Trigonometry fundamentals
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Intro to trig with a lurking mystery about cos(x)2 Full playlist: https://www.youtube.com/playlist?list...Home page: https://www.3blue1brown.comBrought to you by you: https://3b1b.co/ldm-thanks----------------- Contents: Introduction -
Q1 Graph of (cos θ)² -
Q2 Translations of cos θ to (cos θ)² -
Q3 f(2x) = f(x)² -
Intro to Trig -
Q4 sin(3) & cos(3) -
SohCahToa -
Q5 Leaning Tower -
How to Compute Trig Functions? -
Q6 sin(π/6) -
Q7 cos(π/6) -
Q8 Trig of -θ -
Computing Trig Functions -
Q9 cos(π/12) -
Adv Trig Functions -
Q10 Graph of tan(θ) -
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“The most exciting part of the lecture” 
------------------ The live question setup with stats on-screen is powered by Itempool.https://itempool.com/The graphing calculator used here is Desmos.https://www.desmos.com/Curious about other animations?https://www.3blue1brown.com/faq#manimMusic by Vincent Rubinetti. Download the music on Bandcamp:https://vincerubinetti.bandcamp.com/a...Stream the music on Spotify:https://open.spotify.com/album/1dVyjw...If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribeVarious social media stuffs: Website: https://www.3blue1brown.comTwitter: https://twitter.com/3blue1brownReddit: https://www.reddit.com/r/3blue1brownInstagram: https://www.instagram.com/3blue1brown...Patreon: https://patreon.com/3blue1brownFacebook: https://www.facebook.com/3blue1brown
00:00
my end goal for this lecture and the
00:02
following lecture is to draw for you a
00:04
connection between trigonometry and
00:06
complex numbers and I really do think
00:08
when you understand that connection
00:09
between the two it strengthens your
00:11
understanding of both of them so for
00:14
this lecture we're just going to go back
00:14
to the basics on some of the
00:16
trigonometric functions how to think
00:18
about them what they mean and to dive
00:20
into it what I want to first do is
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imagine that you don't actually know any
00:24
trigonometry yet but show how just
00:26
playing around with a graphing
00:27
calculator can lead you to come to some
00:29
pretty non-trivial facts about trig so
00:33
for example let's say you've never heard
00:34
of the cosine function but then one day
00:37
you know it comes up you hear someone
00:39
talking about physics and they mention a
00:40
sinusoidal wave and the cosine function
00:42
and things like that so you go to a
00:44
graphing calculator or something like
00:45
desmos and you just poke it in and say
00:47
okay what does this thing look like
00:49
now before seeing its definition you can
00:51
understand oh yeah I can see how this
00:53
might be relevant to waves and wave type
00:55
dynamics that people might want to study
00:57
it's got kind of a cycling phenomenon to
00:59
it so maybe anytime people are studying
01:01
something associated with cycling you
01:03
could play around with the graph and do
01:04
all the things that you can with any
01:06
normal function if I tweak the input it
01:08
kind of squishes it right so as I put a
01:10
bigger constant in front of that input X
01:12
it squishes things gives it a higher
01:14
frequency okay
01:16
that's all well and good you could
01:17
stretch it out in the Y direction by
01:19
messing with the output now what I want
01:22
to do though is point out something very
01:24
interesting about the cosine graph that
01:26
is something you might want to take note
01:28
of so what I'm gonna do is square it and
01:31
I don't want to show you what it looks
01:32
like yet I want you to predict what it's
01:35
going to look like when we take cosine
01:36
of x squared okay and in particular I'm
01:40
just gonna have you focus in on the
01:41
region from 0 up to 2 pi so again if
01:45
you're just playing around what you
01:46
would notice is cosine is clearly
01:47
related to PI and probably circles in
01:50
some way so it takes a dip down 2 pi
01:53
down to negative 1
01:55
at pi excuse me and then it comes right
01:57
back up to 1 at the input 2 pi so what I
02:00
want you to do is predict what it looks
02:02
like when we take cosine of x squared
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and let's actually do this as one of our
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fun live question dynamics ok so getting
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rid of our Randi which was a very
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informed
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thing for me let's dump into our very
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first genuinely mathematical quiz
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question what does it say here what will
02:20
the graph of f of X equals cosine of x
02:23
squared look like okay and it looks like
02:26
a as an option has these sort of valleys
02:29
where in all of these it might be a
02:31
little small to read but the range is
02:32
going from zero to about two pi so in a
02:35
it hits one and then it goes down and
02:37
hits one again at the input PI and kind
02:39
of goes down with these flat valleys B
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has a similar behavior for when it hits
02:44
one and zero but it's a little spike
02:46
here so it's kind of bouncing right off
02:48
that x-axis see looks just like a
02:52
sinusoidal wave okay it looks like the
02:55
same kind of wave but it's all positive
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and it's it goes from one down to zero
03:00
then up to one in the span of pi and
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then D kind of gets scrunched together
03:05
in the X direction and starts going more
03:08
rapidly as you move farther to the right
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okay so it looks like we've got a lot of
03:12
answers coming in a fair bit of
03:14
consensus towards the top but it's not
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definitely not a unanimous decision on
03:19
this one where people are going so it
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seems like there's one most popular
03:23
answer and then to that kind of tie
03:25
closely for the silver and the bronze
03:28
medals so I'm going to give you just a
03:30
couple more seconds here if you want to
03:32
change your mind or if you want to go to
03:34
3 B 1 B KO slash live and answer this as
03:37
we're going the link is also in the
03:39
description if you just want to click it
03:41
there
03:41
clink click it there and alright I think
03:46
that's probably enough time let's go
03:47
