table, square rooting it, and writing it over 2. (eg for sin 60, look at sign and 60, check the cell they share, that being the number 3, then you have sin60=sqrt(3)/2 without a calculator)
It sounds long writing it out like this, but in actual fact once you know how it is constructed, you can picture the table in your head and get the values from there, which is a huge time saver in the exam. And in case you forget it you can always draw it out again.
Note: the way to get the Tan values from here (since they also have to be memorized) is using the identity tan = sin/cos
Thank you for failing to explain why the pattern is actually meaningless! I got to enjoy this entire video as self-satire.
Totally agree it shouldn’t be used as learning tool or by anyone who hasn’t understood trigonometry to a good degree, but I think this is a good mnemonic for the people who know the meaning well and have to just remember the value quickly. I still have to draw on paper (or in my mind, not my fastest skill) the 30-60-90 or 45-90 triangles to infer the right values, and from now on I’ll begin using this method, because it is faster in terms of recalling time. I won’t forget the mathematical concept, but it’ll be faster for me to get the right one