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Obsolete trig functions and why we don't use them anymore (2013)
- chris_wot 6y agoI strongly recommend looking at the geometric proof of the angle sums formulae for sine and cosine. https://en.wikipedia.org/wiki/List_of_trigonometric_identities#/media/File:AngleAdditionDiagramSine.svg https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...
- anonytrary 6y ago"goobledegook = e + 5.6. Wow you didn't know about goobledegook? Let me teach you about this revolutionary concept." I majored in mathematics and can say I've never heard of versine and have probably used it many times without knowing. I don't think everything needs a name.
- desertrider12 6y agoI think the point of these was for when they read low-precision values out of a printed table and then multiplied them using slide rules. Even though the article doesn't say it, I can't think of any other practical reason to multiply by adding logarithms.
- todd8 6y agoGenerally, slide rules were only good for rough calculations (3 significant digits if you were lucky), but they did contain multiple scales providing multiplication, division, exponential, log10, ln, roots, sin, cos, tan, and hyperbolic trig functions. Notably one couldn’t do addition or subtraction with a slide rule; we used pencils for that. When more precision was needed, I and my fellow engineers would break out our copies of CRC’s reference tables. The better log tables allowed calculations to 5 or 6 significant digits. Using these were not unusual and all of us learned to use them in high school (before 1970 at least).
- lmm 6y ago> Notably one couldn’t do addition or subtraction with a slide rule; we used pencils for that. Huh? It's trivial to make a slide rule that you can add and subtract on, you just rule linear scales on it.
- todd8 6y agoOf course you are right, the slide rule works with logarithmic scales that are “added” together to do multiplication. Linear scales could be used to add, but the low number of significant digits (no more than three on standard slide rules) meant that it wasn’t worth it, and no engineering slide rule that I ever saw had scales for addition.
- gshubert17 6y agoBack in the 1970s I used a slide rule when 2-3 place accuracy was enough; or 5-place or 7-place log tables and a pocket mechanical adding machine (rather than by hand) when more accuracy was required. I found a 2-minute video of the sort of pocket adding machine I used, https://www.youtube.com/watch?v=ryST18JJ7VU https://www.youtube.com/watch?v=ryST18JJ7VU (The device is actually easier to use than shown. Using the colors next to the digits: if silver, pull the stylus down; if red, pull up and over the curve at the top of the column to the next digit. No extra or wasted motions are needed.) Five places were usually enough, but seven places were needed for some astronomical calculations such as eclipses. The $7 I paid for the log tables in the late 1960s would purchase about $50 today; and the adding machine was probably a couple of dollars back then.
- stan_rogers 6y agoThe log tables let you work without slide rules, which were themselves horribly expensive not too terribly long ago in the grand scheme of things. But yes, this was all about practical calculation, not mathematics for its own sake, that needed to be carried out quickly by people who wouldn't have handy access to machines that keep the dirty arithmetic details (and potential for error) hidden from the operator, whether that's your ship's navigator or the machinist with nothing but a sine bar and gauge blocks setting up a precision operation, and if you needed precision using a four-figure table book, the only way you were going to get it was if the derived operations were in the tables.
- bregma 6y agoThe converse is true: slide rules let you do calculations quickly and easily (since they hang in the leather holster on your belt) without having to carry an expensive and heavy copy of the rubber bible or similar work and stop and look up numbers then scratch the values down on paper and work the problem.
- stan_rogers 6y agoAgain, you're thinking of an era where slide rules were relatively affordable. That's mid-twentieth century stuff (although the key word is relatively; my first good slide rule cost more than a month's rent at the time).
- gumby 6y agoSlide rules are literally the physical embodiment of a log table (or several, if you had a multilog rule). You used them by physical addition.
- desertrider12 6y agoOK, now I understand that the logs of these values are for pencil+paper addition and subtraction, not slide rules.
- Waterluvian 6y agoIt kind of helps a non Mathie like me doing trig. But I'll ask fof one better: I would love an application that lets me plug in what I have (coords and angles), what I want, and for it to show me how to get it.
- anonytrary 6y agoDoes it? No one does this: var STRAWBERRY_PI = 1 + Math.PI; For example, no one doing math will use τ (2𝛑) and 𝛑 simultaneously. Choose one or the other, otherwise you are increasing the number of symbols someone has to learn to understand your math. Naming everything is overwhelming. Mathematics is about similarities, not differences. If the transformation is trivial, then don't bother creating a new symbol. It will just confuse you more.
- exmadscientist 6y agoI don't agree at all. This happens all the time. For example, electrical engineering has f, ω, and s, all only differing by basic factors of 2π or i. As far as I can tell, the main reason this gets done is to give commonly recurring quantities single symbols, to reduce errors in lengthy calculations. When those recurring quantities are meaningfully different, in ways that help prime one's intuition, that's all the more reason to do it.
