Trigonometry Redefined without Sines And Cosines 966
Spy der Mann writes "Dr. Norman Wildberger, of the South Wales University, has redefined trigonometry without the use of sines, cosines, or tangents. In his book about Rational Trigonometry (sample PDF chapter), he explains that by replacing distance and angles with new concepts: quadrance, and spread, one can express trigonometric problems with simple algebra and fractional numbers. Is this the beginning of a new era for math?"
No sines and cosines? (Score:5, Funny)
Re:No sines and cosines? (Score:3, Insightful)
(a^2 + b^2)^(1 / 2) = c
to:
a^2 + b^2 = c^2
With a lot of just applying that paradigm to every aspect of trig. Pretty nifty time saver, but I fear the unique insights from this method may be few.
Re:No sines and cosines? (Score:4, Interesting)
What did I miss?
Re:No sines and cosines? (Score:3, Funny)
Rumour has it that one year a freshman ran out of time on a trig exam, stuck on the first question, trying to write out the square root of two..
Re:No sines and cosines? (Score:3, Insightful)
Re:No sines and cosines? (Score:5, Insightful)
Re:No sines and cosines? (Score:5, Insightful)
Parent is factually incorrect (Score:4, Informative)
He didn't replace distance with angles, nor is one of the two "the fundamental element of trig" - they are together the fundemental elements of (standard) trigonometry.
What he actually did was that he replaced distance with distance squared and angles with sine squared.
Distance-squared and dx/ds, dy/ds (Score:3, Interesting)
I have done programming involving coordinates and trig from time to time - originally, stuff like finding where a line is clipped by a polygon. I can remember and do SohCahToa, but that was close to the limit of my knowledge of trig.
The big problems that I found, while trying to write the code, were positive versus negative angles, infinite-angle of vertical lines, and having to calculate
Re:Read the Article (Score:5, Interesting)
Ironically, you've actually harmed, not helped, anyone who hadn't read the article. The main point of this work is that distance AND angle are the wrong things. The quadrance (square of distance) and spread (effectively, the square of the sine of the angle -- though help points out 'angle' is a handwaving concept to begin with) should be the fundamental elements and makes the trigonometry meaningful and easy. That quadrance and spread are (independently) functions of distance and angle is trivially obvious, but it is exactly the difference that is his point that makes the math so much simpler.
Perhaps a more specific example for those programmers out there. No more need for lookup tables or Taylor expansion for trig functions unless a human at some point needs an angle value at which case it can be converted in the final step before output. (Internally, programs would never need them.)
Re:No sines and cosines? (Score:5, Funny)
Re:No sines and cosines? (Score:5, Funny)
Re:No sines and cosines? (Score:5, Funny)
Re:No sines and cosines? (Score:3, Funny)
Re:No sines and cosines? (Score:5, Funny)
Now ... (Score:3, Funny)
Re:Now ... (Score:5, Interesting)
This reminds me of a test in grad school (Score:5, Interesting)
I got no points at all for the question - despite solving the parts relevant to the class - because I didn't know off the top of my head that the integral of that is the arctan function.
I love abstract math but I hate trig.
Re:This reminds me of a test in grad school (Score:3, Informative)
If you have some experience in solving integrals of that sort, the substution x = t is pretty standard.
In this case letting x = tan t is very productive. Working through the algebra one finds that (TeX notation)
Just remembering $\tan = \sin/\cos$ and $\cos^2 t + sin^2 t = 1$, on can work out the following:
We have $1/(1+x^2 = 1/(1+\tan^2 t = \cos^2 t$
Also $dx = 1/cos^2 t dt$, therefore
\[
\int_0^a \frac{1}{1+x^2} = \int_0^{\tan^{-1} a} 1 dt = \tan^{-1} a
\]
So you don't have to remember the form of the integ
Re:Now ... (Score:5, Interesting)
Re:Now ... (Score:4, Informative)
Re:Now ... (Score:5, Interesting)
e^(ix)=cos(x)+i*sin(x)
=> cos(x)=(e^(ix)+e^(-ix))/2
=> sin(x)=(e^(ix)-e^(-ix))/(2i)
Once you know this, then integration and derivation of all sin/cos and derived functions boils down to algebra and derivation and integration of e, which is trivial.
