An even greater secret: sinh, cosh, tanh. Don't tell anyone. This message will automatically self destruct in 5.. 4.. 3..
I came on here to say this too. If you need to hang something up with more than two wall hooks, cosh is your friend.
Obsolete trig functions and why we don't use them anymore (2013)
131–140 of 144 posts
Re: Obsolete trig functions and why we don't use them anymore (2013)
#132Earlier quoted context omitted.
This seems to imagine that the only thing being taught/worth teaching on the mathematics curriculum is number sense. Firstly, that's not the case. If you want students to answer a problem with any practical application (e.g. money/finance, statistics), or more complex problems where the point is to follow the logic, being able to use the calculator in your pocket is great. A perfectly reasonable assigned problem migh…
I think you are misunderstanding me / underestimating students. The slide rule teaches in a very direct physical way how logarithms work and how they can be used. This is something that most people never learn, but is extremely valuable. > This seems to imagine that the only thing being taught/worth teaching on the mathematics curriculum is number sense First of all, number sense is extremely important. Probably the…
> This is a reasonable problem to teach students about for like 1 week at age ~10.
I'm afraid you have a completely unrealistic expectation of the median student (which is reflected in your other comments as well). I have taught hundreds of students and I would think a handful of them could answer this at age 10.
This type of question first appears on the Khan Academy in Grade 7, i.e. targeted at 12-13 year-olds.
There is a significant body of prerequisite work in the earlier grades, some of which is limited by what is developmentally appropriate.
Students will need dedicated practice to recall this and most won't remember it after seeing it or mastering it on one occasion. It needs to be supported in the rest of the curriculum. You might be able to intensively teach it earlier, but it is hardly worth it, because they will completely forget when you intensively teach the next topic.
Many 15-year-olds will continue to struggle with this and, if they are well-supported, might be taught it as a step-by-step procedure in order to best score marks in the exam they need to. These students are unlikely to finish without a good concept of what they were doing - and might be more likely to pick it up as adults out of necessity.
Re: Obsolete trig functions and why we don't use them anymore (2013)
#133Earlier quoted context omitted.
I think the point of these was for when they read low-precision values out of a printed table and then multiplied them using slide rules. Even though the article doesn't say it, I can't think of any other practical reason to multiply by adding logarithms.
Generally, slide rules were only good for rough calculations (3 significant digits if you were lucky), but they did contain multiple scales providing multiplication, division, exponential, log10, ln, roots, sin, cos, tan, and hyperbolic trig functions. Notably one couldn’t do addition or subtraction with a slide rule; we used pencils for that. When more precision was needed, I and my fellow engineers would break out…
Back in the 1970s I used a slide rule when 2-3 place accuracy was enough; or 5-place or 7-place log tables and a pocket mechanical adding machine (rather than by hand) when more accuracy was required.
I found a 2-minute video of the sort of pocket adding machine I used, https://www.youtube.com/watch?v=ryST18JJ7VU (The device is actually easier to use than shown. Using the colors next to the digits: if silver, pull the stylus down; if red, pull up and over the curve at the top of the column to the next digit. No extra or wasted motions are needed.)
Five places were usually enough, but seven places were needed for some astronomical calculations such as eclipses. The $7 I paid for the log tables in the late 1960s would purchase about $50 today; and the adding machine was probably a couple of dollars back then.
Re: Obsolete trig functions and why we don't use them anymore (2013)
#134Earlier 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)"?
As indicated in the other reply, think of the minus as changing the sign of i instead of x.
Re: Obsolete trig functions and why we don't use them anymore (2013)
#135Earlier quoted context omitted.
Generally, slide rules were only good for rough calculations (3 significant digits if you were lucky), but they did contain multiple scales providing multiplication, division, exponential, log10, ln, roots, sin, cos, tan, and hyperbolic trig functions. Notably one couldn’t do addition or subtraction with a slide rule; we used pencils for that. When more precision was needed, I and my fellow engineers would break out…
> Notably one couldn’t do addition or subtraction with a slide rule; we used pencils for that. Huh? It's trivial to make a slide rule that you can add and subtract on, you just rule linear scales on it.
Of course you are right, the slide rule works with logarithmic scales that are “added” together to do multiplication. Linear scales could be used to add, but the low number of significant digits (no more than three on standard slide rules) meant that it wasn’t worth it, and no engineering slide rule that I ever saw had scales for addition.
Re: Obsolete trig functions and why we don't use them anymore (2013)
#136Earlier quoted context omitted.
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…
Yes, I know about the unit circle.
But that's just a way to embed some of group theory. It doesn't actually help you much.
(It's similar to how you can use eg set theory to construct the integers. It's possible, but doesn't actually help you prove anything about interesting about integers that you wouldn't have been able to prove without embedding them like this.)
> 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.
Yes, but that task wouldn't be made easier by this approach compared to starting from just the group theory axioms instead.
Re: Obsolete trig functions and why we don't use them anymore (2013)
#137Earlier quoted context omitted.
What kind of exam requires you to memorize trig identities? Were they trying to prepare you for a world without paper or computers to look them up on? Or to be a smidgen faster at hand calculations?
> Were they trying to prepare you for a world without paper or computers Yes, most high schools spend a few months teaching “trigonometry” as a system for simplifying formulas that was primarily useful historically to save time for human calculators working with pen/paper and a printed lookup table. In the modern world this is entirely anachronistic and the content could be profitably compressed down to a couple week…
I think it would be enlightening but rarely done in school, is focus on the geometry of it and how the relationships are connected, such as https://math.wikia.org/wiki/Unit_circle?file=Unit_circle_ind...
Re: Obsolete trig functions and why we don't use them anymore (2013)
#138Earlier quoted context omitted.
Well, as far as I know, cosinus and sinus have relatively little to do with exponential and imaginary numbers, just like imaginary numbers have intrinsically little to do with the 2D plane. I just think of it as a convenient notation since those obey similar composition rules (that probably stem from their odd and even-ness, and taylor series decomposition from there). So you need at least that one, since it is by co…
how can you say imaginary numbers have intrinsically little to do with the 2D plane? it is entirely possible and equally correct to define imaginary numbers geometrically. It is nice and convenient that this is equivalent to the algebraic definition.
for anyone interested, a great text on this is https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needh...
Re: Obsolete trig functions and why we don't use them anymore (2013)
#139Earlier quoted context omitted.
I think you are misunderstanding me / underestimating students. The slide rule teaches in a very direct physical way how logarithms work and how they can be used. This is something that most people never learn, but is extremely valuable. > This seems to imagine that the only thing being taught/worth teaching on the mathematics curriculum is number sense First of all, number sense is extremely important. Probably the…
> This is a reasonable problem to teach students about for like 1 week at age ~10. I'm afraid you have a completely unrealistic expectation of the median student (which is reflected in your other comments as well). I have taught hundreds of students and I would think a handful of them could answer this at age 10. This type of question first appears on the Khan Academy in Grade 7, i.e. targeted at 12-13 year-olds. The…
«Add 7.1% sales tax to the $19.95 ticket» (or whatever) is a 1-step (or generously 2-step) arithmetic problem, of the kind most students in many parts in the world learn to understand conceptually at age 6–7 with no problem. It additionally involves multi-digit multiplication, of the type people typically learn by about age 9–10 (?).
If students have not practiced solving problems enough to handle combining these things until age 13, and many are still struggling with it at age 15 (with a calculator!), that’s a serious indictment of the entire education system.
Re: Obsolete trig functions and why we don't use them anymore (2013)
#140[deleted]