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?"
Cool (Score:0, Insightful)
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.
Hopefully (Score:3, Insightful)
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.
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.
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: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)?
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.
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:Redefinition? (Score:2, Insightful)
Re:Redefinition? (Score:5, Insightful)
spread is the square of the sine of an angle.
Great for eighth grade, but ... (Score:5, Insightful)
Re:huh? (Score:1, Insightful)
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.
I don't see how this is "easier" (Score:5, Insightful)
Re:Don't worry... (Score:4, Insightful)
Re:Don't worry... (Score:5, Insightful)
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:Don't worry... (Score:5, Insightful)
Simply because you choose a profession does not use it, does not mean it doesn't have value.
Not just physicists or engineers use trig.... (Score:5, Insightful)
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:Don't worry... (Score:2, Insightful)
People need to stop dissing math in their k-12 education as "not something I'd use in real life". That is so not true. Learning that math opens you up to opportunities otherwise unavailable. Kinda like reading, that's useful right? Ok so reading and math are on different levels, but I believe I sorta hinted at my point.
Re:Redefinition? (Score:2, Insightful)
No, the spread isn't directly proportional to the angle. It actually turns out to be the square of the sine of the angle -- which just looks proportional if all you look at are 0, 45 and 90 degrees.
It still isn't revolutionary, though. You're just working directly with the sines and cosines, since you have nice algebraic relationships between them, and never looking at the actual angles, which would take you into the transcendental domain.
I'm sure mathematicians and physicists do this sort of thing all the time. Probably nobody thought to write a whole book about it. Guys, say hello to the next Stephen Wolfram.
Re:Don't worry... (Score:2, Insightful)
Without trigonometry, my friend, you'd still be thinking the Earth is flat and we that we live in the center of the Universe.
Some of us are actually glad that we have progressed since cave men and do our best to further improve society. Education is the starting point.
Re:Don't worry... (Score:3, Insightful)
A lot of the point of learning math is keeping your options for the future open.
Re:Wow (Score:5, Insightful)
And you are wrong about algorithms. Algorithms is math. No ifs, ands or buts about it.
Re:No sines and cosines? (Score:3, Insightful)
agreed (Score:4, Insightful)
Re:No sines and cosines? (Score:2, Insightful)
That said, it might be an interesting way to teach/explore trig. By doing away with the trig functions (which are just the distance->angle mappings,) he gets to solve many simple problems with just algebra and a final square root. Because the sqrt is explicit, this approach might give students a better mental model for trig: as things stand, most students just treat sin, cos, etc, as black boxes, and apply the rules (SOHCAHTOA) by rote.
I'd like to see how his method stacks up when applied to, say, an entire beginner level book or classic text, on geometry/trig.
Re:Not just physicists or engineers use trig.... (Score:2, Insightful)
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.
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....
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.
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.
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.
You've (Score:2, Insightful)
Try proving that a sorting algorithm is O(log n) without using any mathematics. Then you'll understand why mathematics is important.
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: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 politicians. Don't even get me started about the professional merits of Art classes for non-artists.
Math is so generally useful, that I think people are attacking a subject for which there is no rationale merit to attack. I can only speculate this is due to a perceived subjective difficulty of the subject.
Yet it all doesn't matter if there is limited professional scope, and all the material is still very important to be taught well. The purpose of elementary schooling may be misunderstood -- it is not to prepare one for a particular professional but to prepare one for life, which can include many things, despite people having specific plans.
Certainly, the thing that will define what a person's plan will do in the future should not be excluded by something like the difficulty of the kind of math done in math classes. Once learned, a difficult subject could be easy and pleasurable -- if nothing more, having knowledge of trig, places students in a kind of elite: just having the knowledge may be advantageous.
Even people who do not ultimately or intend to choose a technical career may need to talk with scientists and engineers. They may regret it if due to a lack of even high-school knowledge, they cannot be conversant enough on a subject to discuss anything interesting.
By not teaching things like calculus, trig, chemistry, or biology early on, we would rob our youth of a basic knowledge pool -- our future scientists and engineers might never have discovered their favorite subject. Future scientists, etc, could accomplish more in life, get going faster, by learning the basics (which anyone should be able to understand) early on.
Just because a subject's hard to learn or painful at first and therefore encourages some learners to complain or be taken aback by the subject, doesn't mean it's of less value or knowledge will not later be useful. Particularly when surprising things happen in life.
Yeah, most of us may not be scientists and engineers, and most of us don't get stranded on desert islands either. Who says we always get a choice of what knowledge we will need in life? Consider things like mountain climbers... etc... it is rather possible that having or not having technical knowledge of mathematics or physics or not becomes a life-or-death matter. You just never know if some basic tidbit may happen to be extremely helpful or not.