I cannot tell you how angry I was with them for not teaching me this until well after integral calculus.
Re:Now ... (Score:5, Informative)
For example, instead of working on a Euclidean affine coordinate system, by using "Quadrance" as he calls it, the coordinates would not be translation invariant, and you will be forced to attach a non-trivial measure to make integrals work out. So while the integrand might be simplified in the trigonometric identities, you will end up, instead of integrating over "dx", over something like "1/sqrt(x) dx", which hardly makes the integral any more appealing.
Re:Now ... (Score:3, Insightful)
The method doesn't matter, as long as the answer (Score:3, Funny)
Wonderful! (Score:5, Insightful)
I did a stint as a Maths teacher, and it was hard enough trying to convince the kids that it was worth learning Trigonometry then. They'll be even more determined to be ignorant if they hear of this.
Don't worry... (Score:3, Insightful)
Now sometimes while slogging through my day, paying bills, shopping, working and things like that I sometimes say "Man I really should calculate the cosine of this electric bill", but as of yet I haven't been harmed by not actually doing that.
Re:Don't worry... (Score:4, Insightful)
Not just physicists or engineers use trig.... (Score:5, Insightful)
agreed (Score:4, Insightful)
Re:Not just physicists or engineers use trig.... (Score:4, Insightful)
Number one: The people using the device still have to know the math. Suppose you want to calculate compound interest using the formula:
[P(1+r/n)^(nt)]-P
Several times I have seen college students fail to produce the correct answer even armed with a textbook, this formula and a calculator. Why? Because they don't understand the math.
Number two: Calculators lie virtually always. Often they produce answers that are "good enough". However, without an understanding of maths, "good enough" typically means "whatever the calculator reported as an answer". Example: what is the sum
10000000 + 0.00000001 ?
The correct answer is 10000000.00000001
The calculator's answer is 10000000.
Re:Not just physicists or engineers use trig.... (Score:4, Insightful)
The correct answer is 10000000.00000001
No, the correct answer is 10000000. Each term has only one significant figure, so after truncating to the correct precision you get the calculator's displayed answer. Although many calculators will have the inaccurate figure (10000000.00000001) rounded to the nearest base-2 floating point in memory and a long-enough fixed-point format will display it.
Now, if you had said 10000000.00000000 + 0.00000001, then the correct answer would indeed be 10000000.00000001.
Re:Not just physicists or engineers use trig.... (Score:3, Insightful)
Sometimes you can take precision is implied but you have to be very carefull with doing so especially with integers after all is 100 a number given to 1 SF or a number given to 3 SF where two of the digits happened to be zero? you can't tell!
When using a calculator or computer you have to understand that the response won't be an exact answer for most functions and you need to know if t
Re:Not just physicists or engineers use trig.... (Score:3, Insightful)
Re:Not just physicists or engineers use trig.... (Score:4, Informative)
So your complaint basically boils down to this: "carpenters don't need to know trignometry, they only need to know Rational Trignometry".
Re:Don't worry... (Score:5, Insightful)
Luckily its a great store for Physici...
Do you need a cart sir?
Re:Don't worry... (Score:5, Insightful)
Re:Don't worry... (Score:3, Insightful)
That actually happened about the time when high school was created. The masters of determining cirriculum were standing in the balance - they could create a model of school that encourages kids to think critically, and to focus on the process of thought and reasoning.
Or they could take the "student as shoe, randomly-assembled array of facts and figures as foot, us as shoehor
Re:Don't worry... (Score:5, Insightful)
Simply because you choose a profession does not use it, does not mean it doesn't have value.
Re:Don't worry... (Score:5, Insightful)
As an aside: I'm always amazed how many people who do sciences and other technical stuff are always interested in many things, like music, politics, aesthetics, social structure...but hardly any political science or sociology student has even a passing interest in the sciences. I'm starting to believe that the latter are just to stupid to realise how much of an impact those things have on their life.
Re:Don't worry... (Score:5, Informative)
Yes. But then, I live in the Netherlands. Our system is different from the US; instead of (basically) lumping everyone in the same school and making sure they all pass, we send people to different types of high school, based on how they perform on a test at the end of primary school. I went to the highest types of high school, where you get at least 3 years of math IIRC; I took math for the full six years of the program. In the other other types of high school, you get less math because (1) they last shorter, and (2) they tend to focuse more on practical issues than on theoretical ones.
