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# Trigonometry Redefined without Sines And Cosines

#### CowboyNeal posted more than 8 years ago | from the numbers-and-stuff dept.

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?"

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#### Anonymous Coward 1 minute ago

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### No sines and cosines? (5, Funny)

#### Joey Patterson (547891) | more than 8 years ago | (#13584108)

Perhaps Dr. Wildberger is trying to take geometry off on a weird tangent.

### Re:No sines and cosines? (3, Insightful)

#### biryokumaru (822262) | more than 8 years ago | (#13584149)

Actually, it does look like just a tangent of traditional trigonometry. After reading the first chapter, most of his math seems to be the switching forms of the Pythagorean theorem from:

(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? (0)

#### Anonymous Coward | more than 8 years ago | (#13584193)

Sort of like the "few unique insights" of Einstein looking at Maxwell's equations in a "simplified" manner.

### Re:No sines and cosines? (0)

#### Anonymous Coward | more than 8 years ago | (#13584232)

It might be better to start people with geometrical algebra concepts. A good introduction is this site: http://www.xtec.es/~rgonzal1/treatise-a.pdf [www.xtec.es]

### Re:No sines and cosines? (3, Interesting)

#### SilverspurG (844751) | more than 8 years ago | (#13584243)

a^2 + b^2 = c^2
That's the way that I learned it and we still had traditional trig.

What did I miss?

### Re:No sines and cosines? (1)

#### biryokumaru (822262) | more than 8 years ago | (#13584309)

Whenever you needed to get an answer in trig class, could you give it in the form of "c^2" or did you have to take the square root? My teachers always wanted the non-squared version... thus, the practical form of the Pythagorean Theorem that I actually used was "(a^2 + b^2)^(1 / 2) = c." I really only meant that as a base comparison between this and traditional trig.

Although, you could argue that changing the form thusly leads to the creation of the convoluted mathematics behind sine and cosine, et al. You know, I think that's Dr. Wildberger's point. Heh =]

### Cool (0, Insightful)

#### Anonymous Coward | more than 8 years ago | (#13584114)

I never understood that crap anyway.

### Now ... (3, Funny)

#### LordKaT (619540) | more than 8 years ago | (#13584115)

If only he could redefine Calculus to use simple algebraic expressions.

### Re:Now ... (0)

#### Anonymous Coward | more than 8 years ago | (#13584128)

You can represent integrals and most functions (including sin/cos/tan) as a series.

### Re:Now ... (4, Interesting)

#### NoTheory (580275) | more than 8 years ago | (#13584161)

Actually, i think this is a perfect point. My major problem when learning integral calculus was my utter loathing for trigonometric identities (and my inability to remember them) required for solving all sorts of weird integrals. I'm curious how this stuff plays with calculus. If it does, maybe this'll make my life easier if i ever go back and attempt calculus again. anyway, reading TFA, hopefully it says something regarding this :)

### Already done (0)

#### Moderator (189749) | more than 8 years ago | (#13584239)

This has already been done by a man named Karl Weierstrass [wikipedia.org] who came up with a way to express continuity in algebraic terms. You know, the "epsilon-delta" definition you learn in your first week of Calculus. In a nutshell, before this definition, everyone knew that Calculus worked, but no one was sure *why*.

### Re:Now ... (-1, Flamebait)

#### Anonymous Coward | more than 8 years ago | (#13584293)

Been there, done that - it's called Laplacian Transforms.

Mark Edwards
--
Proof of Sanity Forged Upon Request

is 42

### Re:The method doesn't matter, as long as the answe (1)

#### helioquake (841463) | more than 8 years ago | (#13584220)

tan(88.6361deg) = 42.

Is that what you wanted?

### Wonderful! (5, Insightful)

#### h4rm0ny (722443) | more than 8 years ago | (#13584118)

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... (3, Insightful)

#### tgd (2822) | more than 8 years ago | (#13584196)

As a math teacher, it may not be as obvious to you, but as someone who learned trigonometry in school and isn't a math teacher now I can say for certain. When people say they'll never use that in the real world, they're absolutely right.