The apparent difficulty of a subject for some, or lack of effective presentation is no reason to obscure the basics or stop classes from existing -- it's reason to find better ways of getting people to come to learn the importance of these subjects.
Re:Don't worry... (Score:5, Insightful)
Luckily its a great store for Physici...
Do you need a cart sir?
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: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.
No, I think you miss the point (Score:3, Insightful)
In fact, you give away your attitude a bit by mentioning Prince Charles. I'm a republican myself but I think you misrepresent his views. He appears to believe that people should live in human scale buildings designed to give a sense of community, and not live in buildings designed to return the maximum income to the shareholders in developers (who don't have to live in the buildings that bring in that income, and wouldn't want to.)
He also appears to believe that many architects produce stuff intended to glorify themselves and their clients regardless of their fit with the landscape and the environment. This is a neat parallel to the idea that mathematics teaching for the majority should reflect concepts that tie back to the real world so they will be useful, and that people who propose radical changes are often more interested in their own fame and reputation than the real benefits.
Re:change it all... um-hmm (Score:2, Insightful)
It's hilarious - look at Europe with dozens of little legal systems, languages, political fiefdoms and no constitution and two world wars but goddammit WE ARE BETTER BECAUSE WE ARE METRIC.
I'll get OT and bite -- yes, I'd rather live in a continent whose culture extends more than 300 hundred years back in history, does not have a constitution that ensures the same bad politicians always stay in power and act against the interests of the population and looking into pleasing the big corporations, and does not use freedom as an excuse to bring war to other parts of the world. Any questions?
That said, I don't really understand what one thing has to do with the other. The advantage of the metric system is simply to ensure that everyone uses the same rigorously defined units and the potential for confusion is reduced -- much like your pseudo-argument that the USA are somehow better for having a single language, except that in this case you should look into other factors such as cultural diversity.
Ergo, you must be a troll.
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 has not changed. Metric's benefits still have not outweighed the costs of changing all rulers, indicators, speedometers, odometers, signs, etc. -- and as well, the internal rulers in people's minds that have used feet and miles for centuries.
P.S. The number 60 has these low factors: 2 3 4 5 6. That means that if you use the number 60 for measuring, it's easy to divide whatever you're doing into 2 to 6 parts, and each part is another integer. The number 10 only has factors 2 and 5. Arguably, the number 12 (having factors of 2 3 4 6) is more useful than 10. 10's usefulness is that it matches the base (10) that we use
P.P.S. If there was some real political will behind it, the USA might be able to change all typing keyboards to Dvorak from Qwerty. But that would be a social "Apollo Project" in scale, hence it's never going to be done. Keyboards will probably change format when the entire concept of a keyboard changes, like if pervasive voice recognition or neural connections arise. But then the change will be invoked as the standard becomes nonstandard, and the keyboard fades away.
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 shoehorn" approach and force a bunch of crap into kids brains on the swallow-and-regurgitate model. As a kicker, they could make the model one that would encourage independent acts of swallow-and-regurgitate by rewarding it with good grades for minimal mental flip-flops. And they could put a cherry on top by discouraging independent thought by making critical thinkers who try to form their own opinions have to defend their work much more vigorously in order to get good marks (i.e., making "not what the teacher thinks" a synonym for "incorrect").
Guess which one I think models our primary education system?
(And no, I don't blame teachers. The problem is completely systemic, and I see the teachers I've gotten to know as victims of this brain-crushing system, too.)
Re:No sines and cosines? (Score:5, Insightful)
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.
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 mathematics without any reference to application.
The problem is that, in highschools, it is usually presented as a jumbled mess of formulas with almost no logical stucture to it at all.
There are huge gaps in the reasoning, partly owing to the fact that calculus is left entirely to the end, and then largely mistreated. You can't talk about angles without first talking about limits, and you can't really talk about limits until you understand what the real numbers are (hint: if you were confused about the 0.9999... = 1 thing, you've probably never been given a proper definition of the real numbers).
Angles need some notion of arc length, which needs at least the concept of a limit superior. (If not an integral.) The book in the article shows how to accomplish the tasks normally associated with trigonometry without needing the concept of an angle (or really anything from calculus or analysis).
If you look at the things that students have trouble with, it's usually the curriculum's fault for not explaining things in a reasonable logical order.
One of the things many people have trouble with in highschool is the whole issue surrounding the logarithm and exponentiation with a positive real exponent. The reason why they struggle is that these things get defined circularly. Nobody ever really tells you what the expression 2^(sqrt(2)) or 5^pi is supposed to represent. You need to know things about limits and convergence of series in order to define a^b where a is real, and b > 0 is real.
