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."
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. An angle can be calculated using an equation that relates the corners of a triangle to the length 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. 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.
"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."
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. An angle can be calculated using an equation that relates the corners of a triangle to the length 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. 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.