Re:Don't worry... (Score:3, Insightful)
A lot of the point of learning math is keeping your options for the future open.
Re:Don't worry... (Score:3, Insightful)
Maybe someone should make a list of professions you rule out doing well at if you don't learn about trig -- I don't think it's just scientists, carpenters, surveyors, engineers, mathematicians, navigators,.. that need trig.
I don't understand why Math gets singled out so badly. How many people need to use the details of history in life, after all, and Literature classes , Speech classes are also of dubious merit, after all -- most people won't be historians, professional writers, speakers, or politici
Re:Don't worry... (Score:3, Funny)
Re:Don't worry... (Score:4, Insightful)
This sounds like a variant of trig calculations that you often use in computer algorithms, where it's much faster to calculate the square of something than the root. If you do it right, you can avoid roots completely for comparisons, and only do one at the very end of the calculation for actual lengths and distances. Sine and cosine usually appear as the quotient of lengths of sides of a triangle--you rarely calculate sin(x) or cos(x). The one place where roots are unavoidable is normals, which are just so damn handy. But even there you can sometimes get away without normalizing for comparisons in things like backface culling.
Re:Wonderful! (Score:3, Insightful)
I think one big problem is that people are given the impression that mathematics has something to do with the real world, and that it's supposed to be "useful". (Well it is, but not for the obvious reasons.)
Mathematics really just consists of a bunch of structures. These structures can be really quite beautiful on their own, and if it's presented the right way, people should see some reason to study mathema
Figures. (Score:5, Funny)
The "New" has an initial capital for a reason (Score:3, Informative)
Wow (Score:4, Interesting)
If this book pans out, it would ultimately change Calculus for the better and might allow me to pass my classes- I wonder if I figured out how to apply this book to my classes and came up with the correct answers despite having worked the problems with his methods, if I'd still pass the class?
Re:Wow (Score:5, Insightful)
If you want to be the kind of engineer that implements other engineer's ideas then, by all means, blow off your math classes. But if you want to be someone who your peers turn to when they need help, do yourself a favor and learn the math.
All of the engineering sciences are founded on math (this is espescially true of computer science). If you can out code your instructors, that means you can probably out math them too. What you are interpreting as an inability to memorize functions, is probably really just disinterest.
This disinterest may stem from a feeling that what you are studying has little utility, it may stem from a personal dislike of an instructor, it may stem from the notion that math geeks are all squares and smell funny.
Whatever the reason, you need to get past it. A thorough understanding of the math behind engineering will make your life MUCH easier on down the road.
Re:Wow (Score:5, Insightful)
And you are wrong about algorithms. Algorithms is math. No ifs, ands or buts about it.
Re:Wow (Score:3, Interesting)
None (well, very little) of this needed maths.
Algorithms are *not* maths. Why should they be? Anyone can derive something like a bubble sort from first principles without the use of a calculator. A binary search is intuitively obvi
Re:Wow (Score:4, Interesting)
The only reason people don't realize this more is because most of the really hard stuff is already worked out for them. If you were stuck coding in assembler with no libraries to help you out, you'd realize how much math there is under the hood.
Re:Wow (Score:3, Insightful)
You could apply this to pretty much any field with just as much stretching. The music scale works on logarithmic principles, and don't get started on how complex the calculations can be for what makes a good instrument, or good acoustics. That doesn't mean you average cello player need to know anything about sines and cosines.
Re:Wow (Score:3, Insightful)
Some of them sucked.. some of them were really good (IMO).
And that is all you can get without using maths to prove that they are good or bad, be it in the average, best or worst case.
Re:Wow (Score:5, Interesting)
But first, this small reminder:
sin x (vertical component)
cos x (horizontal component)
tan x = sin x over cos x
sec x = 1 over cos x
csc x = 1 over sin x
cot x = cos x over sin x
-> sctsct
Now we substitute these trig functions with simple symbols:
I = sin x (vertical component)
II = cos x (horizontal component)
III = tan x = sin x over cos x
IV = sec x = 1 over cos x
V = csc x = 1 over sin x
IV = cot x = cos x over sin x
Think of them as one, two, three, four, five, and six. Now it boils down to remembering simple combinations of numbers:
integral{ I } = -II
integral{ II } = I
integral{ III } = ln | III + V |
integral{ IV } = ln | IV + VI |
integral{ V } = ln | III |
integral{ VI } = ln | I |
Once you write down more of these combinations, you'll discover patterns in it and from there on it should be easier than ever to remember trig integrals. And when you know most of the trig integrals, you will know most of the trig derivatives, too!
