
How Radians Work in 30 Seconds - _pius
https://lh5.googleusercontent.com/-dJsRfi7_Crw/Utl_miUi3II/AAAAAAAA8jM/2ODyIK015WI/s450-no/How+radians+work.gif
======
ronaldx
This is by LucasVB, who is prolific at animating math for Wikipedia:
[https://en.wikipedia.org/wiki/User:LucasVB/Gallery](https://en.wikipedia.org/wiki/User:LucasVB/Gallery)

His other works are also worth checking:
[http://1ucasvb.tumblr.com/](http://1ucasvb.tumblr.com/) (for example, he has
a super-nice explanation of Fourier series).

~~~
Patrick_Devine
The first time I was able to properly visualize sine and cosine was from an
old 8mm film reel produced in the 1950s by IBM which showed essentially this
same animation:

[http://1ucasvb.tumblr.com/post/43524004530/drawing-
process-f...](http://1ucasvb.tumblr.com/post/43524004530/drawing-process-for-
the-sine-function-since)

After seeing that, trig really just "clicked".

~~~
bluecalm
I was introduced to sine in elementary school. Then we had 6 months of
trigonometry in high school. We even touched infinite series at the end of
high school. Then at univ we had some trigonometry again but not much, some
infinite series and stuff.

Nowhere along the way I was shown that sine is actually how high above the
ground you are when traveling around the circle. It was always some length of
side in right triangle divided by length of another side. That was technically
the same thing but nowhere close as intuitive and natural. Even when at some
point I got interested in math and start thinking about sine as y coordinate
it was always in relation to angle of a triangle (rooted in the center of the
circle, one point at radius, the other at x axis below it) and not in relation
of actual distance traveled along the circle. At some point radians appeared
and we were just taught that's other way to measure angles, so you know: 0
degrees is 0, 90 degrees is 1/2 of PI and 360 degrees is 2 PI. I thought it
was kinda weird to this that way but w/e probably people have their reasons...

The concept is so beautiful and natural yet it was made as artificial as
possible during math education I had. No wonder so many kids don't get or like
math :(

~~~
karlshea
> sine is actually how high above the ground you are when traveling around the
> circle.

That makes total sense, and I wish it would have been explained like that to
me, too.

My math teacher in HS spent an hour and a half explaining how derivatives work
by just working through stuff on the board after he insisted no one take
notes, and it made everything so clear. And reading through the Tau Manifesto
finally made me understand what was going on in trig at a basic level.

I totally agree with you about how artificial math education can feel, and I
think if larger concepts were introduced more like that many more people would
be able to get much better at it.

I read something a couple years ago about how some school was experimenting in
a math program starting in 8th grade where they would teach algebra, geometry,
and calculus concepts all at the same time. I guess the students were really
picking things up fast, and understanding how everything fit together better.

~~~
TeMPOraL
>> _sine is actually how high above the ground you are when traveling around
the circle._

I learned my trigonometry when trying to figure out how to rotate and move
objects around in 2D space for a computer game. I learned then, that _sin_ is
responsible for y axis, and _cos_ for x axis.

But I never ever phrased this as "how high above the ground you are when
traveling around the circle". I like the elegance of that; things would
probably have clicked for me immediately back then, had I heard a sentence
like this (and ditto for cosine being "how far to the right are you").

Bottomline, I'm in agreement with both OP and GP here.

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haberman
It's not an animation, but when I saw the visual illustration of
eigenvalues/eigenvectors on this page I was blown away:

[http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors](http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors)

I studied linear algebra in college and computed plenty of
eigenvalues/eigenvectors, but until I saw that graphic I had no idea what they
actually were. I can't believe in retrospect that a textbook would explain
them _without_ an illustration like this.

~~~
elteto
I was also blown away the first time I saw that animation, it really helped to
clarify the _notion_ of what eigenvalues and eigenvectors are. However, that
is only a geometrical interpretation which is a very particular case, as
eigenvalues/vectors appear in many many other applications which don't really
have a geometrical representation or meaning. That's probably why books tend
to stay away sometimes from particular examples, even though I have to say
that this one specifically is very useful.

~~~
haberman
Isn't the geometric formulation equivalent to the algebraic one? What makes
the geometric representation any less general? Of course in higher dimensions
you can't formulate it in terms of a drawing, but for 2 it seems to capture
the entirety of the concept.

~~~
elteto
Degrees of freedom don't necessarily have to have a geometric meaning in some
(many) systems! For example, in basic analysis of mechanical vibrations you
can use modal analysis to decompose a coupled system (two harmonic oscillators
with viscous dampers in series for example) into two independent systems. The
eigenvalues here correspond to the natural frequencies and the eigenvectors
are the modal shapes, which give you an idea of how the oscillators actually
end up moving after some disturbance. There is some geometrical significance
here but is not quite as easy to interpret.

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dbbolton
Not to be dismissive or condescending, but do a lot of people have trouble
understanding radians? I was under the impression that they were standard
material in high school trig/physics.

~~~
reuven
I got excellent grades in high-school math. I took a year of math in college.
I knew how to calculate with radians, but _never_ understood their basis until
viewing this animation, despite asking people on a number of occasions what
the benefits were vs. degrees. (They never gave me a real answer to that
question, and certainly never derived the basis for radians.)

