F**k trigonometry

[TexasTowelie]

I meet a lot of people, at airports and music festivals. I’m a friendly, talkative kind of person (at least when I haven’t slept in a wet tent for two nights straight). When they ask me what I do for a living, I often say I’m a mathematician. Then they pretty much always tell me how much they hate math. When I ask why, they often suggest it was trigonometry that killed any interest they might have had for the subject.

I agree, trigonometry sucks. Fuck trigonometry. It’s a terribly unmotivated subject, and as a student you are expected to memorize double angle formulas with no proofs. It’s never clear why you’re learning it, except for the possibility of later memorizing the integrals and derivatives of said functions. It’s almost a case study in how to make someone feel like math is meant to be mysterious. A few comments before you decide I’m being unfair.

First, yes, trigonometric functions are needed in Fourier Analysis, which is hugely important nowadays for the sake of music files and information compression. But by the time you’re working with Fourier Analysis, you have more mathematical technology, and in particular you know the magic formula which makes all mysterious double and half angle formulas super easy, namely Euler’s Formula:



Understanding this formula, the Fourier Analyst is equipped to work with trig functions without all the mystery, and absolutely no memorization. In fact my mom did me the favor of explaining the above formula to me when I was in high school so I could avoid all the memorization. It helped, but even then I had memorized it, and it wasn’t until college that I understood it.

Read more: http://mathbabe.org/2015/06/22/fuck-trigonometry/

[font color=330099]This was one of the theorems that I had to prove starting with the unit circle on the complex plane when I took my oral final for my math degree.[/font]

[Spitfire of ATJ]

1. Must have...

[TexasTowelie]

2. I suspect that shirt would have been a violation of the honor code

if I wore it to a math class.

The prudes on the student judiciary would have to kick some ass, YES INDEED!

[TreasonousBastard]

3. Curiously, trig was my favorite math topic, since...

it seemed to have actual uses in, well, measuring things. Even better than plain ol' geometry. Celestial navigation is a handy trig project, and IIRC, about the only reason the cosecant exists.

Never could figure out what WTF a quadratic equation could be useful for.

[Spitfire of ATJ]

4. Orbital mechanics...

But let's not get hyperbolic.

[TexasTowelie]

6. It looks like I found the topic du jour.

However, my observation could be imaginary.

[xocet]

23. That being said, Tait and Thompson would prefer that you fully embrace noncommutativity.... n/t

[xocet]

24. P.S. Since nobody mentioned this fairly intuitive method...

etc.

[lastlib]

29. Methinks we're getting a tad irrational here.........

Don't feed me your pi-in-the-sky malarkey, TxT!

[TexasTowelie]

30. Just lay it on me

so that I end up with a complex.

[lastlib]

31. I'm sure we can get to the root of it in time........

[eppur_se_muova]

16. Any function with a local minimum can be approximated by a quadratic fnxn.

We use this all the time in computational chemistry -- the Morse Potential energy curve of a chemical bond can be approximated by a parabola near its minimum, and the latter is computationally simpler to handle. Once you have three data points to work with -- the energy at three different distances, OR the energy at two distances plus a derivative, OR the energy at one point plus the first and second derivatives, you can easily find a quadratic fnxn to fit (all three!) and thus predict where the equilibrium distance, i.e. the distance at minimum energy, will be. Now calculate the energy at that point (plus maybe a first or second derivative) and add that to your previous data points (dropping the earliest one(s)) to get a new, better fit to a parabola. Continue until there's no appreciable change in energy or distance, and you have the minimum. Note that actually calculating the minimum distance by analytical methods is an insoluble problem for anything more complicated than H2 (two nuclei, two electrons).

Of course, every study of sound and vibration relies at some point on the Linear Harmonic Oscillator (LHO) model, which gives a quadratic energy function -- and when solved by the methods of differential equations, leads naturally to sinusoidal solutions, thus showing that trigonometry is not necessary for sine and cosine functions to be required!