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... (0)

#### Anonymous Coward | more than 8 years ago | (#13584256)

I dunno... I recently wanted to replace my four-by-three panel with a widescreen one of the same height. Given I was going to order online, so couldn't just look at them and get a feel for what was right, trig came in quite handy.

### Re:Don't worry... (4, Insightful)

#### anderm7 (68050) | more than 8 years ago | (#13584271)

I think it kind of depends on what you do for a living. If you work at Wal-Mart, you probably don't need it. If you are and Physicist or Engineer (like my wife and I), you probably do. And thats hard to know in High School

### Not just physicists or engineers use trig.... (4, Insightful)

#### Ellis D. Tripp (755736) | more than 8 years ago | (#13584312)

Even kids who go into trades like carpentry would benefit from a knowledge of the fundamentals of trig. Laying out angles for roof rafters, staircase stringers, etc....

### Re:Don't worry... (5, Insightful)

#### Dr_LHA (30754) | more than 8 years ago | (#13584281)

Everyone complains how trig is not useful, but perhaps its useful because it is hard and is one of the few things left in schools today that actually mentally challenges students.

### Re:Don't worry... (4, Insightful)

#### PocketPick (798123) | more than 8 years ago | (#13584296)

I take it you taught math in elementary school (K to 5th) then, as your point is completely wrong. For a physicist or computer scientist, the principles of trigonometry are invaluable. All those games you might play. All those electronic devices (cell phones, tv, etc) you use on a daily basis. Much of the theory used to devise how they could possibly work was done with *gasp* trigonometry and to a greater extent, calculus.

Simply because you choose a profession does not use it, does not mean it doesn't have value.

### Re:Wonderful! (-1, Flamebait)

#### drsquare (530038) | more than 8 years ago | (#13584311)

Things that you don't actually need in the real world:

Trigonometry
Calculus
Physics
Biology
History
Geography
Square roots
Imaginary numbers
Graphs
IT
Foreign languages
Algebra

Things like that should be taught at university to people who actually want to learn them, not school.
I've never used any of those things outside of school, ever.

### Figures. (5, Funny)

#### Musteval (817324) | more than 8 years ago | (#13584119)

He does this the year after I take Algebra II/Trig. Bastard.

### Yeah (1)

#### Evanisincontrol (830057) | more than 8 years ago | (#13584120)

It's the era where I don't fail Calculus IV because I never went to math in 8th grade.

### Units? (1)

#### mrhale (872484) | more than 8 years ago | (#13584121)

I'd be sceptical if it will take off - 'Wildbergers' just isn't as catchy as 'degrees' or 'radians'....

### Re:Units? (4, Insightful)

#### TheRaven64 (641858) | more than 8 years ago | (#13584292)

If you'd R'd TFC then you would know that spread is a unitless quantity. It is a ratio between two quadrances (lengths squared), and as long as the quadrances are homogenous with respect to their units then they cancel out.

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:Units? (1)

#### tantrum (261762) | more than 8 years ago | (#13584304)

blockquote> I'd be sceptical if it will take off - 'Wildbergers' just isn't as catchy as 'degrees' or 'radians'....

deg, grad, rad and WILD guess what i'd like to use on my calculator

### The "New" has an initial capital for a reason (3, Informative)

#### Bewbewbew (871127) | more than 8 years ago | (#13584123)

The "New" refers to South Wales, rather than the uni. Might wanna read a bit closer next time.

### Wow (3, Interesting)

#### Loconut1389 (455297) | more than 8 years ago | (#13584124)

I'm a failing engineering student at Iowa State University- or at least I was.. My failing point: math. I can out code some of the instructors in my classes, but I can't do math or memorize functions and derivitives of trigonometric functions.

