exp(ix) = cos(x) + i*sin(x)
[cos(x)]^2 + [sin(x)]^2 = 1
i*i = -1
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))
exp(i*2x) = cos(2x) + i * sin(2x)
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)
cos(2x) + i * sin(2x) = [cos(x)]^2 - [sin(x)]^2 + 2i * cos(x) * sin(x)
cos(2x) = [cos(x)]^2 - [sin(x)]^2
sin(2x) = 2 * cos(x) * sin(x)
Most often they are used for simplifying expressions. For that purpose it's not a matter of deducing what they are, but recognizing them in the expression, and spotting the opportunity to use the identity on the path to eliminating terms.
For that purpose one needs to have them memorized in order to spot the opportunity -on the carefully derived homework problem- in the wild.
If I'm stuck with a trig function I don't know what to do with, I just look on Wikipedia for a trig identity and don't worry about derivations or memorization.
This trick is good for exam settings though! My memory is terrible so I always relied on tricks like this in school
Yes, most high schools spend a few months teaching “trigonometry” as a system for simplifying formulas that was primarily useful historically to save time for human calculators working with pen/paper and a printed lookup table. In the modern world this is entirely anachronistic and the content could be profitably compressed down to a couple weeks, with the saved time spent on something else. The main remaining utility is as rather cumbersome, unpleasant, and unenlightening algebra practice.
In the same sense that doing this work by hand will build depth of understanding, I know that remembering the trig identities came before really understanding their relationships. And that also came far before I could truly understand Euler's formula, radians, etc.
"i know two things:
[cos(x)]^2 + [sin(x)]^2 = 1
i*i = -1
but can re-derive all the rest"
He said it relates the most important numbers and constants of the universe. He had it embroidered and framed on his office wall
The part that makes the expression flashy is that the rotation is written via its logarithm, which can be represented as the length of a circular arc (“angle measure”).
It’s a useful fact to know, but is vastly overrated.
So you need at least that one, since it is by convention, and not really derivable:
exp(ix) = cos(x) + i*sin(x)
[cos(x)]^2 + [sin(x)]^2 = 1
 OK, you can look at the taylor series expansion, but you need to remember the derivatives for this.
Because x^2 + y^2 = 1 defines a unit circle, if you perform x^2 + i * y^2 = 1, you get a unit circle over the real/imaginary planes.
Or to put it in a picture: https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Eu...
Remember the definition of sin / cos: https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and...
Once you realize that "the imaginary axis" is just an arbitrary 2-dimension extension field (and that "imaginary" is a very bad name for it), it becomes way easier to see.
Now if you had a 3-dimension graph (real, imaginary, and time), and then you project the unit-circle over the real + time axis, you get a sin (or cos) respectively.
Or to put it another way: there's a periodic nature of i:
* i = i
* i^2 = -1
* i^3 = -i
* i^4 = 1 -- Cycle-length 4
* i^5 = i
* i ^ x = i^(x mod 4) (In general)
This period itself forms a circle, moving from real-to-imaginary and back. (e^i^x) = (e^(ix)) therefore is periodic as well. And the most natural periodic cycle is a circle (which is of course, just sin/cos).
To make this "period 4" into a proper circle, multiply by pi/2.
i^(pi/2)^x == period 3.141592.... or the "circumference" of the unit circle.
e^i^(pi/2)^x == e^(i * pi/2 * x). Done.
(Though, yes, there's a bit of a connection with group theory, but not enough to help you.)
From there, you get extension fields from real vs imaginary already. (Ex: you can form a new extension field from x + y*j, where x and y are complex numbers), which forms a new periodic cycle.
I mean, deriving it all is hard because group theory is hard. I'm not sure if its because the tools "aren't there". Some super-AI or super-human probably can derive it all from those given facts.
But that's just a way to embed some of group theory. It doesn't actually help you much.
(It's similar to how you can use eg set theory to construct the integers. It's possible, but doesn't actually help you prove anything about interesting about integers that you wouldn't have been able to prove without embedding them like this.)
> I mean, deriving it all is hard because group theory is hard. I'm not sure if its because the tools "aren't there". Some super-AI or super-human probably can derive it all from those given facts.
Yes, but that task wouldn't be made easier by this approach compared to starting from just the group theory axioms instead.
e^(iθ) times e^(-iθ) = e^(iθ-iθ) = e^0 = 1
= (cosθ+isinθ)(cosθ-isinθ) = cos²θ + sin²θ
Why do people even have to memorize stuff like this? I simply don't understand what's wrong with referencing stuff you don't remember during a test. That's what we do in real life!
