Trigonometry Redefined without Sines And Cosines

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?"

Comments

Faster calculations ??

[AeiwiMaster]

I am wondering if this could be used to make faster calculations
in raytracers and 3D engines by using integer numbers.

This reminds me of a test in grad school

[zzyzx]

I was taking a real analysis class in my first semester of grad school. I did a good job on the first question, figured everything out, and got that the answer was the integral of 1/(1 + x^2).

I got no points at all for the question - despite solving the parts relevant to the class - because I didn't know off the top of my head that the integral of that is the arctan function.

I love abstract math but I hate trig.

Re:Wow

[chris_eineke]

Here's an easy way to remember the integrals and derivatives of trigonometric functions.

But first, this small reminder:
sin x (vertical component)
cos x (horizontal component)
tan x = sin x over cos x
sec x = 1 over cos x
csc x = 1 over sin x
cot x = cos x over sin x

-> sctsct

Now we substitute these trig functions with simple symbols:

    I = sin x (vertical component)
  II = cos x (horizontal component)
III = tan x = sin x over cos x
  IV = sec x = 1 over cos x
    V = csc x = 1 over sin x
  IV = cot x = cos x over sin x

Think of them as one, two, three, four, five, and six. Now it boils down to remembering simple combinations of numbers:

integral{ I } = -II
integral{ II } = I
integral{ III } = ln | III + V |
integral{ IV } = ln | IV + VI |
integral{ V } = ln | III |
integral{ VI } = ln | I |

Once you write down more of these combinations, you'll discover patterns in it and from there on it should be easier than ever to remember trig integrals. And when you know most of the trig integrals, you will know most of the trig derivatives, too! :)

If anyone is interested in some more documentation on this, then by all means contact me. I am in the process of writing these things down. They should be available some time next week on my homepage.