If anyone is interested in some more documentation on this, then by all means contact me. I am in the process of writing these things down. They should be available some time next week on my homepage.
Or you could just sketch the functions (Score:3, Informative)
Re:Wow (Score:4, Insightful)
I don't know quite how to put this, so I am just going to say it.
The degree doesn't make you an engineer. The MATH makes you an engineer. The degree is just your univerity vouching that you have completed your math and other engineering studies competently.
In my opinion, I think the author of this book is a quack and all I had to see was the first paragraph on the first page of his web site where he states that he has dispensed with (geometric) axioms. You cannot do anything in mathematics without axioms. Period. Math is not capable of proving something from nothing.
UNSW .. not South Wales (Score:5, Informative)
Re:UNSW .. not South Wales (Score:2)
Maybe "People can't think" is better. But I am leaning more to "People can't critically think". If they did .
Hopefully (Score:3, Insightful)
Just Wait... (Score:4, Interesting)
Better LInk (Score:2, Redundant)
Re:Better LInk (Score:2)
Redefinition? (Score:3, Insightful)
He redefined a side of a triangle with a Quadrance - a square of distance. He claims this removes the square root, but guess what? d^2 has the same effect.
He also defined a spread, which is the relationship between two lines. The catch? It is PROPORTIONALLY EQUAL to an angle. 90 degrees is 1, 0 degrees is 0, 45 degrees is 1/2.
I haven't read the full book, but from what I can tell, all this is doing is redefining the constraints of trigonometry, causing nice even numbers to be used.
Re:Redefinition? (Score:2, Insightful)
Re:Redefinition? (Score:5, Insightful)
spread is the square of the sine of an angle.
Re:Why are there 360 degrees? (Score:3, Funny)
Re:Why are there 360 degrees? (Score:3, Insightful)
The base 60 number system of the Babylonians was successful enough to have worked its way through time to appear in our present day modern world. We still have 60 minutes in an hour, 60 seconds in a minute, 360 degrees in a circle and 60 minutes in a degree. Even our 24 hour clock is a legacy from the ancient Babylonians.
Re:Yeah, I looked it up a while back (Score:3, Insightful)
There are proposals to change time, but like changes to the English alphabet, the benefits have to outweigh the almost impossibly large costs of transforming. Look at the English/Metric systems. The USA still
Re:Why are there 360 degrees? (Score:3, Informative)
Uh... (Score:2)
Uh... that's not just redefining trig, that's totally redefining mathematics and logic. I find that hard to believe. Is it just marketing talk? Or did this guy revolutionize the axiomatic system upon which we built all human knowledge? I find the latter doubtful.
And it shows how to apply this new theory to a wide range of practical problems from engineering, physics, surveying and calculu
Re:Uh... (Score:2)
SOHCAHTOA and abstract survery results (Score:5, Insightful)
Cos = Adj/Hyp
Tan = Op/adjacent.
By as someone who's done some surveying it (which uses angles and distance), its easy to see why people use angles and distance. They are fairly easy to measure, so usefull for plans etc..
Replacing angles and distance with abstract quadrance and spread is exchanging one difficult thing (tan,cos,sin) for another (quad, spread)
Quandrance = distance ^2
Spread hard to see.
Re:SOHCAHTOA and abstract survery results (Score:3, Informative)
(1) take any point A on the first line L1. Denote qudrance between O and A is Q.
(2) compute the shortest distance between A and the other line L2 and square it for quadrance (call the quadrance R)
(3) spread between L1 and L2 = s(L1,L2)=R/Q
Calcula
Faster calculations ?? (Score:5, Interesting)
in raytracers and 3D engines by using integer numbers.