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 (1)

#### DarkPixel (570153) | more than 8 years ago | (#13584170)

As long as you show your work! :)

### Re:Wow (0)

#### Anonymous Coward | more than 8 years ago | (#13584177)

Listen brother if you are failing math then engineering is not your thing. Switch to a less math intensive dicipline, like culinary arts.

### Re:Wow (-1, Troll)

#### Anonymous Coward | more than 8 years ago | (#13584226)

When you say you can "out code some of the instructors", I'm assuming this is essentially a competition that involves you and your instructor, but your instructor hasn't been informed that he's participating?

I think this is a classic case of beginner programmer. I can hack some c code, so I must be brilliant.

### Re:Wow (5, Insightful)

#### lobsterGun (415085) | more than 8 years ago | (#13584230)

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 (0)

#### Tony Hoyle (11698) | more than 8 years ago | (#13584283)

That's an old idea, largely discredited... from the days when the math teachers where the computer science teachers.

Programming isn't maths.. maybe simple algebra, but it's a lot more about creativity and logically solving problems. I've been programming professionally now for 15 years and never needed more than rudimentary maths knowledge - nor can I imagine any situation when I would actually need it.

Algorithms are worth learning, but algorithms aint maths either.. they're just the 'known best' way to solve problems.

### Re:Wow (1)

#### technoextreme (885694) | more than 8 years ago | (#13584274)

I'm a failing engineering student at Iowa State University- or at least I was.. My failing point: math. I can out code some of the instructors in my classes, but I can't do math or memorize functions and derivitives of trigonometric functions.

Oooo come on. This will not make things any more especially when you can't memorize with subjects like differntial equations. Those equations you have to memorize what to do on top of figuring out what type of equation to use.

### UNSW .. not South Wales (4, Informative)

#### OzPeter (195038) | more than 8 years ago | (#13584126)

As per the article .. Dr Wildberger is from UNSW, the University of New South Wales .. in friggin' Australia. South Wales is somewhere all together different. But as always people don't RTFA

### Re:UNSW .. not South Wales (1)

#### Gothmolly (148874) | more than 8 years ago | (#13584136)

'Editors' don't always read the SUBMISSION you mean.

### Re:UNSW .. not South Wales (1)

#### OzPeter (195038) | more than 8 years ago | (#13584206)

How about "People don't read" ?? Though this does assume that editors are people.

Maybe "People can't think" is better. But I am leaning more to "People can't critically think". If they did . .editors would edit.

### Re:UNSW .. not South Wales (0)

#### Anonymous Coward | more than 8 years ago | (#13584269)

While your statement should be correct (IMHO), you don't really need to critically think to observe that blatant error. A simple "diff" would show the obvious mistake, and it doesn't take more than the mind of a trained monkey to do this.

### Re:UNSW .. not South Wales (0)

#### Anonymous Coward | more than 8 years ago | (#13584275)

The editor and submitter probably just found it easy to believe that nothing of intellectual consequence comes out of Australia... as do most people, to be honest.

### Hopefully (3, Insightful)

#### JasonEngel (757582) | more than 8 years ago | (#13584127)

This sounds promising, but I have two educational concerns: 1. Is this just a dumbed-down version of trig? ...and on the opposite end of the spectrum... 2. Would this lead to bombarding students with too much math as the requirement shifts from alg/calc/trig to alg/calc/trig/trig2?

### Re:Hopefully (0)

#### Anonymous Coward | more than 8 years ago | (#13584208)

1. Yes - sine and cosine are just expressions of the ratio of x&y for the unit circle. Redefine your angle as a vector from the origin of length 1 in a certain direction, pick out your X and y and divide... Duh.

2. huh? I got trig before calculus. It's not that hard. Hard is applying it to probability, physics, engineering dynamics, chemistry, etc. The language of math and solving stuff is way easier than figuring out what is the right math to solve.

