Trig Functions Your Math Teachers Never Taught You | Hacker News Reader

Trig Functions Your Math Teachers Never Taught You

blogs.scientificamerican.com

111 pointsRougeFemme27 comments

Your math teacher never taught you these because they're just labels attached to quantities expressible in terms of sine and cosine. They don't aid mathematical understanding, and they don't let you solve any problem that you couldn't before.

...they don't let you solve any problem that you couldn't before

But they did have a purpose, years ago. Before pocket calculators, when people used slide rules and log tables, a table of haversines helped avoid nasty loss-of-precision errors near the roots of a function.

To emphasize your response, the article goes into depth on this exact point:

> I must admit I was a bit disappointed when I looked these up. They’re all just simple combinations of dear old sine and cosine. Why did they even get names?! ...

> The secret trig functions, like logarithms, made computations easier. Versine and haversine were used the most often. Near the angle θ=0, cos(θ) is very close to 1. If you were doing a computation that had 1-cos(θ) in it, your computation might be ruined if your cosine table didn’t have enough significant figures.

For serious. Why wasn't math class just "Look, here's an empty set. Go forth and do math."?

> Your math teacher never taught you these because they're just labels attached to quantities expressible in terms of sine and cosine.

... which is why I plan to ignore tangent from now on.

And also sec, cosec, cot.

But notice they are at least not polynomial in cos and sin, whereas the functions introduced in this article are.

To be erudite about it, there are essentially no other trig functions than cos and sin, in the sense that cos and sin (aka x and y) generate the ring of polynomial functions on the circle, ℂ[x,y]/(x²+y²-1).

Versine etc. are elements of the ring and therefore they are generated (as polynomials) by cos and sin, whereas tan etc. are not elements of the ring because they sometimes go to infinity (they're elements of the larger function field).

Or conversely, sine and cosine are just labels attached to quantities expressible in terms of coversine, versine, etc.

That would apply to tangent, cotangent, secant, and cosecant as well, and yet you still learn those (even though, at least in my experience, high school trig didn't attempt to give any foundation for what the hell any of those are (other than tangent), besides their relationships to the other functions.

I don't know if it's a difference in education systems or the individual teachers, but I didn't learn about any of those other than tangent until I entered university, and then only as a side note in a calculus text book.

Tan, cotan, sec, cosec have practical value in calculus.

It might be a little unfair to expect your trig teacher to prove that to you.

If they're going to be taught, it is fair to expect the motivation for teaching them.

Given that:- sin(x) = cos( pi/2 - x ) cos(x) = sin( pi/2 - x ) you only need one of them.

Hello.

I am surprised that none of you mentioned that some of those functions, at least the haversine function saves precision. -Which is important if you do your calculations with something that doesn't have the long doubles and such.

I code some in AppleScript, and I love the haversine function since I have a precision of only 12 digits or so. Have a look at http://en.wikipedia.org/wiki/Haversine.

You can derive the identity 2 sin^2(θ / 2) = 1 - cos(θ) mentioned in the article from the double angle cosine formula,

    cos(2 Φ) = cos^2(Φ) - sin^2(Φ)

Since cos^2(Φ) + sin^2(Φ) = 1, substitute 1 - sin^2(Φ) for cos^2(Φ) in the above and you have:

         cos(2 Φ) = cos^2(Φ) - sin^2(Φ)
                  = (1 - sin^2(Φ)) - sin^2(Φ)
                  = 1 - 2 sin^2(Φ)
       2 sin^2(Φ) = 1 - cos(2 Φ)

Let θ = 2 Φ and the above becomes:

    2 sin^2(θ / 2) = 1 - cos(θ)

If you don't remember the double angle formula I used in the parent, but you remember the angle-addition formulas:

    cos(ψ + Φ) = cos(ψ) cos(Φ) - sin(ψ) sin(Φ)
    sin(ψ + Φ) = sin(ψ) cos(Φ) + cos(ψ) sin(Φ)

You can just set ψ = Φ and get:

    cos(2 Φ) = cos^2(Φ) - sin^2(Φ)
    sin(2 Φ) = 2 cos(Φ) sin(Φ)

If you don't remember the angle-addition formulas, but you remember that multiplying two complex numbers means adding their angles (arguments) and multiplying their lengths (moduli), you can just pick two unit-modulus complex numbers w = cos(ψ) + i sin(ψ) = a + bi, and z = cos(Φ) + i sin(Φ) = c + di. Multiplying them gives some complex number wz = u + vi, but because these are unit vectors, it must be the case that u = cos(ψ + Φ) and v = sin(ψ + Φ). So

    u + vi = wz
           = (a + bi) (c + di)
           = ac + (bc + ad)i + bdi^2
           = (ac - bd) + (bc + ad)i

Then equating real and imaginary parts, you get

    u = ac - bd
    v = bc + ad

If you substitute the trig expressions for the variables it becomes

    cos(ψ + Φ) = cos(ψ) cos(Φ) - sin(ψ) sin(Φ)
    sin(ψ + Φ) = sin(ψ) cos(Φ) + cos(ψ) sin(Φ)

Math is good for people who can't remember things, because you can always re-derive anything you've forgotten.

Duplicate: https://news.ycombinator.com/item?id=6380140

interesting that it did better as a repost 6 hours later.

seems noon PST / 3pm est is a better time to post than 6am est / 9pm pst

Where's the Gudermannian?

From wikipediophile:

"The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without using complex numbers."

In the article they mention that log functions make some multiplication problems much faster for humans to solve. Can they also make it faster for computers to solve?

In other words, if I'm writing a program (perhaps for data analysis) that has to do a large amount of multiplication, can I get the answers faster by converting them to log and then adding them, and then reverse look up the result?

Or, are addition and multiplication equally fast on most computers so that the time to look up log values would always be slower?

Floating-point multiplication generally isn't much slower than addition on modern machines. Logs are useful for avoiding floating point overflow/underflow, though.

It's interesting, I'm actually studying trigonometry at the moment. I have found that the biggest gap is actually understanding what sine, cosine, tangent, secant, cosecant and cotangent are. Once you realise they are lines on a circle, and the reason for the names, things clicked into place for me!

... unless you tried working out distances between lat/lon points.

Surprised noone has posted http://www.theonion.com/articles/nations-math-teachers-intro... yet.

Probably because it's linked to in the first line of the original article.

You could just as easily say there are zero trigonometric functions and do everything in terms of complex exponentiation. Euler's Formula guides the way:

http://en.wikipedia.org/wiki/Euler's_formula

http://www.mathsisfun.com/algebra/eulers-formula.html

My point is that there are multiple ways to express the underlying concepts here and picking the most appropriate way for the current context is a large part of what you learn when you learn more advanced mathematics.

For the love of god can you people PLEASE stop doing this? Taking a glance at a page, assuming the writer is an idiot, whipping up a quick one-upper, and a bunch of upvotes to go along with it?

LOOK:

I must admit I was a bit disappointed when I looked these up. They’re all just simple combinations of dear old sine and cosine. Why did they even get names?! From a time and place where I can sit on my couch and find the sine of any angle correct to 100 decimal places nearly instantaneously using an online calculator, the versine is unnecessary. But these seemingly superfluous functions filled needs in a pre-calculator world.

It's just a neat informative article, not a challenge to your manhood ffs

> It's just a neat informative article, not a challenge to your manhood ffs

Aside from the completely unfounded assumption I must be male, you completely misread my post. I was just pointing out something tangentially related that I feel to be interesting. Nothing else. Any other emotions you feel are entirely your own.

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