# Trigonometry Redefined without Sines And Cosines

#### CowboyNeal posted more than 9 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?"*

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

## Joey Patterson (547891) | more than 9 years ago | (#13584108)

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

## biryokumaru (822262) | more than 9 years ago | (#13584149)

(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 9 years ago | (#13584193)

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

## Anonymous Coward | more than 9 years ago | (#13584232)

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

## SilverspurG (844751) | more than 9 years ago | (#13584243)

What did I miss?

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

## biryokumaru (822262) | more than 9 years ago | (#13584309)

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 9 years ago | (#13584114)

## Now ... (3, Funny)

## LordKaT (619540) | more than 9 years ago | (#13584115)

## Re:Now ... (0)

## Anonymous Coward | more than 9 years ago | (#13584128)

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

## NoTheory (580275) | more than 9 years ago | (#13584161)

## Already done (0)

## Moderator (189749) | more than 9 years ago | (#13584239)

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

## Anonymous Coward | more than 9 years ago | (#13584293)

Mark Edwards

--

Proof of Sanity Forged Upon Request

## The method doesn't matter, as long as the answer (3, Funny)

## PtrToNull (742886) | more than 9 years ago | (#13584117)

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

## helioquake (841463) | more than 9 years ago | (#13584220)

Is that what you wanted?

## Wonderful! (5, Insightful)

## h4rm0ny (722443) | more than 9 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 9 years ago | (#13584196)

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 9 years ago | (#13584256)

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

## anderm7 (68050) | more than 9 years ago | (#13584271)

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

## Ellis D. Tripp (755736) | more than 9 years ago | (#13584312)

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

## Dr_LHA (30754) | more than 9 years ago | (#13584281)

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

## PocketPick (798123) | more than 9 years ago | (#13584296)

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 9 years ago | (#13584311)

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 9 years ago | (#13584119)

## Yeah (1)

## Evanisincontrol (830057) | more than 9 years ago | (#13584120)

## Units? (1)

## mrhale (872484) | more than 9 years ago | (#13584121)

## Re:Units? (4, Insightful)

## TheRaven64 (641858) | more than 9 years ago | (#13584292)

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 9 years ago | (#13584304)

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 9 years ago | (#13584123)

## Wow (3, Interesting)

## Loconut1389 (455297) | more than 9 years ago | (#13584124)

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 9 years ago | (#13584170)

## Re:Wow (0)

## Anonymous Coward | more than 9 years ago | (#13584177)

## Re:Wow (-1, Troll)

## Anonymous Coward | more than 9 years ago | (#13584226)

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 9 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 9 years ago | (#13584283)

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 9 years ago | (#13584274)

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 9 years ago | (#13584126)

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

## Gothmolly (148874) | more than 9 years ago | (#13584136)

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

## OzPeter (195038) | more than 9 years ago | (#13584206)

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

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

## Anonymous Coward | more than 9 years ago | (#13584269)

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

## Anonymous Coward | more than 9 years ago | (#13584275)

## Hopefully (3, Insightful)

## JasonEngel (757582) | more than 9 years ago | (#13584127)

## Re:Hopefully (0)

## Anonymous Coward | more than 9 years ago | (#13584208)

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 9 years ago | (#13584222)

is notdumbed 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 9 years ago | (#13584129)

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 9 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?

## Re:huh? (1)

## labyrinth (65992) | more than 9 years ago | (#13584195)

## Re:huh? (1, Insightful)

## siplus (796514) | more than 9 years ago | (#13584229)

## Re:huh? (0)

## Anonymous Coward | more than 9 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 9 years ago | (#13584180)

## Re:huh? (4, Insightful)

## HateBreeder (656491) | more than 9 years ago | (#13584189)

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 9 years ago | (#13584278)

## Re:huh? (1)

## Roguelazer (606927) | more than 9 years ago | (#13584291)

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

## DataPath (1111) | more than 9 years ago | (#13584133)

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

## Loconut1389 (455297) | more than 9 years ago | (#13584153)

*fires up vi*

## Better LInk (1, Redundant)

## OzPeter (195038) | more than 9 years ago | (#13584145)

## Re:Better LInk (1)

## OzPeter (195038) | more than 9 years ago | (#13584154)

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

## cryptocom (833376) | more than 9 years ago | (#13584146)

: )

(actually...i would probably say that if there WERE an alternate route...but...eh...)

