WildTrig1: Why Trig is Hard

[ghostly65]

How come you don't mention the law of tangents or law of cotangents?

[didu205]

I'm getting more n more interested about trigonometry proofs, I was amazed seeing examples such as the Heron's formula failure with extreme triangles due to round off.

[absolutemes]

Hi this is really nice and I like The "New Trigonometry"

I think use word like "cheat" is really big Arclength "in circle" you can calculate by angle at in radian time the radius and its really works

Thanks

[IClausius]

So we're supposed to throw away the notion of metric space and Euclidian metric - both based on distance??

[njwildberger]

@IClausius I do not advocate throwing away mathematics. But it is necessary to be precise and careful, and if something doesn't work properly, it needs to be fixed or replaced. The notion of metric space in modern mathematics is over-rated, as well as being improperly defined.

[blue2dada]

If the degree measure of the angles of your triangle is taken to be exact, and the side opposite the 48 degree angle is taken to be exactly 11.8, I find the measures of the remaining sides to be approximately 10.1 and 13.0.

If I treat the sides as being exact I find the approximate measures of the angles (in degrees) to be 44.5, 75.0, and 60.5.

As such it seems the values of your example are not correct.

[taltor727]

Trig is not too hard, generally its beautiful and powerful. I'm going to give this rational trig a chance since it seems really cool, and I'll look into the hyperbolic geometry at the same time.

[hiyaitsbruno]

Why is Trig Even Useful in Life!!!!! GRRRRR -.-

[harvellt]

@hiyaitsbruno The magic of vectors.

[Roger4313]

thanks, i understand nothing.

[baneskrbic]

we are learning this in high school second grade...you have no idea how much it hurts....

[jacksoncrockett]

Thank you very much.

[Junkbox09]

hi norman. just a quick correction that there are in fact more than 6 trigonometric identites: there are the versine/cos, as well as the haversine/cos, and even worse, a coversine/cos and an ex(co)secant... yeah it's a lot worse guys, i am trying to amplify the rational trigonometry argument...

[njwildberger]

Hi Junkbox09 While technically that is true, what is usually meant is that there are 6 main identities. From these six main identities one can, using algebraic manipulations, generated lots of additional secondary relations. Thanks for the comment.

[JayantKumarZ]

i don like eet Q_Q

[freezzertime]

l asked my teacher once ''well, why is that?'' what l got was a shrug and ''just do the work'' as a response. But the truth is that l want to know how these things are related to real life! And l've seen that these functions mirror music and figures in general. They dont teach this type of rational approach, nobody does.

[TheDommeran]

@freezzertime People use trigonometry at sea, to measure things that you couldn't measure with a ruler, and to make buildings.

[treyguy09]

A complete waste of time learning this

[teejmd91]

nice videos! good effort chap

[isilder]

What a bunch of weasel words

Well of course regular trig is related to circles,

His "rational trigonometry" is closely related to circles too!

Humans happen to spend a lot of their time going around in circles... well doing maths on circles, so why shouldn't the text book do a lot of circles ?

Any other use of trig is like surveying or solving perverse contrived situations, I am sure the Prof here can tell you a few perverse and contrived situations where "rational trig" does better, but..

[bosseyedbill]

hi,read the comparison by Michael Gilsdorf. Trig is what turns teenagers off maths, learning the relationship between the identities. RatTrig would be easiear to teach and you would use less time compared to TradTrig.If you was going on to do a subject that needed a higher level of maths (engineering, circular movement) then you would need TradTrig.After learnig RatTrig then you you would already have an aptitude for geometry.TradTrig isnt important to teenagers and they are not going to use it

[ReconOne123456789]

bad geometry

[metalhead3767]

I don't see any practical application for this except maybe something in programming. If someone could make a course for trig applied to programming it would be much more interesting.

[santhoshharidas]

its not that v can solve only those 60/45/45 or 60/30/90 without calculators....majority f india's entrance exams won let us carry calculators at al...stil almost 40% questions are based entirely on trig and trig's applied mostly in al the questions!!so u cant say tat v cant solve other angles without a calculator!!!!!

[ajmaveli]

@santhoshharidas

dude i know... im from inida also and entrance exams with out calc is very hard

[DonFicoMontana]

I hate Maths :@

[luckeycat]

this highschool or university?

[0633767jerishowrulez]

trig is easy if you practice with the formulas as a reference.

[johnfield2009]

Fascinating , but I still don't see the advantages of treating geometry like this instead of with sines and cosines and the like. Does it have any applications in physics say harmonic analysis?