- unishark 6y agoYou also save clock cycles by avoiding the extra calculation each time you compute 2pi or whatever by defining TWO_PI once. So it was lot more common in Fortran and C programming days.
- trevortheblack 6y agoAs an electrical engineer you'll typical use f, ω, and s, in different contexts and you'll learn each of those contexts separately. f: consumer electronics ω: circuits s: frequency response Having multiples of terms for each is definitely a hindrance to education. It makes switching between each use a mental headache.
- framecowbird 6y agoversine clearly needed a name once when it was in common usage. As in, "Hey Greg, can you help me? I can't find this computation in these tables. --- Oh, try the versine".
- deleted 6y ago[deleted]
- Sharlin 6y agoThe article explains pretty well why these named functions (or at least some of them) were once useful and thus had common names.
- acqq 6y ago> I don't think everything needs a name. And you are wrong, at least historically: The article points that without the access to the computers and to achieve numerical precision having precalculated tables of such functions was of direct practical use. Even if you personally and more recently didn't need that, the said names were needed and used, and that's why they exist. Had you been you born, for example, more than 100 years ago, you could have been one of the persons assigned to spend maybe years just carefully calculating (or just verifying) the tables of one of those functions which today are seldom referenced by a name. Or at least with a job which involved using such tables (and the names too). Also, in current computing practice, one can still write programs which produce wrong results if one is not aware of the limits of the partial numerical calculations, including these named. One doesn't have to know the names, but in some specific cases it could be needed to have separate functions calculating exactly the values of the functions named. As we start to discover again that the custom processors could calculate more with less binary digits, knowing this becomes even more important compared to the times where "just use doubles" was enough.
- doomlaser 6y agoatan2() is incredibly useful in game development. You can use it to determine the angle between objects, calculate the angle of a 2d vector, and other useful things. https://ryankubik.com/blog/atan2/ https://ryankubik.com/blog/atan2/
- SamBam 6y agoYup, I felt incredibly stupid about six months ago finding out about atan2 for the first time, given the number of times I've made stupid errors using atan, and then correcting those errors by manually checking what quadrant I was in.
- ajuc 6y agoSame. My favorite "looks simple but is hard" task is "given object position and angle and target position return +1 if the object should turn clockwise and -1 if the object should turn counterclockwise to face the target". I've got it subtly wrong so many times and using atan2 only solves half the problems. Now I just copypaste it from my previous game :)
- OskarS 6y agoIsn’t that just the sign of the cross product of the direction vector and the vector from the object position to the target (or the orientation of the bivector in geometric algebra)? No need for trig there, I don’t think.
- jacobolus 6y agoThis is a question of whether the target is to the right or left of the line where the object is pointing. As OskarS says, you don’t need trigonometry for this at all. Just take the wedge product of the two vectors, and check the sign. That is, if your direction is the vector (a, b), and the displacement to the target is the vector (c, d), you can compute ad – bc. If this quantity is positive turn left, else turn right.
- cmroanirgo 6y agothere is also sincos(), which is useful when you need both, such as building matrices. https://man7.org/linux/man-pages/man3/sincos.3.html https://man7.org/linux/man-pages/man3/sincos.3.html https://home.deec.uc.pt/~jlobo/tc/opcode_f.html https://home.deec.uc.pt/~jlobo/tc/opcode_f.html
- _vmyf 6y agobruh this is just adding constants to regular functions ((((top secret))))
- jhardy54 6y agoDunno if you know where the meme came from, but: https://en.m.wikipedia.org/wiki/Triple_parentheses https://en.m.wikipedia.org/wiki/Triple_parentheses
- _vmyf 6y agoohno I didn't know I swear. I just picked this up from somewhere
- framecowbird 6y agoDid you read the article?
- exmadscientist 6y agoMany of the commenters appear not to have, which is disappointing. I thought the article did a nice job of explaining what they are, and why they're no longer relevant.
- _vmyf 6y agoI actually got around to reading it now, I shouldn't have commented earlier
- kortex 6y agoVers- is just from versus "opposite/against". But versine isn't opposite of sine, if anything it's opposing cosine. I too am curious this how this naming convention arose.
- klodolph 6y agoIt is opposite the sine, it is at a right angle. Look at the circle diagram: the "sinus rectus" goes vertically and the "sinus versus" goes horizontally.
- nwallin 6y agoIt always bugged me that secant is 1/cosine, cosecant is 1/sine, and cotangent is 1/tangent. Like what does co- mean? Trigonometry never really clicked for me. I can remember the formulas and such, but I never really understood what they meant. It was an exercise in remembering but never knowing.
