WildTrig3: Spread, angles and astronomy
By Stars & Astronomy On July 30th, 2010Angles have their origin in astronomy and spherical trigonometry. Here we introduce the rational alternative, called spread, and give examples from ISO paper sizes to the faces of a dodecahedron.
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The spread proctator is nice but his circular shape make the scale not linear. Have you ever asked yourself what shape the proctator should have for the scale to be linear? (I have and I got a strange differential equation). If you found that shape I would really like to see it.
The spread proctator is nice but his circular shape make the scale not linear. Have you ever askey yourself what shape of the proctator would make a linear scale? (I have, and I got a messy differential equation). If you ever built such proctator I would really like to see it.
But ty for making video, is good
spread(alfa)=sin(alfa)
When he says spread of an angle, it means sin(angle), sinA=a/ h, where is A- angle a= opposite catheti of the angle and h=hypotenuse of an triangle.
Hi, No the scale is not logarithmic. If you google `rational trigonometry protractor’ you will find a very pleasant one (actually several) created by Michael Ossmann that you can download.
Is the scale on a spread protractor logarithmic? I only discovered your videos because I made some videos with tags “polar trigonometry” and one of your videos came up as a related video.
Just last week I was tutoring a kid in trig, showing him how the 45-45-90 triangle exemplifies 1+1=2 and the 30-60-90 triangle exemplifies 1+3=4. We wondered: why isn’t 1+2=3 exemplified by a canonical triangle? Seeing the A4 at 6:18 blew my mind. I wanted to go out and buy your book today, but my local Barnes and Noble doesn’t have it! Of course we should use square measures in the plane. Should we use cubic measures in 3D? Is that in the book? Can’t wait to get it. Great job reinventing trig!
Quite interesting, I have been using the concept of spreads and quads in graphics programming (in a limited way) without prior knowledge of your publications. Particularly for things like circle intersection testing and 2D distance calculations. The main motivation was to avoid transcendental functions, square roots, etc. as much as possible, as they are computationally expensive. It never occurred to me to look at these concepts in a more generalised way like you did. I shall keep this in mind.
Hi Prof. Wildberger, thanks for the awesome videos!
Your understanding, while common, is flawed. It is crucially important in mathematics to define all terms. The more fundamental, or `intuitive’ a notion, the more important it is to define it well.
While line, point, plane, circle can be defined completely precisely in an elementary way, `angle’ is really different—there is no elementary definition.
Please see my MathFoundations YouTube series for more about precise definitions. And thanks for the question.
I’ve got a question about the “definition of an angle”. I understand that “intuitive notions” such as line, plane, point, etc should not be defined because a definition breaks something difficult into smaller pieces, and in order to define the intuitive notions most complicated concepts are used. So i wonder if defining an angle is appropiate?
The slopes of two lines are quite easily related to the spread between them. That’s one of the beautiful results, take the x axis, the spread between any positively sloped line heads towards one as the value of the slope increases.
yeah sin^2… I guess after 1 month I am still sleepy
Actually the spread between two lines is sin^2 (angle). That’s assuming you know what an angle is, what the function sin x is, and that you are working over the `real numbers’.
With spread, and its purely algebraic definitions, no transcendental concepts are required. Bottom line—its vastly simpler, more general, and a lot easier to compute with.
ok so I watched it again, and, isnt this “spread” baically tan^2 ?
Sorry, I get excited.
Okay, I just watched again and I see now that slope and spread have little to no relation at all because you are using the hypontenuse for the denominator. hmm. Need to watch a few times more.
I’m going to try to understand this better and will watch a few times. I understand for the most part except how to convert spread into degrees and weather or not spread is equal to slope. I’ll watch again to see if I missed something. Thank you so much for these powerful videos!
Is there any webpage that explains it? My brain somehow rejects this video… maybe I’m just sleepy.
You are missing a minus sign. And it is better to think of the Law of Cosines as simply replaced by the Cross law, which is much simpler and more elegant to state. In particular, no transcendental circular functions, and no waffling about what an angle is.
Yes, I will be releasing a rational trig textbook for students, with lots of exercises, pictures and practical applications. But that is still a few years away. Although angle is deeply ingrained, these are times of change, and I am confident that once students realize how much simpler and natural rational trig is, it will be hard to stop it.
Will you be releasing a “rational trig” textbook for students? If so when? And finally, what do you honestly believe your chances are of making “rational trig” the norm in schools? It seems to me that the concept of angle is so ingrained in people of this generation and past, that it is extremely unlikely to see your new trig methods becoming the norm.
So the Law of Cosines is now
c^2 = a^2 + b^2 + 2ab(1-s)^(1/2)
where a, b, and c are the lengths of the triangle and s is the spread of lines a and b.
You are quite correct. So this will help you in connecting rational trig to ordinary trig. However the notion of spread is more elementary than both the notions of angle or sin x.
Looks to me like spread A = (sin^2 A). Am I wrong? Am I missing something important?