After more than 2000 years of silence there is a current lively discussion about trigonometry. Dr. Wildberger, an Associate Professor in mathematics in Sydney Australia (University of New South Wales) simply states, that

Generations of students have struggled with classical trigonometry because the framework is wrong,

The transcendental functions sin, cos, tan, etc. are complicated, inaccurate and computation expensive, Dr. Wildberger argues in his new book Divine Proportions: Rational Trigonometry to Universal Geometry. He goes into some details in its first chapter, which is available online.

Instead of using distance and angle as fundamental measurable quantities, Wildberger introduces quadrance and spread as first citizen quantities. Quadrance is the distance squared and spread proportional to the sine squared of the angle between two lines.

(1)Q = (x₂ − x₁)² + (y₂ − y₁)²

(2)s(l₁, l₂) = (a₁ b₂ − a₂ b₁)²(a21+ b21)(a22+ b22)

with points

p₁ = (x₁, y₁)

p₂ = (x₂, y₂)

and lines

a₁ x + b₁ y + c₁

a₂ x + b₂ y + c₂

I can see the benefits of rational math expressions here, especially for 2D computer graphics algorithms. But I can't — at least at the time of this writing — see the necessity to do completely without distance and angle.

Especially in mechanics — from an engineer's view — I really do need not so much an angle, but it's derivatives, angular velocity and acceleration. Oscillations also seem to be very hard to handle without angles and distances.

In my mechanics lecture I usually give a short introduction to vector math, where I derive similar formulas to (2) above. I published a short article about this.

Nevertheless I welcome such new ideas and would really like to read Dr. Wildberger's complete book.