- magnio 6y agoThe prefix 'co-' stands for complementary. Two complementary angles add up to 90°. The cosine of a angle is therefore the sine of its complementary angle, etc. For the secant and tangent, it originates from the geometry of the unit circle. The first image in the article explains it well.
- deleted 6y ago[deleted]
- phaemon 6y agoIf you like visual explanations, you might enjoy this from 3blue1brown: https://www.youtube.com/watch?v=yBw67Fb31Cs https://www.youtube.com/watch?v=yBw67Fb31Cs
- jacobolus 6y agoWikipedia explains this clearly, https://en.wikipedia.org/wiki/Versine#Versine_and_coversine https://en.wikipedia.org/wiki/Versine#Versine_and_coversine
- deleted 6y ago[deleted]
- extremelearning 6y agoFor those interested in this topic, I wrote a similar blog post on the geometric interpretation of the various trig functions, which also includes some similar diagrams for the hyperbolic trig functions: sinh, cosh, tanh, sech, etc. http://extremelearning.com.au/trigonometry-visualised/ http://extremelearning.com.au/trigonometry-visualised/
- fuzzybear3965 6y agoA few sentences in. > "Sanskirt for 'chord' Sanskrit, you mean?
- extremelearning 6y agoLOL! tx.
- pacaro 6y agoI'm not that old, but I learned the haversine formula in a marine navigation class I took through (boy) scouts. Our instructor was a retired merchant navy officer who was somewhat upset that the curriculum didn't include it, and deemed that we should know it. My A-level (≈ AP) maths teacher was mildly amused that I knew it and digressed about log tables. We were (I believe) the last cohort to be issued log-tables, but given the choice in examinations between log-tables, slide-rule, or calculator, we all made the same choice.
- pacaro 6y agoI'm also mildly amused that the author failed to mention the hyperbolic functions, of which there is one for each of the regular circular trig functions
- nine_k 6y agoMaybe they are not considered obsolete?
- jacobolus 6y ago> given the choice in examinations between log-tables, slide-rule, or calculator, we all made the same choice Students would benefit greatly if given a slide rule instead of an electronic calculator for their exams. The former is an effective teacher which viscerally reveals crucial insights, while the latter is pedagogically almost useless; using an electronic calculator to solve problems consists of nothing beyond punching keys and looking at the result, given as a disembodied string of digits entirely out of any context. The slide rule probably even ends up slightly more efficient for the type of problems students typically see in high school, after a bit of practice. Precision is more than sufficient. Any assigned problem which needs a calculator would be more effective if the calculator were removed and the numbers involved were simplified to the point students could manage them mentally or with a trivial amount of scratch work. Where electronic computers really shine is in their programmability, and for that students should be working with a full programming language and a real keyboard.
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- netsec_burn 6y agoAn even greater secret: sinh, cosh, tanh. Don't tell anyone. This message will automatically self destruct in 5.. 4.. 3..
- lambdasquirrel 6y agoI came on here to say this too. If you need to hang something up with more than two wall hooks, cosh is your friend.
- waynecochran 6y agoI have an old CRC math handbook with these tables ... once you have calculators there was no need for these tables and all these alternate trig functions...
- lordnacho 6y agoHere's one I came across on Facebook the other day. I had no recollection of it from school, but it's pretty neat and it appeared in someone's cheat sheet of standard trig formulas with double angles and all that familiar stuff. https://en.wikipedia.org/wiki/Mollweide%27s_formula?wprov=sfla1 https://en.wikipedia.org/wiki/Mollweide%27s_formula?wprov=sf...
- sleepy_keita 6y agohaversine is incredibly useful when you're doing quick distance calculations between points on the Earth though...
- enriquto 6y agoYes, I used the haversine formula just a few weeks ago.
- pmiller2 6y agoBada bing, bada boom. I used to work with GIS data quite frequently, and I probably used this at least once a week at the time.
- secondcoming 6y agoBut first everyone must agree on the radius of the earth! And if the distances are known to be small it may be quicker to calculate the chord length.
- np_tedious 6y agoI rely on haversine distance quite a bit and feel sort of disappointed in myself that I never really looked up the background on it before now
- deleted 6y ago[deleted]
- brazzy 6y agoAnyone can explain the meaning behind this: > When the Onion imitates real life, it's usually tragic. linking to an article titled "Man Puts Glass Of Water On Bedside Table In Case He Needs To Make Huge Mess In Middle Of Night"? Is the word "tragic" in the OP article here used jokingly, or is it a reference to an actually tragic event I don't know about?
- deleted 6y ago[deleted]
- rgoulter 6y agoIt's a joke, not about an actual tragedy. A good reason to put a glass of water on bedside table is because you might be thirsty in the middle of the night. You might accidentally knock the glass over and cause a huge spill instead. So "in case he causes an accident" is silly. If in real life someone knocked over a glass of water that they put on their bedside table, it's just like the joke, and "tragic". (And an example of life imitating art).
