WildTrig3: Spread, angles and astronomy

[footstep002]

Hello, Prof. Wildberger. Thanks for your quick answer.

I have another question. You said spread are all same in four sides because it is defined to be between lines. But I still don't understand. So what you are saying is that no matter what spread side you mark with, the narrower side's spread is always is the defined one?

Then how do you define the wider one?

To me even if spread is all about between the lines, wider one seems different if you define from that side of the lines.

[njwildberger]

@footstep002 It is not so elementary to discuss the four regions formed by two intersecting lines. However later in this series I discuss the Triangle Spread Rules, which distinguish between convex obtuse and acute sectors of a triangle (it is really in the context of a triangle that this distinction is most valuable). For now, just try to get used to the idea that spread is between lines, not associated with any wider or narrower sides.

[footstep002]

In the video at 4:30 there's a illustration about spreads in four sides, that are equal. I get that oppsite side spread is same. What I don't get is that spread between quadrance R and P is equal to the spreads that are adjacent(the wider one).

Please explain why they are same.

[njwildberger]

@footstep002 In rational trigonometry, the spread is defined between lines, not rays.

[MrTanookiMario]

@footstep002

In more conventional terms, it's exactly the same reason that you'd get identical solutions to sin(45°) and sin(135°) (or any other such pairs that sum to 180°).

Spread isn't the same thing as angle; it's about ratios--not "width."

[isilder]

More propaganda.. spread is just sine(angle), so we really see that you are just copyring trigonometry with just some bare faced propaganda..

eg There is nothing wrong with infinity as the ratio for parallel (conicident for gap=0) lines... your spread also does not convey the distance between parallel lines..so parallel lines are rules out from spread, and yet you used it to argue against using ratios and angles ! You did no better but just claimed it was better ..only through a smokescreen .

[TheBooTya]

@isilder

spread is actually a sin squared. anyway, it is just another way of seeing things.

[nfwu]

@isilder The entire point of this series of videos (as I understand it) is to completely avoid invoking transcendental functions. i.e.: trigonometry without requiring calculators/trig tables because sine, cosine and tangent are simply unnecessary.

[danodet]

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.

[kojelja0]

But ty for making video, is good ;)

[kojelja0]

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.

[hanko2001]

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.

[njwildberger]

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.

[hanko2001]

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!

[AntiProtonBoy]

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.

[llliiilll222]

Hi Prof. Wildberger, thanks for the awesome videos!

[Viridian85]

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.

[iwantcoolname]

ok so I watched it again, and, isnt this "spread" baically tan^2 ?

[njwildberger]

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.

[iwantcoolname]

yeah sin^2... I guess after 1 month I am still sleepy :(

[iwantcoolname]

Is there any webpage that explains it? My brain somehow rejects this video... maybe I'm just sleepy.

[acmrom]

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.

[njwildberger]

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.

[masterjrm]

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.

[njwildberger]

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.

[danieloreto]

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?

[njwildberger]

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.

[ukbraintrainer]

Looks to me like spread A = (sin^2 A). Am I wrong? Am I missing something important?

[njwildberger]

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.

[IcyNami]

you went pretty fast in the last one.

[Bloodsaberxy]

you shouldve been my teacher

[opinionhead444]

I like the videos so far. I teach secondary mathematics here in the States. How likely is it for all the math communities (secondary on up) to accept rational trigonometry over the standard trigonometry we love and use today?

Furthermore, how would this relate to calculus? To integrate certain radical expression, one needs to use the circular trig functions. How do you approach that problem?

[njwildberger]

It will be a while before rational trig replaces classical trig, but I believe it will happen. The circular trig functions are indeed important for integration---but that is a more advanced subject than the geometry of triangles. In a future video I'll show how many calculus problems that involve circular trig functions can also be solved with quadrances and spreads.

[opinionhead444]

Thank you very much for answering my question, Mr. Wildberger.

[sikory]

but having a second thought about it, the problem I see isn't trigonometry, but the way I see it, it can't replace the classical concepts everywhere

[njwildberger]

Angles are needed primarily in mechanics: studying uniform circular motion. They are also important in harmonic analysis, and some other more advanced areas. My point is that they are not needed for most elementary geometry. An angle is a sophisticated concept!

[sikory]

this is a great concept, however the spread simply can't be useful in a lot of fields of science where it is needed to indicate angles(mostly rotations ) of more than 2/π since every spread(except 0 and 1) has, over the range of 2π, 4 values that have the same spread and but a different angle. But I appreciate it a lot that people like you try to question the fundamentals of today's mathematics and really come up with something useful, since I do think(not having read your book) that it's useful

[rainforestguy1]

This is exactly what I was hoping to find, thanks!!!

[otherchaz]

Would like a link to protractor and book ordering info, either in the video itself or in the description.

[njwildberger]

Please Google `Spread protractor' and `Rational trigonometry' for additional links. There are also supporting papers at my UNSW site.