Posted on 09/18/2005 8:41:47 AM PDT by cloud8
Mathematics students have cause to celebrate. A University of New South Wales academic, Dr Norman Wildberger, has rewritten the arcane rules of trigonometry and eliminated sines, cosines and tangents from the trigonometric toolkit.
What's more, his simple new framework means calculations can be done without trigonometric tables or calculators, yet often with greater accuracy.
Established by the ancient Greeks and Romans, trigonometry is used in surveying, navigation, engineering, construction and the sciences to calculate the relationships between the sides and vertices of triangles.
"Generations of students have struggled with classical trigonometry because the framework is wrong," says Wildberger, whose book is titled Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg books).
Dr Wildberger has replaced traditional ideas of angles and distance with new concepts called "spread" and "quadrance".
These new concepts mean that trigonometric problems can be done with algebra," says Wildberger, an associate professor of mathematics at UNSW.
"Rational trigonometry replaces sines, cosines, tangents and a host of other trigonometric functions with elementary arithmetic."
"For the past two thousand years we have relied on the false assumptions that distance is the best way to measure the separation of two points, and that angle is the best way to measure the separation of two lines.
"So teachers have resigned themselves to teaching students about circles and pi and complicated trigonometric functions that relate circular arc lengths to x and y projections – all in order to analyse triangles. No wonder students are left scratching their heads," he says.
"But with no alternative to the classical framework, each year millions of students memorise the formulas, pass or fail the tests, and then promptly forget the unpleasant experience.
"And we mathematicians wonder why so many people view our beautiful subject with distaste bordering on hostility.
"Now there is a better way. Once you learn the five main rules of rational trigonometry and how to simply apply them, you realise that classical trigonometry represents a misunderstanding of geometry."
Wild Egg books: http://wildegg.com/ Divine Proportions: web.maths.unsw.edu.au/~norman/book.htm
Source: University of New South Wales
So9
"I was a victim of New Math, and have never fully recovered."
You and me both. I remember sitting in math class in what must have been 1st or 2nd grade. The principal came in and with great pomp, told us about how we were going to learn "New Math" and how it would make us so superior to all of those who came before.
I struggled through school and failed college calculus several time having never learned some pretty basic principles of math. Finally had a great teacher who turned on the light.
Gee, thanks guys!
We were allowed calculators in college. . .but in high school. . . the most we were allowed was a slide rule.
Ah, for the good old days of a big bamboo Decitrig hanging in a scabbard off your belt: the sign of the nerd LONG before the Pocket Protector was invented. . .
I hated Trig. Still do. In the 35+ years since I took Trig in high school and later college (3 times to get a "C") I have never once had any use whatsoever for what I learned. To me there's no logic or flow to Trig as it was taught to me. Maybe there is but no one ever made that clear at all.
And I have Math ability. I took Calculus 1 and got a B so I know it's not me. I sat my son down after he graduated from public school after he did horribly in Math on his SAT test and made a 19 on his first college test in a basic "refresher" (non-credit) course. I taught him how doing algebra was just like working a puzzle and how simple it can be by just figuring out the puzzle based on what you already know. This semester he's now taking Differential Equations and Calculus Based Physics. Again, someone can have the ability, but its all how its taught that determines how well someone does in Math.
But I still wouldn't know a sine from a cosine if they bit me in the butt. I do know what a tangent is. That's about it.
Some Old Hags
Cackle All Hours
Till Old Age
Sine = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent
Who, however, was not so bold as to claim that you could do without angles, an angle being the thing that is measured with a sextant.
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It appears to be a press release to sell a book. I'd like to see this "new" method before I decide to apply it (if that's even possible).
So is politics.
On a unit circle, that is a circle with radius 1, 1 being in any units such as inches or radians or no units, place a point anywhere on the circle. The sine is the vertical distance from the horizontal line that divides the circle in half top and bottom to a point on the circle and the cosine is the horizontal distance from the vertical line that divides the circle in half left and right to the point on the circle. Or, just remember that sine is the vertical. The triangle should be drawn from the center of the circle, along the horizontal to the point below [or above if the point is on the lower half of the circle] the point on the circle, then along the perpendicular from that point to the point on the circle, then back along the radius from the point on the circle to the center of the circle. The sine is the length of the vertical distance and the cosine is the length of the horizontal distance if the circle is a unit circle.
Considering you can do most of the computation with integers, yeah! I've seen 70's era Fortran using this technique.
Besides this is sort of a MATH thread rather than English usage. Mathematicians/Scientists commonly use their own notation so as to confuse as many as possible and I didn't attend Harvard or Cambridge.
So, EXCUUUSSSSSEEEEEE MMEEEEE!!!!
Thank ya vera much.
Use it to quickly find integer pairs near a given ratio so one can select tooth counts for gear sets.
Ahhhhh Heeeelp Nooooo - It's 1966 and I'm back in HS sitting in Mr Barry's Mechanical Drafting class!!!!
Just kidding.
I had Mechanical Drafting as a Major and made it and Engineering a career :-)
However 1966 was the last time I had to draw a gear.
e^(i*x) = cos(x)+ i*sin(x); % Euler's Rule
where i == sqrt(-1)
Complex exponentials are the eigen (characteristic) functions of linear systems. They can have my complex exponentials when they pry them from by cold calmy calculator.
Ha! When I took trig, there were no calculators.
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