[njwildberger]

Hi meichenl:

I think I am fair to the status quo. Most students would struggle with triangles that were quite different from the 90-45-45 and 90-60-30 examples.

Your example is a cosine. What triangle are you thinking of?

The reality is that Rational Trig is simply a lot better than the current approach. Please keep watching more of the series, and find out why.

[meichenl]

Professor Wildberger,

I agree that from what I've seen so far, rational trig is both interesting and easier for students to learn.

As for a specific triangle, we could have a 90 - 67.5 - 22.5 triangle, for example. If the hypotenuse has length 1, the other two sides have lengths 1/2*sqrt(2 + sqrt(2)) and 1/2*sqrt(2 - sqrt(2)).

[njwildberger]

Yes I agree.

[taltor727]

@njwildberger Hi, I'm really interested in this, but does it have practical applications in Calculus, Linear Algebra, and Differential Equations?

[meichenl]

It's not true that there are only two triangles students can work out by hand. I worked out by hand just now that

cos(pi/8) = 1/2*sqrt(2+sqrt(2))

I'm a fan of this new rational trig stuff, but you should at least be fair to the status quo.

My guess is that your ideas will catch on better if you say, "Hey, here's an interesting new way to think about trigonometry," rather than "I figured out the correct, proper, elegant, superior formulation of trig and everyone else in the world is wrong."

[sexyjmanxx]

i hate trig !!!!!!!!!!!!!!!!!!!!!!!!

[bottle2lip]

my head hurts

[jad594]

Hi what do you reccomend for me in American high school, im really good with polynomials, but anyway theese are my choices.

Algebra

Trigonometry

Statistics

Geometry

What do you think is the easiest?

[QuantumMaths]

Geometry.

[alexander92648]

trigonometry is hard. don't take that. algebra is the easiest.

[0633767jerishowrulez]

algebra is the easiest, and you will use it the most in future math classes

[TexTK]

im lucky to have a cool trig teacher and she makes the subject more fun the way she teach it

[Tiboroun]

I don't find trig difficult, just intolerably boring. Maybe your way will be more fun to do, assuming it can work in place of "irrational trig."

[MedwayMan1945]

I think it was tantamount to blasphemy in ancient Greece to say that there were such things as irrational numbers which is why they avoided them.

Also you do not need calculus to define the length of a curve. It can be defined as the least upper bound of a set of approximations to its length by the lengths of curves made up of line segments.

Also I do not find trigonetry at all difficult.

[njwildberger]

A simple example: how would you compute the arclength of the parabola y=x^2 from (0,0) to (1,1) without calculus?

[MedwayMan1945]

Let a curve be defined parametrically on a closed interval [a, b] with parameter t with x = x(t) and y = y(t) being continuous function on [a,b]. There is no requirement that x(t) and y(t) be differentiable.

Can you give a definition of the length of the curve using calculus that does not assume differentiability of x(t) and y(t)?

The definition in terms of approximations by the lengths of line segments is rigorous and does not assume differentiability.

[MedwayMan1945]

First approximation = distance between (0,0) and (1,1).

Second approximation = distance between (0,0) and (0.5, 0.25)) + distance between (0.5,0.25) and (1,1).

Third approximation = distance between (0,0) and (0.25, 0.625)) + distance between (0.25, 0.625) and (0.5,0.25)) + distance between (0.5,0.25) and (0.75,0.5625) + distance between (0.75,0.5625) and (1,1).

And so on forming an infinite sequence the limit of which is the length of the curve between (0,0) and (1,1).

[njwildberger]

Yes, but to actually calculate this limit, and to show that it is not exceeded if we use some different subdivision of the curve, is much harder.

I think modern mathematics is too quick in defining concepts without proper consideration of how you actually compute them. Perhaps such a `definition' as you propose might better be called a `description'?

[MedwayMan1945]

It is easy to prove. Given epsilon greater than 0. Take a fixed subdivision.  Say t0 is the parameter of an end point of one of the line segments. Then there is a delta0 such that if |t t0| less than delta0 then d(t, t0) less than epsilon where d(t, t0) denotes the distance between the points corresponding to t and t0. Let delta be the minimum of the delta0s for all the end points of the subdivision. Say there are n such points.

[MedwayMan1945]

There should be a minus sign between t and t0 near the end of line 3 in my last post.