Incidentally, we can derive [cos(x)]^2 + [sin(x)]^2 = 1 from the fact that cos and i * sin are even and odd components of exponentiation with some base, like so: 1 = exp(ix) * exp(-ix) = (cos(x) + i * sin(x)) * (cos(x) - i * sin(x)) = [cos(x)]^2 + [sin(x)]^2.
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.
I learned it while reading the Wikipedia article Slide Rule, it mentions that two math educators David B. Sher and Dean C. Nataro invented a Prosthaphaeresis slide rule in 2004 - it's what a slide rule may look like in an alternative universe. Unfortunately, I still haven't seen it - I couldn't find the original paper, it doesn't seem to have a DOI, can't use Sci-Hub.
If you want to multiply two numbers, you can treat them as rotations, take the logarithm of each (i.e. find the angle measure), add the two, then take the exponential.
Mathworld says Prosthapharesis formulas "convert a product of functions into a sum or difference" , which resembles a property of logarithms (i.e. log(ab) == log(a) + log(b)). However I don't quite get how angle measurement itself is a type of logarithm.
Maybe something to do with Euler's formula?
This is a very handwavey statement. I would say a more precise statement is that the proper exponential and log maps are the "correct" thing to use to map multiplication to adidtion and vice-versa. They are exactly the maps that do this properly. The fact that cosine is a sum of exponentials means that you can write the inverse of cosine as a (slightly ugly) formula in terms of logarithms (arcos(z) = -i log(sqrt(1-z^2) + z). This is the sense in which the map to angle measures is a "type of log".
If you want to compute the product of two rotations, you can take logarithms and rewrite it as a sum of angle measures:
[x + i√(1 − x²)][y + i√(1 − y²)] = exp(i acos(x) + i acos(y))
[x + i√(1 − x²)][y − i√(1 − y²)] = exp(i acos(x) − i acos(y))
Now if you just look at the real parts, you can write:
xy − √(1 − x²)√(1 − y²) = cos(acos(x) + acos(y))
xy + √(1 − x²)√(1 − y²) = cos(acos(x) − acos(y))
Taking the sum on each side:
2xy = cos(acos(x) + acos(y)) + cos(acos(x) − acos(y))
This method is rather more cumbersome (due to rescaling and then converting the factors to rotations as an intermediate step) but fundamentally based on the same concept as:
xy = exp(log(x) + log(y))
So now we use that algorithm and you've just calculated where the ship was a few hours ago :-)
I could imagine these could be useful to be defined by computer libraries, as (see article) it would help reduce large reductions in accuracy for some calculations (e.g. by subtracting similar numbers, or wanting to calculate sin^2).
Oh and by the way the log tables supplied by the exam board also had some useful formulae printed in them we'd otherwise have to memorise. Not that that had anything to do with it...
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.
Firstly, that's not the case. If you want students to answer a problem with any practical application (e.g. money/finance, statistics), or more complex problems where the point is to follow the logic, being able to use the calculator in your pocket is great.
A perfectly reasonable assigned problem might be: add the 7.1% sales tax to this ticket price, do you have enough cash?
And secondly, number sense is easier to teach without depending on particular algorithms.
This is the same complaint as people grumbling about "common core" math, or saying we should go back to the basics of just rote learning times tables and long division.
Students are learning about number sense by different routes now - routes that don't end up with most students blindly trying to follow a particular algorithm with no sense of what is actually happening.
I wish I had learned times tables properly as a kid though. I struggle with things like 6*7 on a near daily basis for my work and general life. Not really for number sense but for not having to shift my attention to a basic math problem while I'm working on something else.
> This seems to imagine that the only thing being taught/worth teaching on the mathematics curriculum is number sense
First of all, number sense is extremely important. Probably the most important thing taught in primary/secondary math courses. But it is certainly not the only important thing.
Students can learn to use an electronic calculator in very little time. A person of average intelligence who understands the relevant math should be able to learn to use their calculator pretty much independently, and become fluent at it with a tiny bit of practice over a short time. There are very quickly diminishing returns to teaching “calculator skills”, because those are extremely shallow.
> And secondly, number sense is easier to teach without depending on particular algorithms.
I really have no idea which ‘particular algorithms’ you are talking about. Have you ever used a slide rule? It is a very flexible general-purpose tool.