I now feel enlightened; too bad it has been more than 20 years since I last
needed to calculate with them! However, it'll probably come in handy when I
help my children with their math homework in the coming years...

~~~
derekp7
So here's a question -- is it just the animation that helped, or is it the
combination of the animation and being 20 years older? Or, more specifically,
having the luxury of being able to throw brain power at the subject without
having to worry about learning material for several other subjects, and many
sub-topics within all those subjects, all at the same time.

~~~
reuven
I'm sure that it all helped. But I definitely remember asking, on several
occasions, why we would ever want to use radians rather than degrees, and why
they seemed to be in such weird increments. And no one ever really told me.

The animation was helpful in that it explained everything very quickly and
obviously, obviating the need for a numeric or verbal explanation.

------
keithpeter
Very nice.

Practical: take a 15cm ruler and drill small holes at each end. Thread a 1.2m
ish loop of string through the holes so there is a radius of 57cm. Tie the
string together about 20cm from the ruler.

The result lets you measure angles on the horizon, e.g. width of Moon ( _not_
Sun with children please!) and the altitude of navigation stars near rise and
set.

[http://en.wikipedia.org/wiki/Kamal_%28navigation%29](http://en.wikipedia.org/wiki/Kamal_%28navigation%29)

[http://kaloujm.com/page_kamal_uk.php](http://kaloujm.com/page_kamal_uk.php)

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cjdrake
Needs a version with tau: [http://tauday.com/](http://tauday.com/)

~~~
shurcooL
Thanks for this. Tau seems nice, I'll try using it from now on and see if it
makes things simpler (judging from what I've seen so far, I'm optimistic about
it).﻿

~~~
shurcooL
Looks good so far.

[https://github.com/shurcooL/Conception-
go/commit/78c4bad7219...](https://github.com/shurcooL/Conception-
go/commit/78c4bad7219ccf92c5a217a274f1ba1f3ef4f3c3)

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eulern
Nice visualization. Many confusions could have been avoided had "radians" been
named "radiuses". Measuring angles in spanned "radiuses" minimizes
abstractions and explains the 2*pi factor. (Radii works too.)

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dhughes
I muddled through it learning electronics not realizing I was not quite
understanding it, this helps.

Who was it that said something along the lines of " If you can describe a
complex thing to someone briefly in simple terms that shows knowledge"
Feynman? Einstein?

Whoever said that I find it profound because most people can learn something
eventually but to _know_ something, knowledge, I find close to impossible,
very hard.

~~~
hcrisp
This is probably not what you are thinking, but C.S. Lewis said, "Any fool can
write learned language: the vernacular is the real test. If you can't turn
your faith into it, then either you don't understand it or you don't believe
it."

~~~
dhughes
Similar and it's the gist of it. Understanding something well enough to teach
it to someone who doesn't, some people seem to be able to learn enough to get
by but not teach others what they are supposed to know.

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chris_wot
Nice! They should use this in schools.

~~~
Coincoin
Some do, that's how I learned them.

edit: I still think the most confusing thing about radians when learning them
is the 2 pi. This seems so arbitrary.

We now use Tau at work and it makes everything easier to understand.

~~~
RogerL
Why arbitrary? C=2pi*r. It's degrees that are arbitrary.

~~~
sp332
C = tau * r. Why divide the circle in half?

~~~
rtkwe
A = 2 _tau_ r^2. Why divide the circle in half?

The early use of pi was calculating area and it was derived based on that. At
least as far as history can tell us. Records are a little sketchy that far
back.

~~~
shawnz
The argument in favour of tau in the formula for the area of a circle is this:
To find the area of a circle with radius r, you integrate the circumference of
every circle with radius between 0 and r. The integral of tau*r with respect
to r is 1/2 tau r^2. Since 1/2 ax^2 is such a common form when dealing with
integrals, it can be easier to remember this way.

~~~
rtkwe
That's a weak argument. Integrating 2a * x is also insanely common in dealing
with integrals. And really if you're deriving the area of a circle and
integrating 2pi * r gives you pause you're not on a good road to begin with
and removing a 2 from some equations isn't going to help much.

edit: forgot that * would make everything italicized.

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ivan_ah
For those who prefer a text definition:

    
    
       radians = arc length in a circle with R=1

~~~
judk
Length of which arc?

~~~
morpher
The measure of an angle in units of radians is equal to the length of the arc
subtending that angle in units of radii.

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colund
This article is so rad!

I love when something is explained in such an excellent simple way and was a
good refresher for me. This is how things should be taught.

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alok-g
Interesting to note that the author, LucasVB, uses a custom PHP library using
GD to make these animations. He explains why he finds alternatives like
Processing, Matplotlib, etc. insufficient here:

[http://1ucasvb.tumblr.com/faq](http://1ucasvb.tumblr.com/faq)

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espeed
According to Carl Munck in "The Code"
([http://www.youtube.com/watch?v=Xw9lTB0hTNU](http://www.youtube.com/watch?v=Xw9lTB0hTNU)),
radians were the unit of measure used to build and position the megaliths.

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MatthewWilkes
30 seconds? More like 206264.

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leke
What is that little angle after the 3rd r that completes the pi-r?

~~~
edtechdev
pi is 3.141...

so it's the .141... part

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tommydiaz
Man...why wasn't I shown this gif in calculus?

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forlorn
Rocket science.