ahead and grade this what do you think
03:51
it will be all right it looks like a
03:53
majority of you went with C which is the
03:55
one that looks like a sinusoidal wave
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this is the second most common one was a
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which has a similar shape but it's kind
04:02
of more flattened out on the bottom
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third most common was B and then the the
04:07
last most common was D and I think
04:09
that's pretty good instincts because a B
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and C which were the top three choices
04:13
there if we go back and look at our
04:14
actual cosine graph itself what you'll
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notice is that at the input 0 its 1 so
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when you square that 1 squared should
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stay 1 similarly negative 1 when you
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square that that should come up
04:25
as well and then positive one at the end
04:29
of its cycle again that's one so the
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idea that no matter what you're going to
04:32
get something that dips down to zero
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then up to one down to zero and up to
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one should make sense but the specific
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shape is what we don't know turns out as
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you just saw the answer is C so if we go
04:43
over here and reveal it looks just like
04:46
a little cosine wave which is weird if
04:49
you think about it because what we're
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doing when we square something is
04:52
affecting the output it's affecting
04:54
stuff happening in the Y direction right
04:56
but to get another cosine wave that's
04:59
oscillating more quickly it's not at all
05:01
clear that that's what would happen
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and using this even before you know what
05:06
the cosine of X means if you're just
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playing around being kind of childlike
05:09
with your graphing calculator you can
05:12
discover a pretty non-trivial fact about
05:14
trigonometric functions and then as we
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start defining them and getting into
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them you will appreciate just how
05:19
non-trivial it is because it really is
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not obvious so for this I want to take
05:24
us to a second question okay and I would
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look at this graph hard K remember that
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it looks like a scaled-down version of
05:31
the cosine graph and what I'm going to
05:33
want you to do is basically come up with
05:36
an identity based on that okay so
05:39
question number two what does it say
05:42
working from the assumption that the
05:44
graph of cosine of x squared looks like
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a scaled down version of the graph of
05:49
cosine of X which of the following
05:51
equations is true okay now because this
05:55
might take a little moment for you guys
05:56
to work out pencil and paper highly
05:58
encourage you to actively work it out
06:00
I'm just gonna give you know maybe 30 to
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60 seconds to start penciling through
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noodling it out and let's uh let's
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introduce a little bit of pause and
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ponder music just to put us in the
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problem-solving mode I think so which
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one of these reflects the fact the
06:19
cosine of x squared is kind of a scaled
06:20
version of itself
07:02
so it looks like we still have a fair
07:04
bit of answers rolling in and I don't
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want you to feel rushed so I'm gonna
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give you time to figure it out for
07:09
yourself if you want to while you're
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answering this it looks like we've got
07:16
ourselves a question that's rolled in
07:18
okay so Diego Mathemagician love the
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name asks what is the new music great
07:25
question so the guy who usually composes
07:27
the music for the channel his name is
07:29
Vince Vince Ruben Eddie I told him that
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we were doing these streams and I
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mentioned I wanted kind of a jeopardy
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here I'll turn it down to it I can talk
07:37
over it Vince your music is great but I
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want to talk about it a little bit more
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I wanted just a Jeopardy type theme but
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mixed with the usual three blue and
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brown vibe so a little clock ticking and
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I think it really puts you in the
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problem-solving mood now that is
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probably enough to noodle this one out
07:51
so I'm going to go ahead and grade this
07:53
and I'm guessing the majority have
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gotten it correct okay great job so to
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the 3165 of you that answered B which is
08:02
that it's 1/2 cosine of 2x plus 1
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congratulations you're absolutely
08:06
correct but let's let's think about how
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you might do that in a step-by-step way
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and this is one of those things that
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again if you're just noodling around on
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a graphing calculator you can come to so