### Re:Hopefully (1)

#### superwiz (655733) | more than 8 years ago | (#13584222)

I just read the sample chapter. It is not dumbed down. Its concept of "spread" is closely tied to how angles are viewed by mathematicians on abstract manifolds. I.e., the only property of the angle that really matters is its cosine. Essentially the angle is "named" for its cosine. This works both in simple Euclidean space, multi-dimensional Euclidean space, AND in spaces with non-euclidean metrics. So learning trigonometry in this way will make understanding of subjects like Tensor Calculus much more natural. It's a good attempt. But, of course, it must be carefully ironed out before we start comitting generations of children to this as the world view.

### huh? (1)

#### vapor22 (410851) | more than 8 years ago | (#13584129)

are trigonometric functions REALLY that hard to learn?

is trigonometry one of the root causes of the layman's hatred for math?

that's doubtful and even if it was true, his version of trigonometry still requires algebra which has a far greater hatred among joe sixpack.

### Re:huh? (1)

#### PakProtector (115173) | more than 8 years ago | (#13584168)

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?

C-- ???

### Re:huh? (1, Insightful)

#### siplus (796514) | more than 8 years ago | (#13584229)

if you've ever progrmmed a for-loop, then you have basically integrated! ;)

### Re:huh? (0)

#### Anonymous Coward | more than 8 years ago | (#13584246)

I guess I could just say, "I fucking hate integration!" Who's with me?

The Ku Klux Klan. I've heard they prefer segregation to integration.

### Re:huh? (1)

#### promatrax161 (913597) | more than 8 years ago | (#13584180)

You are right, it is either sines and cosines or square roots and fractions (I am guessing here, it may be something else)...in any case a lot of algebraic computation is rather unpopular among pupils :(

### Re:huh? (4, Insightful)

#### HateBreeder (656491) | more than 8 years ago | (#13584189)

You're wrong.

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:huh? (1)

#### OzPeter (195038) | more than 8 years ago | (#13584278)

I just read the first chapter of his book, and while I agree that he does away with calculus for dealing with angles (no more sines and cosines) one of his sample solutions ends up with the square root of 7. Now how do you evaluate that with out a recourse to numerical method?

### Re:huh? (1)

#### Roguelazer (606927) | more than 8 years ago | (#13584291)

Which is why you break down complex angles into half-angle and double-angle identities, which lets you get the precise answer (with lots of addition of irrational radicals, but whatever)... Take the funky angle sin 22.5 for a simple example. Your calculator will tell you that the answer is 0.38blah. However, since you know that cos45 = sqrt(2)/2, and you know the half-angle identity for sin is sin(x/2) = sqrt((1-cosx)/2), then you know that sin(22.5) = sqrt((1-sqrt(2)/2)/2) = sqrt(2-sqrt(2))/2. Using repetition of this method, you can find out the exact values of very small angles, which you can then add together to find precise values of larger angles. Of course, it takes a long time, but how is that worse than finding the "spread" of an angle, particularly in curves, where you need the angle itself to calculate arclengths?

### Just Wait... (4, Interesting)

#### DataPath (1111) | more than 8 years ago | (#13584133)

Just wait until someone reimplements many software functions that rely on sine, cosine, and tangent, using these new concepts, and get a patent on them.

### Re:Just Wait... (1)

#### Loconut1389 (455297) | more than 8 years ago | (#13584153)

Now -that's- an interesting idea...

*fires up vi*

### Better LInk (1, Redundant)

#### OzPeter (195038) | more than 8 years ago | (#13584145)

DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry by N J Wildberger [unsw.edu.au] is a link to the advertisment for his upcoming book, which also has a PDF dowbload of the first chapter.

### Re:Better LInk (1)

#### OzPeter (195038) | more than 8 years ago | (#13584154)

Oops .. just realised that *I* didn't RTF summary .. my bad

### new era? nah.... (1)

#### cryptocom (833376) | more than 8 years ago | (#13584146)

just a new way of coming to the same solutions. that's like saying the transportation industry has been revolutionized because an alternate route has been found between your house and where you work...
: )
(actually...i would probably say that if there WERE an alternate route...but...eh...)

