Rational Trigonometry

[InferiorWang]

Rational Trigonometry:
http://en.wikipedia.org/wiki/Rational_trigonometry
http://web.maths.unsw.edu.au.nyud.ne...orman/book.htm

Anyone heard about it, yet? I read the released portions when the book was released and just found the wikipedia page for it.

Anyone have any thoughts on it? Anyone care? I think that this is an opportunity to improve on our system of mathematics and our understanding of the universe.

[Litew8stoner]

oh man I suck at math

[hippiedude]

i dont think its all that great,
basiclly its just manipulating the value of a daistance/measurment and then changing it in the end. Isn't that what algebra is? taking an undefined value (x) and with that eliminating everything else around it to find the value of x, then taking that value and putting it back into the equation it started in?

[InferiorWang]

One strength of this system is that it is algebreically representable. One problem with classical trigonometry is that our functions (sin, cos, tan) are based on what's called a power series.

i.e.
sin(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + (x^9)/9 ... and on and on

as you already know, this results in nasty decimal numbers that we have to do calulations with. Those decimal numbers are just approximations and get more accurate the further the equation is worked out. Take the sin(45*). We know that it is sqrt(2)/2. Sqrt(2) is an irrational number. We can not present it as a fraction. Therefore, we can't take an exact value to use in our calculations. With rational trigonometry, relations between lines are represented as fractions, allowing much more versatility in computation.


In addition, this system has found new relationships. They aren't really new, but mathemeticians have been searching for relationships between lines for thousands of years. As you know, in our classical trigonometry, emphasis is placed on length and and angle. Have a slanted line? Want see how far up it goes? Multiply the length the sine of the angle. How far across? Multiply the length of the line by the cosine of the angle. How many people actually understand what sine and cosine are?

This new system seeks to change the way we view trigonometry by presenting us with relationships between lines. Have you ever noticed how man trigonometric relations require squaring of our data? At the most basic level, we have sin^2 + cos^2 = 1 which is an adaptation of the pythagorean theorem a^2 + b^2 = c^2. It should then seem obvious that our data SHOULD be squared and represented differently. That's where rational trig comes in.

Instead of distance and angle, this system introduces what is called quadrance and spread. In terms of classical trig, quadrance is the distance squared and spread is the sine squared. While this appears to be a renaming of the way we already do things, possible even a simplification, but it is really much more than that. We can measure the spread of any angle by simply adding a line, to make a right triangle, then take a ratio of the quadrance of a specific two sides. This ratio, as previously stated, is equal to the sine of the angle squared and doesn't require to punch a single thing into a calculator to get the results. This finally gives us a relation between lines that can be applied anywhere and worked out by hand.

I'll come back later and fill in a little history of classical trig. Learning a little about this history of our math can put a lot in perspective about why we have to do things the way we currently do and put new ideas into perspective. Math is simply a study of relations and sometimes it takes hundred or thousands of years to put ideas together in a way that sheds new light on our world.

[Joint]

I feel like I should understand what's on that page, being a computer sci major and all, but 3 years of calculus has destroyed my brain. Oh well, back to linear algebra.

[InferiorWang]

lol just finished my second round of cal 3. I too am a CS major, but with a Math minor. I'm taking the 400 level history of math this semester and it got me thinking about rational trig again. And I have to retake my linear algebra next semester. I got over-stressed and f'ed it up.

[Joint]

*high fives for CS majors all around*

I've heard mixed things on linear algebra. We'll see how this semester goes. A lot of people in CS that I know are getting math minors since it's only a few more classes, but since I wasted my first 1.5 years of college taking engineering classes (I switched majors) I don't really have time for any minor if I want to be done in 4 years total.

[InferiorWang]

That's the only reason I'm getting mine. 3 more classes. I thought about doing engineering, but it seemed like a lot of work. Like CS is much better. I should have chosen a bs major so I could enjoy more of college. And have girls in my classes.