Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Obsolete trig functions and why we don't use them anymore (2013)

blogs.scientificamerican.com

Post reply on HN
Print view View original

101–110 of 144 posts

Re: Obsolete trig functions and why we don't use them anymore (2013)

#101

Post by dragontamer »

eru wrote: post #77
immmmmm wrote: post #69

Earlier quoted context omitted.

As the math teacher at the first year of university once said: "i know two things: [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 but can re-derive all the rest"

Good luck deriving group theory from that. :) (Though, yes, there's a bit of a connection with group theory, but not enough to help you.)

Well, unit-circles over a finite field are a periodic "square". So you gotta redefine the norm from norm = sqrt(x^2 + y^2) into norm = max(x, y).

From there, you get extension fields from real vs imaginary already. (Ex: you can form a new extension field from x + y*j, where x and y are complex numbers), which forms a new periodic cycle.

I mean, deriving it all is hard because group theory is hard. I'm not sure if its because the tools "aren't there". Some super-AI or super-human probably can derive it all from those given facts.

Re: Obsolete trig functions and why we don't use them anymore (2013)

#102

Post by publicola1990 »

doomlaser wrote: post #4

atan2() is incredibly useful in game development. You can use it to determine the angle between objects, calculate the angle of a 2d vector, and other useful things. https://ryankubik.com/blog/atan2/

In many places though it is clearer to to use vector dot product to calculate the angle than use atan2.

Re: Obsolete trig functions and why we don't use them anymore (2013)

#103

Post by seanc »

benrbray wrote: post #66

Let me teach you a secret: You don't need to memorize any trig identities, you just need to know Euler's formula and one or two more facts: exp(ix) = cos(x) + i*sin(x) [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 That is, the complex number exp(ix), which lies on the unit circle in the complex plane, has real component cos(x) and imaginary component sin(x). Consequently, the line from 0 to exp(ix) in the complex plane makes…

The trouble is that it is rare that trig identities are used for simplifying computations.

Most often they are used for simplifying expressions. For that purpose it's not a matter of deducing what they are, but recognizing them in the expression, and spotting the opportunity to use the identity on the path to eliminating terms.

For that purpose one needs to have them memorized in order to spot the opportunity -on the carefully derived homework problem- in the wild.

Re: Obsolete trig functions and why we don't use them anymore (2013)

#104

Post by pacaro »

willyt wrote: post #58
jacobolus wrote: post #37

Earlier quoted context omitted.

> given the choice in examinations between log-tables, slide-rule, or calculator, we all made the same choice Students would benefit greatly if given a slide rule instead of an electronic calculator for their exams. The former is an effective teacher which viscerally reveals crucial insights, while the latter is pedagogically almost useless; using an electronic calculator to solve problems consists of nothing beyond…

When I did my A-levels, we were taught to give answers as surds the exams usually asked for this when we were doing trig and it was generally accepted to give answers as fractions so I think I ended up using my expensive graphic calculator once or twice in 6 hours of maths exams.

This matches my recollection too. I had a calculator, I used it almost exclusively for "wasting" time programming in class, I don't believe I ever used it in an exam.

While the graphing calculators were available we weren't told we needed them so, being on a tight family budget, I didn't have one. My sister, a few years later was told that she had to have a graphing calculator, which she claims she never used

Re: Obsolete trig functions and why we don't use them anymore (2013)

#105

Post by matheusmoreira »

benrbray wrote: post #66

Let me teach you a secret: You don't need to memorize any trig identities, you just need to know Euler's formula and one or two more facts: exp(ix) = cos(x) + i*sin(x) [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 That is, the complex number exp(ix), which lies on the unit circle in the complex plane, has real component cos(x) and imaginary component sin(x). Consequently, the line from 0 to exp(ix) in the complex plane makes…

> This trick came in handy on calc 3 exams whenever I couldn't remember the silly trig identities that were required to perform integrals.

Why do people even have to memorize stuff like this? I simply don't understand what's wrong with referencing stuff you don't remember during a test. That's what we do in real life!

Re: Obsolete trig functions and why we don't use them anymore (2013)

#106

Post by benrbray »

seanc wrote: post #103
benrbray wrote: post #66

Let me teach you a secret: You don't need to memorize any trig identities, you just need to know Euler's formula and one or two more facts: exp(ix) = cos(x) + i*sin(x) [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 That is, the complex number exp(ix), which lies on the unit circle in the complex plane, has real component cos(x) and imaginary component sin(x). Consequently, the line from 0 to exp(ix) in the complex plane makes…

The trouble is that it is rare that trig identities are used for simplifying computations. Most often they are used for simplifying expressions . For that purpose it's not a matter of deducing what they are, but recognizing them in the expression, and spotting the opportunity to use the identity on the path to eliminating terms. For that purpose one needs to have them memorized in order to spot the opportunity -on th…

I generally just know that any trig formula can be converted to quite a few different other ones visa trig identities. Whether a certain expression is more or less complex than another is a matter of perspective and context.