On going from linear equations to quadratics, the phenomenon of a local minimum or maximum is introduced for the first time, marking a fundamental conceptual change which carries through to all higher polynomials. So most of what we study of higher polynomials in high school or college math involves treating them like quadratic equations, "only more so". Many, many college students know the quadratic formula from memory, but how many have ever even used the formulas for a cubic equation (they can be found in most math handbooks or tables of formulas)? Very few, because the gain in problem-solving capacity hits the point of sharply diminishing returns after quadratics. In fact, polynomial equations of degree higher than five are generally unsolvable analytically.

It may not have been obvious from high school algebra why quadratic equations are special, but they really are. Just as some phenomena emerge in three dimensions that are meaningless in two, so quadratic equations have the minimum degree of complexity for which the notions of local minima and the physical phenomena of optimum or equilibrium values can be dealt with.

[cantbeserious]

5. I Found Trig Easy - Follow The Rules - Crank The Formulas

eom

[6chars]

7. i am partial to arithmetic

i use it often.

[rogerashton]

8. In the fourth grade,

I had a teacher who assigned long division problems as punishment. And being who I am, I got a lot of punishment.

My dad taught me to "cheat" using a table of logarithms, and it was all smooth from there.

That will give you an idea how old I am. I actually had a mini-slide rule in my pocket protector.

[TexasTowelie]

9. I'm 50, but I did learn how to use a slide rule.

The slide rule competition my freshmen year in high school and I never competed in it since I was in number sense that year. I also learned how to use a table of logarithms so I had a decent mixture of old and new methodologies for math play.

[Android3.14]

10. Loved trig, but hated my trig teacher

I found solving identities absolutely fascinating as well as many other aspects. Mrs. Thompson, on the other hand, was a mean bitch who threw erasers at people, prowled up and down the rows of desks and belittled anyone whose eyes weren't glued to her. She did everything she could to crush a love of mathematics, but she failed.

BA in mathematics, beeyotch.

[bananas]

11. I was taught trig with proofs

as were some of the commenters on her blog:

DS
June 22, 2015 at 9:57 am

Trig is great. Your complaint is only about how it is taught (memorization). Teach it with a focus on proofs (the way I was taught, and the way geometry is typically taught), and it’s a great place to develop and exercise proof-writing skills and let students experience the elegance and beauty of mathematics.
...
[hr]
Helix
June 22, 2015 at 12:19 pm

I agree with this comment. Trig was part of our geometry curriculum and all formulas including the double angle formulas were taught as proofs from first principles.
...

[Erich Bloodaxe BSN]

12. I remember strggling with it for some large portion of the year, then suddenly having an epiphany

and being able to start cranking out 3 or 4 page proofs with ease.

Nowadays, I remember squat of any of it, which is sad. So many things learned, so many things forgotten over the decades from disuse.

[MichMan]

13. Couldn't disagree more

As a degreed Mechanical Engineer, I took a LOT of math in college

I have spent my career working in Quality for auto parts suppliers in manufacturing, and basic Trigonometry (along with statistics) has been by far the most useful math of anything I ever took.

Just the basics on determining triangles, and using sines, cosines, and tangents to find unknown dimensions and angles is very useful in machining and inspection. It was used daily when I worked in a machine shop by our entire skilled trades work force.

[MannyGoldstein]

14. Interesting.

Learned trig in HS, Euler's in college for Fourier transforms. I never thought about doing trig with Euler's, I need to think on that a little.

[Igel]

15. Need trig.

Teach high school physics, algebra-based. And no trig.

I look at the Russian equivalent textbooks and I see vector algebra. Work is defined as a dot product.

They don't live in a universe in which everything is either graphical vector addition and decomposition or at 0, 90, 180, or 270 degrees in a Cartesian plane. I tried using unit vector notation to help the kiddos do some vector addition and subtraction, but in the end finding the coefficients of i, j proved just as painful.

When they get to charges moving in a magnetic field, well, of course those particles are deflected. There's a cross product involved.

