just the other day a meme made it to the front page of reddit which originated in the math memes
subreddit this is the meme it's a variation on the drake format meme where the thing at the top is
what you don't want you're shunning that the thing at the bottom is the better version you do want
in this case we're looking at values of the sine function for different angles i will say i'm aware
that memes particularly one like this that's been around for a couple years now can evolve into a
more ironic sense and occasionally this is used to show that the bottom one looks superficially
better but actually isn't however i'm pretty sure this was posted in earnest given the title
of the post was isn't it awesome way to learn the values to be honest and the whole thing was like
organic marketing for a private tutoring firm particularly if you look through the comments
that the original poster was putting below i think they were genuinely trying to say this is a better
way to write the values of a sine function and surely it is it's got a really nice pattern in it
if you look at the sine values for 0 degrees 30 45 60 and 90 and you're at the traditional way
the values seem to have nothing to do with each other but if you rearrange them all
you get this fantastic sequence that goes from the square root of zero on two all the way up to the
square root of four on two and that obviously that's what mathematicians love that kind of
unexpected pattern just hiding in plain sight and suddenly you see it so i took this meme
and i shared it in a private whatsapp group for mathematics communicators and they hated it oh
almost universal a massive wave of dislike and disdain if i may for this meme and the reason
they all got so emotional is that the pattern in this meme doesn't exist yeah it's not real
the major red flag that something is a mess here is that a lot of this
is not in simplified form not only that it's in arbitrary form i mean who writes
zero as the square root of 0 divided by 2. i mean you can have all sorts of crazy equations
put a 0 in there the whole thing equals 0. it doesn't give us any additional insight
into the notion of 0 that you can take it square root and divide it by 2 and it's still
0. and likewise 1 who writes 1 is the square root of 4 divided by 2 i mean it's just it's the square
root of a thing divided by the square root of the thing again you can write loads of equations
and divide them by the same thing and get one what are we learning and then my poor sine of 30 i mean
a lot of this could be justified if by putting it in a more complicated form we were able to
see some kind of interesting pattern get that in a moment but now we're just we're just hiding
otherwise interesting values i love sine 30 sine 30 equals a half it's so good and yet it's now
sine of 30 equals the square root of 1 divided by 2 i mean why would you do that to sine 30.
you know what let me show you why the sine of 30 degrees is so amazing using this geogebra
file made by ben sparks link below so i'm going to turn on the sine value which just to recap
as i move this dot around the circle the sine value is just how far above the axis it is so
here we're slightly above the axis and so the sine value is currently 0.18 and as i move this
all the way around it goes all the way up until it's one it's a height of one above the axis
and when i get to 30 degrees which is about there so it's like a third of the way give or take
around that quarter you can see it hits a half and i love that a third of a quarter
is a half it's so good and then by time we go like so we're halfway across or two-thirds up
that all lines up again because of symmetry and then you get the point over here and so the whole
nature of sine and angles and circles once you hit 90 degrees your quarter turn it's one because it's
the top of the circle and when it's zero it's zero and it's such a shame to have lost that with the
way they do it and here you're getting way more detail so you can see initially the plot is pretty
much a straight line and then by the time you get up to the top here then it starts to curl out and
these are all the bits that we've got in um in the original meme but why stop there i mean you've
just it's like you've come in part way through the straight line and then it's just starting to get
interesting you need to go all the way around look at that back down now we get the full symmetry of
back down to zero and then the same thing repeats but it's flipped everything's now negative because
we're a negative amount above the axis we're below the axis so i can fill in the rest of the sine
wave just a bit let's get that bit over there ah and then back and so this for me this is the real
beauty of the sine function these cool values and and the nice it makes sense looking at it
geometrically why a third gives you a half and all these things so good but yet they've all been lost
we better get some pretty amazing insights into the patterns behind the values coming out
of a sine function to justify this reckless observation of the true values we all know
and love and sadly we don't this sequence is ridiculous for a start you've got random jumps
between the values we go from 0 to 30 and then we only go up 15 to 45 and then 60 and 90 we're
what is up with that and there's no nice pattern between the results i mean it looks like it
but there isn't they're not multiples of each other you're not adding the same amount each time
they look visually similar but there's no i mean because of the square root that pesky square root