## Lousy analogy (0)

## Anonymous Coward | more than 9 years ago | (#13584166)

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 9 years ago | (#13584150)

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 9 years ago | (#13584205)

## Re:Redefinition? (1)

## AndreiK (908718) | more than 9 years ago | (#13584215)

## Re:Redefinition? (1)

## ab8ten (551673) | more than 9 years ago | (#13584255)

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 9 years ago | (#13584294)

## Re:Redefinition? (5, Insightful)

## sameerd (445449) | more than 9 years ago | (#13584213)

spread is the square of the sine of an angle.

## Re:Redefinition? (1)

## ab8ten (551673) | more than 9 years ago | (#13584235)

## Re:Redefinition? (1)

## AndreiK (908718) | more than 9 years ago | (#13584308)

## Why are there 360 degrees? (1)

## Colin Smith (2679) | more than 9 years ago | (#13584259)

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

## Skirwan (244615) | more than 9 years ago | (#13584285)

## Re:Redefinition? (1)

## thrashbasket (880168) | more than 9 years ago | (#13584303)

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

## Anonymous Coward | more than 9 years ago | (#13584155)

(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 9 years ago | (#13584167)

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

## eyebits (649032) | more than 9 years ago | (#13584181)

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

## freewaybear (906222) | more than 9 years ago | (#13584251)

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

## mikael (484) | more than 9 years ago | (#13584187)

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

## drooling-dog (189103) | more than 9 years ago | (#13584266)

## Uh... (1)

## Comatose51 (687974) | more than 9 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 newWait... 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.theoryto a wide range of practical problems from engineering, physics, surveying and calculus.Maybe someone much more knowledgable can explain this for me.

## Re:Uh... (1)

## maddhatt (149310) | more than 9 years ago | (#13584183)

## Re:Uh... (1)

## gonerill (139660) | more than 9 years ago | (#13584201)

the axiomatic system upon which we built all human knowledgeYou think your knowlege of where you parked your car is built on an axiomatic system?## Re:Uh... (1)

## starwed (735423) | more than 9 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 9 years ago | (#13584164)

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 (1)

## AndreiK (908718) | more than 9 years ago | (#13584190)

## Faster calculations ?? (5, Interesting)

## AeiwiMaster (20560) | more than 9 years ago | (#13584176)

in raytracers and 3D engines by using integer numbers.

## Non-Linear Angles (1, Interesting)

## Anonymous Coward | more than 9 years ago | (#13584203)

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 9 years ago | (#13584228)

## fractional numbers? (1)

## snoig (535665) | more than 9 years ago | (#13584178)

## Three words (0, Redundant)

## DrXym (126579) | more than 9 years ago | (#13584179)

Remember them and trigonometry is a doddle.

## Re:Three words (1)

## DarkPixel (570153) | more than 9 years ago | (#13584247)

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

## caffeined (150240) | more than 9 years ago | (#13584192)

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 9 years ago | (#13584223)

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

## Anonymous Coward | more than 9 years ago | (#13584280)

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

## levin (170168) | more than 9 years ago | (#13584225)

## easier for computers? (1)

## El_Muerte_TDS (592157) | more than 9 years ago | (#13584244)

Would be very nice to have a performance boost at the math level for 3D calculations.

## Several points: (1)

## Ruie (30480) | more than 9 years ago | (#13584253)

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

## Curmudgeonlyoldbloke (850482) | more than 9 years ago | (#13584254)

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

## freewaybear (906222) | more than 9 years ago | (#13584287)

## Great, but (1)

## RAMMS+EIN (578166) | more than 9 years ago | (#13584265)

## New beginning for maths? (1)

## pedicabo (753738) | more than 9 years ago | (#13584273)

## Yes, (1)

## lobsterGun (415085) | more than 9 years ago | (#13584290)

OrangeHipposAlwaysHaveOrangeAngleswhich yields...

OppositeHypotenuse = sin thetaAdjacentHypotenuse = cos thetaOppositeAdjacent = tan theta## A bit crackpotty? (1)

## geordieboy (515166) | more than 9 years ago | (#13584295)

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 9 years ago | (#13584310)