- throwaway_pdp09 6y agoI used haversine to good effect in a commercial product a few years ago. I am not fond of calling this stuff 'secret'.
- Mauricebranagh 6y agoInteresting the article mentions using Log Tables and links to a video on their use. Log tables where printed booklets used as an alternative to slide rules . http://www.sliderules.info/a-to-z/log-tables.htm http://www.sliderules.info/a-to-z/log-tables.htm I did get taught them in a mech engineering class - this was just at the inflection point (late 70's) when everyone switched from slide rules to calculators
- scottlocklin 6y agoFun book I read earlier this year, which steps you through the history of trigonometry applied to navigation: "Heavenly Mathematics; the forgotten art of spherical trigonometry." https://www.researchgate.net/publication/276103942_REVIEW_of_Heavenly_Mathematics_The_Forgotten_Art_of_Spherical_Trigonometry https://www.researchgate.net/publication/276103942_REVIEW_of...
- bobbybabylon 6y agoFunnily enough, haversines and the lot can be found in Dover's Trigonometry Refresher. Had.to do some revision myself for back-to-school. O.G. Artillery measurement and spherical navigation can be revised in that book, while spherical trig. can be practiced in the 1954 edition Schaum's Plane and Spherical Trigonometry. Good stuff.
- benrbray 6y agoLet me teach you a secret: You don't need to memorize any trig identities, you just need to know Euler's formula and one or two more facts: exp(ix) = cos(x) + i*sin(x) [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 That is, the complex number exp(ix), which lies on the unit circle in the complex plane, has real component cos(x) and imaginary component sin(x). Consequently, the line from 0 to exp(ix) in the complex plane makes an angle of x radians with the real axis. Any complex number can be written as an exponent like rexp(ix), where x is the angle and r is the distance from the origin in the complex plane. Using this understanding, we can interpret complex number multiplication as a rotation in the complex plane: (a * exp(ix)) * (b * exp(iy)) = ab * exp(i (x+y)) Ok, the above was a digression, let's get to the trig identities. Let's derive the double angle formula. We can start with the point on the unit circle representing an angle of 2x, and use Euler's formula to simplify it one way: exp(i*2x) = cos(2x) + i * sin(2x) Ok, we can also use the fact that a^(bc) = (a^b)^c to write the same expression a different way, then FOIL it: exp(i*2x) = [exp(i*x)]^2 = [cos(x) + i*sin(x)]^2 = [cos(x)]^2 + (i*i)*[sin(x)]^2 + 2i * cos(x) * sin(x) = [cos(x)]^2 - [sin(x)]^2 + 2i * cos(x) * sin(x) Now, comparing the two expressions, we see that cos(2x) + i * sin(2x) = [cos(x)]^2 - [sin(x)]^2 + 2i * cos(x) * sin(x) Taking real and imaginary parts of both sides, we find that: cos(2x) = [cos(x)]^2 - [sin(x)]^2 sin(2x) = 2 * cos(x) * sin(x) This works for most identities you want to derive, and there is a variation for things like cosh and sinh. This trick came in handy on calc 3 exams whenever I couldn't remember the silly trig identities that were required to perform integrals.
- immmmmm 6y agoAs the math teacher at the first year of university once said: "i know two things: [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 but can re-derive all the rest"
- eru 6y agoGood luck deriving group theory from that. :) (Though, yes, there's a bit of a connection with group theory, but not enough to help you.)
- nautilus12 6y agoScrew you SA for blasting an ad at my sleeping family at 7 am
- jessaustin 6y agoNow is the time to improve your life: https://github.com/gorhill/uBlock https://github.com/gorhill/uBlock
- tracker1 6y agoIt's funny, this article only serves to remind me how much I've actually forgotten in practice. I don't think I've really done much consideration for trig since my sophomore year of HS. I didn't get into programming via a formal education, and the programming I do doesn't involve heavy graphics or physics so it's just not something I think about. Interesting article, and only serves to inspire me to add re-discovering trig/physics/calc to my todo list on reading.
- nashashmi 6y agoThese equations are very useful in circular hydraulics. It’s unfortunate I never learned this stuff.
- cdelsolar 6y agoI used the haversine formula quite recently, I believe in a previous job, for accurately drawing a map given GPS coordinates, or something like that.
- jessaustin 6y agoI'm sure I have things disabled in some particular way that SA site devs haven't envisioned, but it's annoying to click through the "Cookie Settings" modal that obscures half the page every time I open a new SA page. Probably the cookie that stores this decision could be classified as "strictly necessary"...
- deleted 6y ago[deleted]