Re:Faster calculations ?? (Score:3, Interesting)
Yes, for some badly written code (Score:5, Interesting)
If you implement a geometrical function whose input is a bunch of lengths and their ratios and whose output is a bunch of lengths and their ratios then you frequently shouldn't need any trig in your function, instead you should be using algebraic functions (+, -, *, /, nth root etc.). But many people find it easy to solve their problem by converting a bunch of stuff to angles using trig and then converting back. In fact, I optimised someone's rendering code recently by literally looking for every line of code where he'd used trig and replacing it, where possible, with the algebraic version. It worked well and speeded up the code massively. In addition I uncovered what you might call bugs. For example at one point this guy did a 180 degree rotation by converting to polar coordinates, adding M_PI to the angle and converting back. Rotating (x,y) by 180 degrees gives (-x,-y). The guy who'd written it was 'trig happy' and should have stopped to thing about the geometrical significance of what he was doing rather than just doing the obvious thing. If this book encourages people to use trig less it might be a good thing. But you don't need to talk about 'quadrance' and 'spread' to use the principles I'm talking about.
But of course there are times when you need trig. I'd hate to see the guy differentiate rotations without trig.
Re:Faster calculations ?? (Score:4, Insightful)
Furthermore, a lot of what this guy did is kind of a trick. Using 'spreads' may work when given an explicit triangle, but the part he's skimming over is that spreads are missing a REALLY nice property of angles. They don't add. Angles are a very nice parameter for rotation because a rotation of 10 followed by a rotation by 20 is the same as a rotation by 30. This property is implicitly used all over the place in graphics. So, in the end you probably have to use some angle-like measure when doing computer graphics (which is all about transformations, not measurements of unknowns). And in doing so, I'm sure you end up computing sines and cosines to do projections based on those rotations.
In the end, you just can't cheat your way out of the fact that a projection based on a rotation is a transcendental operation that numerically requires computing a truncated infinite series.
Interesting - but not entirlely new (Score:4, Insightful)
I read the article and part of hist concept of spread for an angle is the ratio (in a right triangle) of the side opposite the angle to the hypotenuse of the triangle. As I recall from my high-school geometry classes, though, this is exactly how the sine is calculated. (The cosine is the adjacent side divided by the hypotenuse.)
The interesting thing about his approach, though, is that he defines the concept of spread so that there's no need to refer to the underlying sine concept and the calculations are all (relatively) simple algebra - squaring and addition and subtraction. I would like to read some more and play with it a bit - the fact is I still have bad dreams about those damned trig identity tables, which I never really successfully felt comfortable with!
Interesting.
Great for eighth grade, but ... (Score:5, Insightful)
Re:Great for eighth grade, but ... (Score:3, Interesting)
This stuff is junk. On page 8: Square roots are to be avoided whenever possible.
Followed by page 16: To convert back to distances, take square roots.
He claims that sines and cosines are hard because the poor student can't calculate them by hand. How many here can extract a square root by hand?
I don't see how this is "easier" (Score:5, Insightful)
Bah!! (Score:3, Funny)
Is this silly? (Score:3, Interesting)
quadrance is the square of the distance
spread is the square of the sin angle
If you want to teach trig this way, it will certainly be easier to learn. But then, your students will not appreciate or understand square roots and negative numbers.
It is highly unlikely to lead to any revolutionary new algorithms. It appears that you are changing an angle to something rational, because all most of the popular sine values have square roots in them. However do not be fooled by this, it is just as hard to calculate spread as it is to calculate the sine of an angle. (except in a few cases).
Galileo got there first (Score:3, Interesting)
In the same way, as I understand it based on the sites referenced in the story, this guy makes use of an angle replacement that is not useful to most people. If I want to divide a circular cake into n portions, using spread will not be helpful. Although degrees are a unit with no rational basis(there are not 360 days in a year, and 480 might be better-divisible by 5,3 and powers of 2) they are relatively easy to use for circle division, which is all the circle math that most people need to do most of the time.
Unfortunately, mathematics lectures at Cambridge left me with the permanent belief that mathematicians' ideas of what is simple and what is complex merely illustrate the fact that physics and engineering use math, but they use a useful subset (which changes with time) and do not necessarily buy into ideas which mathematicians regard as self-evident.
I looked at one of his first examples (Score:3, Interesting)
What a silly non-sense (Score:3, Insightful)
Here is what he does: He replaces the distances by its square, and calls it a squandrance. He replaces the angle by the square of its sine and calls it a spread.
