
Trig Functions Your Math Teachers Never Taught You - RougeFemme
http://blogs.scientificamerican.com/roots-of-unity/2013/09/12/10-trig-functions-youve-never-heard-of/?WT_mc_id=SA_DD_20130913
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chrislipa
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.

~~~
mistercow
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.

~~~
FkZ
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.

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McUsr
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](http://en.wikipedia.org/wiki/Haversine).

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picomancer
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)

~~~
picomancer
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.

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greenyoda
Duplicate:
[https://news.ycombinator.com/item?id=6380140](https://news.ycombinator.com/item?id=6380140)

~~~
hayksaakian
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

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mjcohen
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."

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rotten
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?

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

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chris_wot
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!

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

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darth_aardvark
Surprised noone has posted [http://www.theonion.com/articles/nations-math-
teachers-intro...](http://www.theonion.com/articles/nations-math-teachers-
introduce-27-new-trig-functi,33804/) yet.

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

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derleth
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://en.wikipedia.org/wiki/Euler's_formula)

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

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RodericDay
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

~~~
derleth
> 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.

