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

blogs.scientificamerican.com

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

#91

Post by quietbritishjim »

pacaro wrote: post #16

I'm not that old, but I learned the haversine formula in a marine navigation class I took through (boy) scouts. Our instructor was a retired merchant navy officer who was somewhat upset that the curriculum didn't include it, and deemed that we should know it. My A-level (≈ AP) maths teacher was mildly amused that I knew it and digressed about log tables. We were (I believe) the last cohort to be issued log-tables, bu…

We were apparently allowed to ask for log tables in our A-level physics exams, even if we already had a calculator with us (which of course we all did). Our teacher took the step of proactively putting them on our desks before the start of the exam because that seemed to be consistent with the rules and didn't hurt.

Oh and by the way the log tables supplied by the exam board also had some useful formulae printed in them we'd otherwise have to memorise. Not that that had anything to do with it...

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

#92

Post by MayeulC »

immmmmm wrote: post #69
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…

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"

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 convention, and not really derivable[1]:

    exp(ix) = cos(x) + i*sin(x)
I personally just use Euler's notation (using exp) as a tool to derive trig formulas, since I know power composition rules quite well already, and those tend to be more useful in general.

   [cos(x)]^2 + [sin(x)]^2 = 1
Is just the Pythagorean theorem on a unit circle, which also defines sin and cos :)

[1] OK, you can look at the taylor series expansion, but you need to remember the derivatives for this.

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

#93

Post by sriku »

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

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

#95

Post by HeyLaughingBoy »

pacaro wrote: post #16

I'm not that old, but I learned the haversine formula in a marine navigation class I took through (boy) scouts. Our instructor was a retired merchant navy officer who was somewhat upset that the curriculum didn't include it, and deemed that we should know it. My A-level (≈ AP) maths teacher was mildly amused that I knew it and digressed about log tables. We were (I believe) the last cohort to be issued log-tables, bu…

Same here: haversines seemed normal to me since I was trained as a merchant marine officer and we used them in Great Circle calculations. I still have my old copy of American Practical Navigator ("Bowditch") with pages and pages of trig tables.

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

#96

Post by laszlokorte »

filmor wrote: post #86
amelius wrote: post #78

Earlier quoted context omitted.

From these two facts, you can come to the conclusion that sin(x) == 0 and cos(x) == 1, for any x. I.e., you need to know more.

No, you can't. You can come to the conclusion that if there exits an x with sin(x) = 0, then cos(x) = 1. The expression as it stands is true for every x \in R without any additional assumptions.

but you could conclude that cos and sin are both jsut the constant function: x->sqrt(0.5)

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

#97

Post by dragontamer »

MayeulC wrote: post #92
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"

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…

cos(x) and sin(x) are just the unit circles applied over the 2-dimesnional real + imaginary axis.

Because x^2 + y^2 = 1 defines a unit circle, if you perform x^2 + i * y^2 = 1, you get a unit circle over the real/imaginary planes.

Or to put it in a picture: https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Eu...

Remember the definition of sin / cos: https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and...

-------

Once you realize that "the imaginary axis" is just an arbitrary 2-dimension extension field (and that "imaginary" is a very bad name for it), it becomes way easier to see.

-------

Now if you had a 3-dimension graph (real, imaginary, and time), and then you project the unit-circle over the real + time axis, you get a sin (or cos) respectively.

-----------

Or to put it another way: there's a periodic nature of i:

* i = i

* i^2 = -1

* i^3 = -i

* i^4 = 1 -- Cycle-length 4

* i^5 = i

* i ^ x = i^(x mod 4) (In general)

This period itself forms a circle, moving from real-to-imaginary and back. (e^i^x) = (e^(ix)) therefore is periodic as well. And the most natural periodic cycle is a circle (which is of course, just sin/cos).

To make this "period 4" into a proper circle, multiply by pi/2.

i^(pi/2)^x == period 3.141592.... or the "circumference" of the unit circle.

e^i^(pi/2)^x == e^(i * pi/2 * x). Done.

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

#98

Post by HeyLaughingBoy »

segfaultbuserr wrote: post #44
pacaro wrote: post #16

I'm not that old, but I learned the haversine formula in a marine navigation class I took through (boy) scouts. Our instructor was a retired merchant navy officer who was somewhat upset that the curriculum didn't include it, and deemed that we should know it. My A-level (≈ AP) maths teacher was mildly amused that I knew it and digressed about log tables. We were (I believe) the last cohort to be issued log-tables, bu…

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 and you've just calculated where the ship was a few hours ago :-)

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

#99

Post by gmadsen »

MayeulC wrote: post #92
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"

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.

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

#100

Post by HeyLaughingBoy »

jacobolus wrote: post #37
pacaro wrote: post #16

I'm not that old, but I learned the haversine formula in a marine navigation class I took through (boy) scouts. Our instructor was a retired merchant navy officer who was somewhat upset that the curriculum didn't include it, and deemed that we should know it. My A-level (≈ AP) maths teacher was mildly amused that I knew it and digressed about log tables. We were (I believe) the last cohort to be issued log-tables, bu…

> 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…

As I mentioned in a comment upthread, I was trained in navigation as a Merchant Marine officer and we were not allowed to use electronic calculators during exams. The reason given was "what if you're in the middle of the ocean and your batteries die?" or something stupid like that. But the real reason was that our instructor wanted us to think about what we were doing. Celestial navigation involves a lot of steps and errors compound. Forcing you to think clearly about the intermediate results means that you're more likely to realize you pulled the wrong number out of a table, or your star sight was just plain wrong and you're not off the coast of Madagascar, but somewhere near the Azores instead.
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