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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?"
Just wait until someone reimplements many software functions that rely on sine, cosine, and tangent, using these new concepts, and get a patent on them.
Inconceivable!
Actually, i think this is a perfect point. My major problem when learning integral calculus was my utter loathing for trigonometric identities (and my inability to remember them) required for solving all sorts of weird integrals. I'm curious how this stuff plays with calculus. If it does, maybe this'll make my life easier if i ever go back and attempt calculus again. anyway, reading TFA, hopefully it says something regarding this :)
There are lives at stake here!
I am wondering if this could be used to make faster calculations
in raytracers and 3D engines by using integer numbers.
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.
As my Quantum Chemistry professor once said: "Trigonometry is just two formulas: cos^2(x)+sin^2(x)=1 and exp(ix)=cos(x)+i*sin(x). I don't know how they can blow it up into an entire 16 week course"
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.
"All you have to do is be fragile and grateful. So stay the underdog." Chuck Palahniuk, Choke
If you implement a geometrical function whose input is a bunch of lengths and their ratios and whose output is a bunch of lengths and their ratios then you frequently shouldn't need any trig in your function, instead you should be using algebraic functions (+, -, *, /, nth root etc.). But many people find it easy to solve their problem by converting a bunch of stuff to angles using trig and then converting back. In fact, I optimised someone's rendering code recently by literally looking for every line of code where he'd used trig and replacing it, where possible, with the algebraic version. It worked well and speeded up the code massively. In addition I uncovered what you might call bugs. For example at one point this guy did a 180 degree rotation by converting to polar coordinates, adding M_PI to the angle and converting back. Rotating (x,y) by 180 degrees gives (-x,-y). The guy who'd written it was 'trig happy' and should have stopped to thing about the geometrical significance of what he was doing rather than just doing the obvious thing. If this book encourages people to use trig less it might be a good thing. But you don't need to talk about 'quadrance' and 'spread' to use the principles I'm talking about.
But of course there are times when you need trig. I'd hate to see the guy differentiate rotations without trig.
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
Sadly, you had this problem because those bastards never ever let you in on the secret:
e^(ix)=cos(x)+i*sin(x)
=> cos(x)=(e^(ix)+e^(-ix))/2
=> sin(x)=(e^(ix)-e^(-ix))/(2i)
Once you know this, then integration and derivation of all sin/cos and derived functions boils down to algebra and derivation and integration of e, which is trivial.
I cannot tell you how angry I was with them for not teaching me this until well after integral calculus.
Ironically, you've actually harmed, not helped, anyone who hadn't read the article. The main point of this work is that distance AND angle are the wrong things. The quadrance (square of distance) and spread (effectively, the square of the sine of the angle -- though help points out 'angle' is a handwaving concept to begin with) should be the fundamental elements and makes the trigonometry meaningful and easy. That quadrance and spread are (independently) functions of distance and angle is trivially obvious, but it is exactly the difference that is his point that makes the math so much simpler.
Perhaps a more specific example for those programmers out there. No more need for lookup tables or Taylor expansion for trig functions unless a human at some point needs an angle value at which case it can be converted in the final step before output. (Internally, programs would never need them.)