> This is the same complaint as people grumbling about "common core" math, or saying we should go back to the basics of just rote learning times tables and long division.
No, it is precisely the opposite recommendation to those.
> A perfectly reasonable assigned problem might be: add the 7.1% sales tax to this ticket price, do you have enough cash?
This is a reasonable problem to teach students about for like 1 week at age ~10. If they learn how to do it using pen and paper, or a soroban, or mental arithmetic, or a slide rule, or a pile of loose pebbles, they’ll have no trouble accomplishing the same with a calculator. It is not a reasonable problem to spend 5 more years on. We are talking about exams for 15-year-olds.
I'm afraid you have a completely unrealistic expectation of the median student (which is reflected in your other comments as well). I have taught hundreds of students and I would think a handful of them could answer this at age 10.
This type of question first appears on the Khan Academy in Grade 7, i.e. targeted at 12-13 year-olds.
There is a significant body of prerequisite work in the earlier grades, some of which is limited by what is developmentally appropriate.
Students will need dedicated practice to recall this and most won't remember it after seeing it or mastering it on one occasion. It needs to be supported in the rest of the curriculum. You might be able to intensively teach it earlier, but it is hardly worth it, because they will completely forget when you intensively teach the next topic.
Many 15-year-olds will continue to struggle with this and, if they are well-supported, might be taught it as a step-by-step procedure in order to best score marks in the exam they need to. These students are unlikely to finish without a good concept of what they were doing - and might be more likely to pick it up as adults out of necessity.
If students have not practiced solving problems enough to handle combining these things until age 13, and many are still struggling with it at age 15 (with a calculator!), that’s a serious indictment of the entire education system.
Singapore was the only one I found where percentage increase is introduced earlier as a national standard.
It's mentioned in the curriculum for Singapore Primary 6, i.e. 11–12 year olds, for students in the higher of two streams for mathematics.
> There is a tendency among beginners to want to compute resistor values and other circuit component values to many significant places, and the availability of inexpensive calculators has only made matters worse. There are two reasons you should try to avoid falling into this habit: (a) the components themselves are of finite precision (typical resistors are 5%; the parameters that characterize transistors, say, frequently are known only to a factor of two); (b) one mark of a good circuit design is insensitivity of the finished circuit to precise values of the components (there are exceptions, of course). You'll also learn circuit intuition more quickly if you get into the habit of doing approximate calculations in your head, rather than watching meaningless numbers pop up on a calculator display.
While the graphing calculators were available we weren't told we needed them so, being on a tight family budget, I didn't have one. My sister, a few years later was told that she had to have a graphing calculator, which she claims she never used
Is there a difference?
I went to high school well after the rise of calculators. Indeed, only one of my maths teachers had even used a slide rule, so I had to figure the thing out myself. One of the first things was that the multiplication scale on a slide rule is a low precision log-table. The other is why most math and physics problems only asked for solutions to two or three significant figures. (While physics problems do deal with precision, the consistency of the precision is definitely suspect.)
Why game development? It's useful everywhere! I have never found a genuine usage of atan(), it's always atan2() that is natural to use.
[ x, y ] ↦ [ atan2(y, x), π/2 − 2 atan(√(x² + y²)) ]
But atan per se comes up all over the place if you start doing much work with angle measures.
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 :)
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.
The domain of a function is the set of all possible inputs. In particular, the domain of the inverse tangent consists of all real numbers.
The author means the range.
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
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.
For the secant and tangent, it originates from the geometry of the unit circle. The first image in the article explains it well.
And if the distances are known to be small it may be quicker to calculate the chord length.
> "Sanskirt for 'chord'
Sanskrit, you mean?
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.
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).
I found a 2-minute video of the sort of pocket adding machine I used,
(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.
Huh? It's trivial to make a slide rule that you can add and subtract on, you just rule linear scales on it.
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.
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.
var STRAWBERRY_PI = 1 + Math.PI;
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.
f: consumer electronics
s: frequency response
Having multiples of terms for each is definitely a hindrance to education. It makes switching between each use a mental headache.
Perhaps it's just my unconventional education (see username), but when I'm doing something like filter design, I really do end up using all of those at once. I'll do the on-paper design portions with s, the computational section of the design with ω, and then when I build the thing to test it, the scope or spectrum analyzer read out in f (if I'm lucky). So they're genuinely all there at once!
(Press the calculator icon for algebraic view)
> 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?
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).
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.
I am not fond of calling this stuff 'secret'.