If I'm stuck with a trig function I don't know what to do with, I just look on Wikipedia for a trig identity and don't worry about derivations or memorization.

This trick is good for exam settings though! My memory is terrible so I always relied on tricks like this in school

Re: Obsolete trig functions and why we don't use them anymore (2013)

#107

Post by josephcsible »

sriku wrote: post #93
benrbray wrote: post #66

Let me teach you a secret: You don't need to memorize any trig identities, you just need to know Euler's formula and one or two more facts: exp(ix) = cos(x) + i*sin(x) [cos(x)]^2 + [sin(x)]^2 = 1 i*i = -1 That is, the complex number exp(ix), which lies on the unit circle in the complex plane, has real component cos(x) and imaginary component sin(x). Consequently, the line from 0 to exp(ix) in the complex plane makes…

You don't need to treat "[cos(x)]^2 + [sin(x)]^2 = 1" separately either. e^(iθ) times e^(-iθ) = e^(iθ-iθ) = e^0 = 1 = (cosθ+isinθ)(cosθ-isinθ) = cos²θ + sin²θ

Sure, this takes "[cos(x)]^2 + [sin(x)]^2 = 1" off the list, but doesn't it require you to put "cos(-x) = cos(x)" and "sin(-x) = -sin(x)" on it, to get "(cosθ-isinθ)" from "e^(-iθ)"?

Re: Obsolete trig functions and why we don't use them anymore (2013)

#108

Post by evanb »

josephcsible wrote: post #107
sriku wrote: post #93

Earlier quoted context omitted.

You don't need to treat "[cos(x)]^2 + [sin(x)]^2 = 1" separately either. e^(iθ) times e^(-iθ) = e^(iθ-iθ) = e^0 = 1 = (cosθ+isinθ)(cosθ-isinθ) = cos²θ + sin²θ

Sure, this takes "[cos(x)]^2 + [sin(x)]^2 = 1" off the list, but doesn't it require you to put "cos(-x) = cos(x)" and "sin(-x) = -sin(x)" on it, to get "(cosθ-isinθ)" from "e^(-iθ)"?

Only if you forget about how complex conjugation works. (e^-ix) = (e^ix)* = cos x - i sin x

Re: Obsolete trig functions and why we don't use them anymore (2013)

#109

Post by stan_rogers »

bregma wrote: post #88
stan_rogers wrote: post #21

Earlier quoted context omitted.

The log tables let you work without slide rules, which were themselves horribly expensive not too terribly long ago in the grand scheme of things. But yes, this was all about practical calculation , not mathematics for its own sake, that needed to be carried out quickly by people who wouldn't have handy access to machines that keep the dirty arithmetic details (and potential for error) hidden from the operator, wheth…

The converse is true: slide rules let you do calculations quickly and easily (since they hang in the leather holster on your belt) without having to carry an expensive and heavy copy of the rubber bible or similar work and stop and look up numbers then scratch the values down on paper and work the problem.

Again, you're thinking of an era where slide rules were relatively affordable. That's mid-twentieth century stuff (although the key word is relatively; my first good slide rule cost more than a month's rent at the time).

Re: Obsolete trig functions and why we don't use them anymore (2013)

#110

Post by dmurray »

HeyLaughingBoy wrote: post #98
segfaultbuserr wrote: post #44

Earlier quoted context omitted.

I recently learned about the existence of the Prosthaphaeresis algorithm [0], an even more ancient and obsolete trigonometric calculation technique. It allows you to use a trig table to do calculations when you don't have a log table. To calculate a x b , find arccos( a ) and arccos( b ), then use the product-to-sum formula: cos( a ) x cos( b ) = 1/2 x [cos( a + b ) + cos( a - b )]. For example, to calculate 42949672…

This is funny because when I was in school being taught navigation, we were told that celestial navigation would basically tell you where you were half an hour ago, because doing all the math with trig tables and a slide rule or log tables (forced to learn how to do it without electronic calculators), it would take about 30 minutes to reduce a star sight to a position on Earth surface. So now we use that algorithm an…

Is that really true? Maybe it takes 30 min to calculate from first principles, but I'd expect you could have a book of tables that tells you with just one or two lookups.
Post reply on HN