### Lousy analogy (0)

#### Anonymous Coward | more than 8 years ago | (#13584166)

its like saying the transportation industry has been revolutionised because you can now cycle to work instead in walking. Oh wait, it has.
No idea if that book is junk or not though, but you'd be wiser to take a course in rhetoric (or failing that English composition) instead anyhow.

### Redefinition? (3, Insightful)

#### AndreiK (908718) | more than 8 years ago | (#13584150)

Erm, I actually read the sample chapter, and one thing I don't get: What did he do that is so revolutionary?

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? (2, Insightful)

#### DarkPixel (570153) | more than 8 years ago | (#13584205)

If you were a programmer that relied on an implimentation that used traditional trig, you would understand why 'redefining' the route to the correct answer to use simple algebraic expressions would be a good thing...precision. I am a computer graphics enthusiast and I dwell in alot of 3d math that involves calculus (mainly all sorts of complex curves). Guess what, that crap all likes trig! If I can define the formula for a three dimensional sphere without trig, thank you, thank you, thank you. I'm gonna go read this book when it comes out.

### Re:Redefinition? (1)

#### AndreiK (908718) | more than 8 years ago | (#13584215)

Calculating their definition of spread is the exact same calculation as the definition of arctangent, as far as I can tell.

### Re:Redefinition? (1)

#### ab8ten (551673) | more than 8 years ago | (#13584255)

Absolutely. Just think of all the 3d math that goes on behind the scenes of games today. All those lookup tables can be kissed goodbye.
I wonder if this method would actually be faster (even if it were easier to implement) than traditional lookups? Has Carmack read this yet? :-)

### Quaternions (0)

#### Anonymous Coward | more than 8 years ago | (#13584294)

Just use quaternions and be done with it.

### Re:Redefinition? (5, Insightful)

#### sameerd (445449) | more than 8 years ago | (#13584213)

Spread is NOT proportionally equal to an angle. 30 degrees is 1/4 and 60 degrees is 3/4

spread is the square of the sine of an angle.

### Re:Redefinition? (1)

#### ab8ten (551673) | more than 8 years ago | (#13584235)

Spread is not proportional to angle at all. If you read the pdf summary, he states that .5 is 45deg while .5 and .75 are equal to 30 and 60deg respectively. The relatiosnship is not linear, but would plot as a curve with an inflexion at 45 degrees.

### Re:Redefinition? (1)

#### AndreiK (908718) | more than 8 years ago | (#13584308)

It is based on a proportion. I never said it was linear, however. Basically he's redefining the domain of the functions to be neater.

### Why are there 360 degrees? (1)

#### Colin Smith (2679) | more than 8 years ago | (#13584259)

I never really got a good answer to that one from my maths teachers.

### Re:Why are there 360 degrees? (2, Funny)

#### Skirwan (244615) | more than 8 years ago | (#13584285)

There are 360 degrees in a circle because there are 365 (point whatever) days in a year. The ancient Greeks were more primitive than we are today; lacking computers, they couldn't manage a simple off-by-one error, and had to fall back on the less sophisticated off-by-five-and-a-long-decimal error.

### Re:Redefinition? (1)

#### thrashbasket (880168) | more than 8 years ago | (#13584303)

so why is 30 degrees 1/4 and 60 degrees 3/4?

### anybody remember the chant: SOH CAH TOA (1, Funny)

#### Anonymous Coward | more than 8 years ago | (#13584155)

My high school math teacher used to march around class chanting SOH CAH TOA, SOH CAH TOA, SOH CAH TOA!
(and now, thirty years later I still remember)
Sine = Opposite over Hypotenuse (SOH)
Cosine = Adjacent over Hypotenuse (CAH)
Tangent = Opposite over Adjacent (TOA)
(when dealing with right-angle triangles)

TDz.

### Re:anybody remember the chant: SOH CAH TOA (1)

#### AndreiK (908718) | more than 8 years ago | (#13584167)

I'm glad I'm not the only one!