Ok, the relations between the squandrances and spreads of a rectangular triangle are simpler than those between its sides and its angles -- they are just simply obtained from pythagoras' theorem.
However, two way more fundamental relations suddenly become horribly difficult: Say going from town A to C I go from A to B then from B to C. A to B is 5 squandrance, B to C is 3 squandrance, how much squandrance is it from A to C? No, not 8, but 5 + 3 + 2*sqrt(8). The simple addition of distances becomes a square-root function...
Also, say I turn left by 30 degree, and then do that again. Guess what, I turned left by 2*30 = 60 degrees. However, if you were doing this in spreads, you don't want me to tell you the answer. I think there is a reason why sailors have been using angles and not spreads...
As for engineers: Why do they have to learn trigonometry? Not for geometry, you can just do that with coordinates. But because it turns up in waves everywhere, which in turn turn up everywhere. (Aside: the mathematical reason is that sine and cosine are the solutions of one of the most fundamental differential equations.) For example, the voltage of an AC current will depend proportionally on a sine fucntion. The angle you substitute in sine will be proportional to time. The spread you would have to substitute in a rational geometry function instead is NOT proportional to time, but to the square of the sine of time. Too bad my stopwatch measures time in seconds and not in the square of its sine...
Will his approach lead to faster computations? Of course not. Whereever it would help, people of course already knew last year already (and maybe even, gosh, in 2003) to use some trigonometric substitution....
Most of you missing the point. (Score:5, Insightful)
I am a high school mathematics teacher and I train students for mathematics competitions. I think most of you are missing the point of Dr. Wildberger.
Dr. Wildberger is not trying to redefine trigonometry, he is simply trying to give it a new perspective and hopefully, the new perspective would allow new insights into new methods of solving trigonometric problems. Protesting that memorizing the trigonometric functions as side adjacent over side opposite, etc., etc., is very easy and intuitive ignores the fact that in analytic geometry, that is not even how the trigonometric functions are defined!
Yes, really! For example, the sine of theta is defined in analysis as the y component of the radial vector from the origin to a point in a circle of unit radius whose arc distance from the x-axis is theta. The cosine of theta is defined similarly but this time taking the x-component. From this two simple definitions, the entire panoply of the trigonometric identities can be usefully derived!
The analytical definition is certainly not intuitive and not easy to memorize for a high school student! The side opposite, side adjacent trick is just that, a trick that is useful sometimes and certainly useful enough for high school mathematics but it is not a very useful definition as far as analysis is concerned.
For example, computing the derivative of the cosine function is not easy to understand if you restrict the definition of cosine to side adjacent over hypotenuse! Not to mention the fact that most students think there is magic involved in the computation of the trigonometric functions because the method of computation is not in their textbooks. It is only when one studies the calculus that the methods for computing the trigonometric functions are explained!
Dr. Wildberger has an idea that he thinks will make trigonometry more intuitive and I hope he is really onto something here. It would certainly help me with my students. I have read only the downloadable first chapter of the book and the idea is intriguing. Waving off Wildberger's new ideas without reading the entire book and without understanding the mathematics of trigonometry is just tragic.
In times like this I always remember the architect (I forgot the name, help me out here please) who refused to accept an architecture medal because the society that was giving out the medal invited Prince Charles to hand out the medal. That architect said, "I refuse to accept a medal from a person who believes that our grandfathers already know everything there is to know about how to build buildings and that there is nothing we can ever add to that knowledge anymore."
Just my two cents.
No, I think you miss the point (Score:3, Insightful)
Very nice. Makes sense to a game programmer (Score:5, Informative)
Wildberger has put a cleaner theory underneath the kind of geometry game engine developers use. This may turn out to be useful.
Lately I've been doing robot motion planning, which has too much unnecessary trig in it. With enough work, it's often possible to derive a trig-free solution to some of the key problems. Better ways to think about trig-free solutions will help.