because of it there is no nice sequence between each of these terms that's not the pattern we care
about though is it we don't care about the square root or the divide by two it's just the zero
one two three four that's the visually pleasing pattern why this image gets shared so i thought
well let's move away the rest of the stuff we don't need instead of having the sine of some
angle equals the square root of some number n divided by 2 we'll move the 2 to the other side
we'll square both sides and 4 times sine squared of theta that's what gives us the 0
1 2 3 4 sequence from the original image for the angles that they picked but why just those angles
let's plot all of them so i did i've gone from 0 to 90 degrees and those horizontal lines are
the whole like the integer values going up to four because now we're multiplying it by four
so instead of one we go up to four and any continuous function that went from zero to four is
going to go through one two and three in between we can just pick those points and there's your
meme and as you can see the line does different things in between it's there's nothing interesting
about those points other than they're just where it happens to be a whole value and you could argue
you're right they do happen at some slightly interesting points but that was more obvious in
the original top version i will concede one point though it's nice and symmetric which we didn't
have before before we were looking at a quarter of the total wave started straight and then flattened
out whereas now because we've got the square of a sine wave which is just another sine wave because
the square of the sine of an angle equals 1 minus cos twice the angle divided by 2 and that's
regular cos not squared that's a sine wave sine squared is a sine but because it's squashed in we
now have half a cycle we go from the bottom curvy bit all the way up to the top curvy bit technical
way of describing that which is half of the wavelength perfectly symmetric the straight
bit is now in the middle and that's why when we had one of those integer lines crossing
right in the middle the other two are the same distance in each direction so one's 15 degrees one
way like a third of the rest of it and one's 15 degrees the other way so even though they're not
nicely spaced you've got 0 and 90 you've got the middle one and because of the symmetry the other
ones are the same distance n i mean it's it's something so i take back some of what i was saying
about the square root all of that said the square roots and the symmetry is not enough to save this
meme it's both a bad mathematical pattern because it doesn't actually exist and it's a bad way to
learn those values even in the whatsapp group with the other mass communicators when i shared
it someone put in this way of learning the same values the same way where you pick the finger
that corresponds to the angle and the square root of the number of fingers above and below give you
sine and cos and tan that's a superior way to do this if all you want is an opaque terrible
mnemonic with no insight into what's actually going on but i want some insight and so i asked
around on that whatsapp group what people think the bottom bit of that meme should be if it's just
what's the thing which is better than what is at the top what is the version of the top
that we want to embrace and here is what i got the first response i got was from katie stackles
who just said the angles should be radians so there we are that's a lot better mathematicians
love radians partly because yes you do see some extra patterns but largely it's just a purer more
fundamental way to measure the size of an angle and actually we can see another pattern now
we've got pi on two pi on three pi on four then no pi on five then pi on six why did the original
meme maker take out pi on five or 36 degrees as we know it and you can contort that into
the square root of a thing over two except it's five minus root five on two that's what happens
if you go away from there very carefully selected specific examples up next ben sparks drew this
which is pretty good he did say he was a bit rushed and when i spoke to him i was asking
everyone if i could use their submissions he said he wanted to add a circle and given we were using
his jojoba file earlier fine we'll put in the proper version and that's really nice because now
you're going to see the logic behind where these values come from and how the structure and shape
of the wave and the circle change what the values are and how they interact with each
other so i think that's really nice and finally becky warren sent in two triangles and this is my
personal favorite in fact this is how i remember what these values are and if you have to teach
these values just you know if someone has to learn them this is the one i recommend because not only
is it easy to remember you just got to sketch those two triangles which to this day i still do
but it's not just some arbitrary pattern that's a really nice bit of logic and it shows you how
pythagoras links to these values and we've got our root 2 and our root 3 the main roots
on display and that's why i think this is the ultimate version of this ridiculous meme
well thanks so much for watching the video we really appreciate it if you're new here
do subscribe to the channel i want you to do that that's a good thing over here
they'll be hugely appreciated and not all the videos are ridiculous over analysis of memes
that's a good point all the videos are ridiculous in their own separate special
ways so do make sure you check out uh this video over here this is also pretty good