And when they do periodic motion, they are stuck with learning just about nothing except basic wave terminology, wavelength, frequency, wavespeed.

At the end they've been given a fairly large list of equations and learned that the goal of physics is to be able to look up the correct equation on the list, plug in numbers, and get the answer for the test.

(In high school we didn't memorize trig identities. We were expected to backtrack to prove them. My machinist brother used trig almost daily.)

When I think Euler's I think phasors and my (much despised) electrical engineering class.

I like them now. Then again, I live in Houston, home of the Houston Euler's.

[TexasTowelie]

17. Okay, your last sentence made me laugh.

Do you still have a Houston Euler's jersey?

I took trig my senior year in high school when I was 17. What was surprising was when I was 35 my next door neighbor needed some help his trig class and by knowing just a few identity formulas everything came into my head so easily from proofs to phase and amplitude shifts.

My understanding of Euler's formula is actually much better now than what it was when I was a college student. I also used my knowledge of trig in nearly every math class, physics class and even my musical theory class when I was a college student. I was the only student in my music theory class that had the appreciation of the mathematical and physics components which explain arpeggios and dissonance.

[BlueJazz]

18. "I was the only student in my music theory class"

Ha, didn't know you knew music theory. Figures though, now I know you like math.

[TexasTowelie]

19. Yes,

I played trumpet in the symphonic band when I was in college. The degree requirements for BS degree in mathematics called for at least 3 hours of performance art and 3 hours of classroom instruction in the arts so that final semester I took music theory which was an easy "A". In some ways, I view myself as a modernized jack-of-all trades and master of none.

[BlueJazz]

22. Great! I'm decent in Physics (or was). Now that I'm an avid amateur astronomer...

...,I don't use math as much. Sounds strange but the modern and expensive scopes and mounts do (almost) all the higher brain functions...well, for my little stuff.
Played piano and sax for CBS...studio player. Nice job but mechanical and sometimes boring (waiting for some diva to brighten our day with her Royal presence). Was born/cursed with perfect pitch.

Other than those 2 things, I fairly suck at most other stuff.

[Warpy]

20. Yes, I loathed trig with the hot fire of a thousand supernovae

When I got into calculus, I got my math mojo back. Trig had nearly killed it.

[TexasTowelie]

21. I cruised through trig with a 99 average.

I was the first person that my teacher ever gave a 100 when I took analytic geometry the following semester. She would only give a 100 if you made a 100 on every assignment and test; otherwise, she rounded down to a 99.

[gvstn]

25. Trig was the easiest math course that I ever took.

After the two years of algebra classes, trig after you memorized the formulas was like second grade math to me.

I had a teacher that did daily quizzes, so you had to do your homework because three failed quizzes was a letter grade off, no matter how well you did on the tests. But really there are only so many formulas you had to remember, once you had them down it was just a matter of plugging them in. Trig was the first time, I actually got an A in a math class since about 6th grade.

Geometry was just impossible for me. The first D of my scholastic career (and the teacher probably only passed me as a gift). My mind just does not work that way.

[Taitertots]

26. I remember having an argument with a teacher who refused...

To explain what happens in a calculator when you push "Sin(x)".

She probably didn't know. It was just a button that you push when you want a calculation performed.

[Dr. Strange]

28. That's why high school math teachers need to take calculus.

So they can answer the good questions!

[struggle4progress]

27. "I never could do math!" "You must be a genius!"

"I can't balance my check book!" "I've never needed math!" "My worst teacher ever was my high school math teacher fifty five years ago: I hated that class!"

etc etc etc etc



You'd get a better response telling them you were a mass murderer

[Capt.Rocky300]

32. All I remember of trig is.......

Saddle Our Horses, Canter Away Happily To Other Adventures. Or something like that.

[TexasTowelie]

33. The one that we used was:

Oscar had a horde of apples Sally can't take.

[Orsino]

34. ...get money. n/t