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New angle on trigonometry

triangular buildings

Architects use trigonometry to make sure corners and walls are where they should be. But could they be using something simpler? (Image: iStockphoto)


Associate Professor Norman Wildberger, of the University of New South Wales in Sydney, Australia, says his theory of "rational trigonometry" is more like algebra as you can plug numbers into an equation and get an accurate result.

"We're going to look at trigonometry in a new way," says Wildberger.

"We're going to leave sines and cosines to the circular motion part of mathematics and not force it on triangles."

Wildberger says the trigonometry we know and love (or hate) today has its historical roots in the work of ancient astronomers, like Ptolemy, who studied the motion of planets.

In this case, the angle between two points in the sky, as seen from Earth, was a reasonable way of calculating the distance between them, he says.

But, says Wildberger, the problem came when others applied these theories, developed for spherical geometry, to the study of flat triangles.

"It's all very well to do if you're working on a sphere or a circle but when we actually study triangles there aren't any circles there," he says.

New concepts

The key purpose of trigonometry is to understand the relationships between the corners and sides of triangles.

It is used in areas like surveying, engineering and construction today.

Classical trigonometry calls the separation between two lines an "angle", which is the length of a circular arc between two lines.

An angle can be calculated using an equation that relates the corners of a triangle (using the concepts of sine, cosine or tangent) to the length (distance) of the side opposite it.

But Wildberger says that distance is not the best way to measure the separation of two points and angle is not the best way to measure the separation of two lines.

"It's not the concept that leads to a mathematics that is the most pleasant and the most useful and the most accurate," he says.

Instead of distance, Wildberger's trigonometry uses a concept called "quandrance", the square of distance.

Instead of angle, he uses the concept of "spread", calculated by dividing one quadrance by another.

The spread between two lines is a number between 0 (representing parallel lines) and 1 (representing lines at right angles).

Wildberger says it would be possible to make a new protractor that measures spread instead of angle.

You would then plug the values for the quadrance and spread into his set of equations.

More accurate

What's better about the system, says Wildberger, is that all the terms in the equations can be calculated exactly, or are "rational", hence the term for his new theory, "rational trigonometry".

But sine, cosine and tangent, are usually only approximated, he says, making them "transcendental functions".

This means that any complex calculation using classical trigonometry could result in a significant accumulation of errors.

Wildberger says he hopes that "rational trigonometry" will provide high school students with a simpler way of thinking about triangles that is both more accurate and easier to carry out.

And he says the improved accuracy will be important elsewhere such as in GPS surveying or when engineers design devices.

Wildberger says he developed the new theory while studying more complex maths.

"It wasn't that I set out to do this. It was just a fluke, in a way, that I realised 'Hey these ideas can change elementary trigonometry'."

Wildberger's book, Divine Proportions: Rational Trigonometry to Universal Geometry, published by Wild Egg books, is launched this week.