### Re:anybody remember the chant: SOH CAH TOA (1)

#### eyebits (649032) | more than 8 years ago | (#13584181)

I used to teach it. :)

### Re:anybody remember the chant: SOH CAH TOA (1)

#### freewaybear (906222) | more than 8 years ago | (#13584251)

What about this goody: Sine! sine! cosine! sine! 3.14159! (my favorite cheer)

### Re:anybody remember the chant: SOH CAH TOA (1)

#### mikael (484) | more than 8 years ago | (#13584187)

Yes, our teacher did that, but she drew a picture of a volcano, with the right angled triangle, as one of the slopes.

### Re:anybody remember the chant: SOH CAH TOA (1)

#### drooling-dog (189103) | more than 8 years ago | (#13584266)

Oscar Had A Heap Of Apples.

### Uh... (1)

#### Comatose51 (687974) | more than 8 years ago | (#13584158)

It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'.

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 calculus. Wait... This is math. There are no theories. It's either proven or unproven. There might be conjectures waiting to be proven but I've never heard of theories being used in mathematic. Then again, I am not a mathematician.

Maybe someone much more knowledgable can explain this for me.

### Re:Uh... (1)

#### maddhatt (149310) | more than 8 years ago | (#13584183)

I am hoping that he is refering to a Theorem: which is a statement that can be proven via logic. An Axiom otoh is something that is simply generally accepted to be true.

### Re:Uh... (1)

#### gonerill (139660) | more than 8 years ago | (#13584201)

the axiomatic system upon which we built all human knowledge You think your knowlege of where you parked your car is built on an axiomatic system?

### Re:Uh... (1)

#### starwed (735423) | more than 8 years ago | (#13584301)

Yes; of course it is. ^_^

Look up the definition [answers.com] of axiom. One of the basic axiom's we use everday is that our memory of the past is accurate, and that events we remember will influence the present and the future.

### SOHCAHTOA and abstract survery results (4, Insightful)

#### acomj (20611) | more than 8 years ago | (#13584164)

ahh Sin= Op/Hyp

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 (1)

#### AndreiK (908718) | more than 8 years ago | (#13584190)

Spread is the relationship between quadrances. Directly proportional to angle, 45deg = 1/2, 90 deg = 1

### Faster calculations ?? (5, Interesting)

#### AeiwiMaster (20560) | more than 8 years ago | (#13584176)

I am wondering if this could be used to make faster calculations
in raytracers and 3D engines by using integer numbers.

### Non-Linear Angles (1, Interesting)

#### Anonymous Coward | more than 8 years ago | (#13584203)

This is horrible for ray tracing. The angles are non-linear. In computer graphics, it is easy to add anagles 45deg+45deg=90deg. That is the beauty of regular angles.

With his method you can't just add angles line that. You have to do an elaborate calculation.

### Re:Faster calculations ?? (3, Interesting)

#### Anonymous Coward | more than 8 years ago | (#13584228)

AFAIK most 3D engines already use tables with values for different angles and extrapolate for faster trigonometric calculations since you don't need that much precision in a game anyway

### fractional numbers? (1)

#### snoig (535665) | more than 8 years ago | (#13584178)

Isn't that what sines and cosines are in the first place?

### Three words (0, Redundant)

#### DrXym (126579) | more than 8 years ago | (#13584179)

SOH CAH TOA. SOH (Sine(x) = Opposite / Hypotenuse, Cos(x) = Adjacent / Hypotenuse, Tan(x) = Opposite / Adjacent where x is an angle of a right angled triangle)

Remember them and trigonometry is a doddle.

### Re:Three words (1)

#### DarkPixel (570153) | more than 8 years ago | (#13584247)

You have one for remembering the four pages of calc-trig identities in the back of my calc book?

### Interesting - but not entirlely new (4, Insightful)

#### caffeined (150240) | more than 8 years ago | (#13584192)

The professor seems to have found an interesting way to present trig - however, it should also be noted that he does actually rely on sines as an underlying concept.

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.