Continuation of proof:

Now take a sequence of subdivisions such that the minimum distance between adjacent end points tends to 0. Choose a subdivision where the minimum distance is less than delta. Adding the end points of the fixed subdivision to the subdivision would increase the approximation by at most 2n epsilon and would not be less than the approximation for the fixed subdivision.

[MedwayMan1945]

Conclusion of proof:

Since epsilon is arbitrary it shows that the approximation for the fixed subdivision is less than or equal to the limit for the sequence. If you have two such sequences then the approximation for a subdivision of one of the sequences is less than or equal the limit for the other. Thus the limits for the two sequences are equal.

[MedwayMan1945]

Correction to my proof:

In the continuation when I mentioned the distance between adjacent end points I meant the difference between the parameters of those points.

[lkky]

how would you compute that summatory in your video? k from 0 to 30 , -1^k((x^(2k+1))/(2k+1)!

[knowledgepars]

huh europeans measure angles with gradiant?

thats a misleading information. i am a college student at UK and we dont measure angles with grad! we either use degree or radiant or RAD.

just to let you know trigonometry is not hard hehe and this guy is like an average secondary school teacher in UK

[patilnikh]

@knowledgepars Actually, you sound like one, sorry, you don't know who this guy is!He's a Ph.D in mathematics in University of Yale.

[JHPrince]

You are very smart!

[Tidecku]

Sweet video!

[friendschampionship]

great job! thanks from Brazil :)

[cougarwolfhawk]

This was a great help, and explained quite clearly.

[Pebbe496]

One note: Europeans do not measure angels in grads (400gon = full circle). Only surveyors do that (well, I don't know any, but I've read). The confusion is perhaps because the German word for degree is "Grad".

[njwildberger]

Thanks for the clarification.

[Junkbox09]

@Pebbe496 they use it in the islamic compasses... no matter what use, grads, rads, degs just confuses everyone!

[mathdaniel]

This really good.

[sixbillionmorons]

I've been reading about your ideas since I stumbled upon your web site. (I was looking up information on set theory and was delighted to find your criticisms of infinite sets, but that's another matter.) I wanted to know if you're having any luck getting the math community to consider your revisions. I'd be surprised if you did; they've built too much on the current approaches. Is there a way you could entice them to use just part of it, opening the door to get people moving in that direction?

[njwildberger]

Mathematics is a conservative subject, but I am optimisitic that as mathematicians get used to the main new concepts (mostly quadrance and spread), they will start to see the many advantages. We can't expect to change things overnight, but there are a lot of students out there that would really benefit with having a simpler trigonometry.

[noobenstein]

@njwildberger "Mathematics is a conservative subject, but I am optimisitic that as mathematicians get used to the main new concepts (mostly quadrance and spread)"

Good luck. How many years has it taken in the past for such simplifications to occur?

Also, why would the overcomplication occur in the first place? If the greeks had it right?

I don't even like the concept of 0.5 or -1. What is 0.5 of a cat? if I divide 3 cats amoungst 2 people, then what? How do I take a photograph of -1 sheep?

[joshrocks4178]

i couldnt stop starring at the drum stick.....

made me wanna throw my h'workaway and go play music lol. gd vid.

[opinionhead444]

Trig was one of my favorite mathematical subjects. I don't know why it seemed so easy to me then, because when I tried to work out the trigonometric ratios connected to the Golden Mean, it was rather difficult.

[rockyqq]

Didn't Euclid avoid distances and angles because he did everything with a straight edge (not a ruler) and a compass (not a protractor)?

Fascinating work! I can hardly wait to see the "longest broomstick around the corner hallway" problem. To really think in spreads would be a transition.

[njwildberger]

Quite right, the distinction between straightedge and ruler is an important one, also between compass and protractor.

The longest broomstick (or ladder) problem is a good one. I will try to post that sometime. Thanks for the suggestion.

[BanhXeo]

i think functions is harder than trig. Functions with variables in the denominator, things like (x+2)/(x-9). You should post some tutorials.

[Callohcallay]

wow great idea, making trig more understandable, im in it right now, not doing too bad but happy to lean how to do it without fustration

[Nes232]

Cool

[rubixcubesolve]

Nice

[nocturnalpunk]

im wondering why trigo is hard..... trigo is the the subject that i hate the most!

[FizzyCO2]

I love your video =)

[teatown]

Thank You SIR!!!! Your effort is outstanding and the world is better because of your videos. Thanks and prosper!

[otherchaz]

In a future video, please show us how rational trigonometry applies over other fields!

[njwildberger]

Good suggestion, I will do that. Thanks.