I was lucky, and found things to read on my own which described enough of mathematics to me to get me interested, and then went to university for pure mathematics.
The reason why mathematics should be taught in highschool is that people should gain some concept of logic, which is useful no matter where you're headed, and by proving propositions and theorems, one eventually gains an incredible grasp of logic. This isn't currently done though.
Mathematics is basically presented as an awful illogical mess where at best, the students are taught to solve some very specific problems in a mechanical, unthinking fashion, and at worst, their self-esteem is damaged and they come away thinking that they are bad at something which they've never been exposed to. I've seen some very bright people who thought that they were terrible at math, and for this reason avoided going into fields of study that they'd otherwise have been interested in.
I hope we can eventually do something about this because, as a student of mathematics, I can say that the present state of affairs at the elementary and highschool level is terrible, and while I can easily see ways in which it could be made better, actually carrying it out is another thing altogether.
Re:No sines and cosines? (Score:5, Insightful)
Re:Now ... (Score:3, Insightful)
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:2, Insightful)
10000000 + 0.00000001 ?
The correct answer is 10000000.00000001
The calculator's answer is 10000000.
A more interesting example would probably have been 0.00000001 + 10000000 - 10000000 = 0 and not 0.00000001 given that you evaluate from left to right. It is a phenomenon of floating point arithmetics know as catastrophic cancellation.
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 the loss of precision is significant or not.
to take one possible example suppose for a simulation you decided to represent the position of your boat and something it is towing as relative to its home base using a pair of numbers.
Mathematically the reference point is arbitary and so this seems fine. However you then decide to represent those numbers as whatever floating point type is conviniant and suddenly the accuracy decreases as you move away from the reference until it eventually gets bad enough that your simulation breaks in some way.
Nievely written algorithms can lose a huge ammount of precision way beyond the obvious loss from the data type limits when converted from mathematical perfection to computer data types.
Re:Not just physicists or engineers use trig.... (Score:1, Insightful)
Your reasoning sounds like either a know it all student or a bucket chemist
Re:Wonderful! (Score:2, Insightful)
We now have a new model of teaching math, which concentrates mostly on "computing" things; every exercise asks you "blah, blah, a=6, b=8, blah blah blah, x=?". Geometry, trigonometry, algebra, analysis, everything. We call this "evolving to the way the western society does teaching".
When I started really learning math, by this I mean the 5th grade, the exercises were like "Hypothesis: Given A and B _prove_ that C holds". Simple things, things which solved _a whole class_ of exercises with numbers, which later developed into more complex things, which were built with these bricks.
When you put things like this the student has to think of a way to prove C, maybe even be original about it. Maybe prove a few lemmas before proving that C holds. An exercise like this will have a two page solution in which you will never see a number, possibly (I'm exagerating a little, but you all get the ideea). When you find numbers in an exercise you'll be happy to get out of it the easy way: you have solved the problem before, you just filled the dotted spaces, trivial.
Also as an example, when we were shown the formula A^2 + B^2 = C^2 (the Pythagora theorem), we were shown the prof for this and also prof for the reciprocal theorem. When we were told that cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b), in the 9th grade, we were also given the demonstration.
For every theorem I saw during my highschool, if the reciprocal theorem holded, the reverse implication was true, I was shown the prof for that too.
I like math, I think math is easy, or at least not harder than other things that are more difficult than crossing a street.
You might think I was very smart at that time compared to the majority of my generation to be able to understand all this at that age, you are probably right :) (I'm also very modest). But really, all my colegues who found all this much more difficult than me, and who pursued other kind s of carrers which are not related at all with math have now a very well formed way of thinking. Math taught us to think.
We were ofcourse lucky to have gifted math teachers, who could teach us all this the right way, but my final point is math is good for your brain, it allows you to develop good thinking, makes you learn how to think. Even people who study liberal arts (yes, i mean you two liberal-arts-students-reading-slashdot), should have a good understanding of basic math.
As a final word: Math is like sex ... err ... no it's not like it ... math is ... err ... may the ... err ... no that was not it ... err ... "This is a good day for science!" ... yes, this is it, or at least close to it.
Re:Not just physicists or engineers use trig.... (Score:3, Insightful)
Re:Faster calculations ?? (Score:1, Insightful)
Since the focus of graphics in the last 10-20 years has shifted from getting anything at all in 2D to doing photorealism in 3D, angles are arguably not directly useful to modern algorithms. In fact, in most of the work I do, angles are essentially translated into scaled sines or cosines directly from the relevant vectors (cross products or dot products), and operated from there on as pure trigonometric quantities.