But if you graduate at Dr. Wilberger's group... (Score:3, Funny)
Bert
Trig is not hard, it's just taught REALLY badly (Score:3, Informative)
Euler's equation:
e^(i*x) = cos(x) + i*sin(x)
Need a double angle formula? No problem.
e^(i*2*x) = cos(2*x) + i*sin(2*x)
e^(i*2*x) = (e^(i*x))*(e^(i*x))
= (cos(x) + i*sin(x))*(cos(x) + i*sin(x))
= (cos(x))^2 - (sin(x))^2 + i*2*cos(x)*sin(x)
So you can clearly see that
cos(2*x) = (cos(x))^2 - (sin(x))^2
sin(2*x) = 2*sin(x)*cos(x)
All of the trig identities fall out as simply through simple algebraic games when you start with Euler's equation. It also makes looking at the calculus of trig pretty trivial, greatly simplifies the study of waves, etc. I cannot for the life of me understand why everyone doesn't teach it this way.
What a completely silly idea. (Score:3, Interesting)
It is not revolutionary, nor redefined. What it does it to rewrite trigonometry in similar thing as the good old "SOH-CAH-TOA" method taught in American high schools.
Wildberger's sole insight are the following:
- Instead of using the linear norm, he chooses to use the equivalent quadratic norm for distances, thus removing the squares from Pythagorean theorem. (So, for a right triangle, his version would be BASE + HEIGHT = HYPOTENUSE).
- Instead of using angles and calculating sines and cosines from it, he uses the concept of Spread, which is essentially just the sine of the angle squared!!
Well, one immediately sees a problem with the second point when trying to do something more than traditional planar Euclidean geometry: an obtuse angle will have the same spread as one other acute angle, and they share spreads with two other angles greater than pi radians!His argument is foundamentally flawed even in the first chapter when he "explains" the traditional method of measuring angles. He claims it as the following (paraphrasing):
so far so good, but he goes on to argue that That is complete bullshit. When I was in middle school in Taiwan, we were taught that 1) we start with two rays instead of two lines. The angle between rays A and B would be given by taking the arc length of on the unit circle by going counter clockwise from A to B. So, semantically, the angle between A and B, and the angle between B and A, would sum up to a nice 2pi. While colloquially, we often prefer to deal with the smaller of the two measurements.There are obvious problems with the Quadrance and Spread concept when applied to real life. For one thing, you can't just add Quadrances: say I know that points A B and C are on a line, in that order. Say I also know the distance AB and BC, then the distance AC = AB + BC. But if I measure using the Quadrance, sqrt(AC) = sqrt(AB) + sqrt(BC) would hold. He is just passing the buck in his "redefinition". While making things simpler for trigonometry, he is making things more difficult for other bits of Euclidean geometry.
Similar, while traditionally angles can just sum, his definition using a spread cannot sum angles.
In summary, his "getting rid of sines" is just replacing it with its valuation, which is hardly novel: it is something taught standard in American classrooms (remember SAHCOHTOA?).
"Decimal number plane"?? (Score:3, Interesting)
He seems to mean real plane, but with a pejorative connotation of needing decimal numbers to do ordinary trigonometry.
Google the phrase (in quotes); you get exactly one hit - this book.
Re:huh? (Score:2)
I can do C, C++, and Java, and have written a few MUDS with minimum functionality (They were all in C -- part of the reason I switched to C++ and then to Java), but I never got past Algebra in High School. I can't do Trig. Calc makes me want to die.
I guess I could just say, "I fucking hate integration!" Who's with me?
Re:huh? (Score:5, Insightful)
It's not that trigonmetric functions are hard to learn, it's that they rely on transcendal function like sine/cosine which are calculated numerically (as a taylor power series expansion, for example) thus are only approximations to the true values (an accurate number would require the calculation of an infinite series, which isn't practical in given time/space).
The article clearly states that: "Advanced mathematical knowledge, such as linear algebra, number theory and group
theory, is generally not needed." (to use this method)
I think that having a percise, simple (polynomials, rational fractions) alternative to current methods in eucleadian trigonometry, is very welcome.
Wouldn't it be nice to be able to calculate angles and distances without having to use a calculator (for sine/cosine calculations)?
Re:anybody remember the chant: SOH CAH TOA (Score:2)
Re:Units? (Score:5, Insightful)
I don't usually advocate this kind of behaviour, but the chapter is actually well worth reading. Quadrance is a neat hack - use the square of distance instead of distance to eliminate some nasty square roots, but spread is a much more interesting concept. The notion of spread removes the dependence upon circles to define relative direction, which removes a lot of complexity from trig.
Re:Well, not exactly (Score:3, Informative)