### New meanings from now on.. (1)

#### mayhemt (915489) | more than 8 years ago | (#13584223)

sin 90 ...is 90 wrong things u did against will of god 42 is wrong answer why?....its cos 90 tan 90 ...is the new skin care product

### Re:New meanings from now on.. (1, Funny)

#### Anonymous Coward | more than 8 years ago | (#13584280)

HAHA!!! You are not funny.

### Great for eighth grade, but ... (5, Insightful)

#### levin (170168) | more than 8 years ago | (#13584225)

What happens when kids get to math subjects where trigonometric functions are used for more than just calculating the dimensions of geometric figures? How does this "spread" thing represent angles greater than 180 degrees without redefining what you are measuring, and how does this really make a persons life any easier unless someone tells you the spread as in the textbook chapter? If his explanation is to be taken as any sort of indication of how to measure the spread, then you might as well just walk off the dimensions of what you're measuring because you'll have to do that anyway to calculate it. How will you integrate problems that call for constant rotation using spread? This seems like trading a little bit of pain now for a lot more down the road, and I pray that it won't catch on in the US. If students are really having this much trouble with this subject then it should be introduced earlier and in smaller portions, not ignored. The last thing we need is for someone to take another slice out of the already anaemic math programs in our primary schools.

### easier for computers? (1)

#### El_Muerte_TDS (592157) | more than 8 years ago | (#13584244)

Does this make things easier for computers?
Would be very nice to have a performance boost at the math level for 3D calculations.

### Several points: (1)

#### Ruie (30480) | more than 8 years ago | (#13584253)

• The math is correct and the expressions he is using are really just a rational parametrization of the circle - you can find two rational expressions x=R(t) and y=Q(t) that will trace out a circle as t goes along real line.
• However, people use angles for a reason - the angle variable just measures the arc cut out on a circle of unit radius. So, it would be rather hard to compute angular momentum in terms of spread, or say, define a periodic monochromatic signal (for which you are much better off using sin() and cos()
• The most important point is that the worse thing a teacher can do is dumb down the material - the students feel it and are a lot less interested. If someone has trouble accepting trigonometric functions as is, teach them algebra of complex numbers (useful also as miniature version of vector algebra) and then explain Euler's formula.

### I don't see how this is "easier" (4, Insightful)

#### Curmudgeonlyoldbloke (850482) | more than 8 years ago | (#13584254)

Imagine if we'd been using "quadrance" and "spread" for years - and then some bright spark suggested calculated using sines and cosines. Which would people see as easier?

### Re:I don't see how this is "easier" (1)

#### freewaybear (906222) | more than 8 years ago | (#13584287)

Will anyone be serving pi at this function?

### Great, but (1)

#### RAMMS+EIN (578166) | more than 8 years ago | (#13584265)

Ok, so using squares of distances instead of plain distances, and relations between lines instead of angles makes calculations easier. But isn't that shifting the problem? Now measuring becomes more complex, as do calculations based on angular velocity. Still, it's good that someone is trying to provide a new perspective; back when I was doing trigonometry I always thought there must be something simpler underneath.

No.

### Yes, (1)

#### lobsterGun (415085) | more than 8 years ago | (#13584290)

but does it have a simple mnemonic like:

Orange
Hippos
Always
Have
Orange
Angles

which yields...

Opposite
Hypotenuse = sin theta

Hypotenuse = cos theta

Opposite
Adjacent = tan theta

### A bit crackpotty? (1)

#### geordieboy (515166) | more than 8 years ago | (#13584295)

It really seems to me that his concept of "spread" to measure the orientation of two lines is much less intuitive than angle. The concept of angle is just not hard to grasp compared to this weird construction of dropping perpendicular lines.

And it isn't true that you need calculus to understand cosines and sines, you just need some simple plane geometry (right angle triangles inscribed in circles and so on). You can even plot the cosine and sine functions without calculus.

### Bah!! (2, Funny)

#### doi (584455) | more than 8 years ago | (#13584310)

Now THESE are some divine proportions. [ariagiovanni.com]
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