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what I might do is take here I'll just
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I'll go ahead and write a new cosine
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function ok and then I'm going to start
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manipulating it so cosine of X we won't
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look at the original the first thing
08:26
you'd notice is ok we want to make it
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entirely positive because that's what
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the other one is so maybe I'm going to
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shift it up by one ok and that makes it
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all positive but now it's twice as high
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as the other graph we're looking at so
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if we want to make it equal we might
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chop it in half but it still cycles once
08:43
every 2 pi right you have to go from 0
08:46
all the way up to 2 pi before this blue
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graph that we're looking at goes through
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the cycle from 1 down to 0 now back up
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to 1 so the last step is going to be to
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squish it in the x-direction and
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sometimes people find this rainy encode
09:01
in counterintuitive that you scale up
09:04
that input that you multiply it by to in
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order
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to squish it but what's really going on
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if you think it through is we want to
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say instead of taking two pi to go
09:13
through your full cycle I want you to go
09:15
through that cycle as X ranges from zero
09:17
to PI so if we make that input to X then
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the thing being plugged into cosine will
09:22
go from zero to two pi as X goes from
09:24
zero up to PI and what you'll look at is
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the graphs are right on top of each
09:28
other right which is very cool right on
09:29
top of each other and that means you've
09:31
discovered a pretty non-trivial fact
09:33
about trigonometry and it's worth
09:35
writing down because it's one of those
09:37
things that comes up later in a very
09:40
hard to memorize way I always had a
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little bit of trouble actually
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remembering what this thing is so
09:45
writing down what you just found cosine
09:48
of x squared okay is equal to one plus
09:57
cosine of two x which is basically
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saying increase its frequency so we're
10:02
going to double its frequency and then
10:04
add one to it so it's going to shift
10:05
above that y-axis and then we're going
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to divide that by two okay so this is a
10:12
fact that we're going to turn back to at
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the very end of the lecture and as you
10:15
start to get an appreciation for what
10:17
the cosine means it becomes clearer and
10:19
clearer just how non-obvious this is and
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in particular the one thing I want you
10:23
to take note of is the weird fact that
10:25
we've got this function that when you
10:26
square it it's not equal to but it's
10:28
somewhat related to the idea of scaling
10:32
up that input now if you were clever
10:35
there is another function that behaves
10:37
very similarly to this where when you
10:39
square it it's the same as doing
10:41
something to its input and this actually
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takes us to our next question
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so let's go ahead and move past from
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this one okay and I'm going to ask you
10:49
something that seems very unrelated to
10:51
trigonometry and it'll take us until the
10:53
neck like next lecture excuse me to
10:55
realize just how related it is what is
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our question say which of the following
10:59
functions satisfies f of 2x is equal to
11:03
f of x squared okay so when you plug in
11:06
2x that should be the same as f of x
11:08
squared option a square root of 1 minus
11:13
x squared
11:14
option B log of X option C
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x-squared an option D 2 to the X okay so
11:24
I'm gonna give you a couple minutes know
11:26
a little bit to answer that we won't
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take too long on this one no it looks
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like I accidentally created it right
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away well so sorry for everyone who felt
11:36
a little bit rushed there my finger
11:38
slipped
11:39
well in either case it's now been
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revealed that the answer is d 2 to the X
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okay and we can we can see this again if
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we go and just play around a little bit
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in our desmos graphs so what we might do
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is pop over here and let me plug in 2 to
11:54
the X look at what that graph looks like
11:56
so let's not look at our cosines anymore
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2 to the X and indeed if I take this
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whole thing and I square it on the one
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hand squaring usually does a kind of
12:11
stretching motion in the Y direction but
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what's neat about this shape is that
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it's exactly the same as taking 2 to the
12:16
2x and if you know anything about
12:18
exponents this kind of makes sense
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because that 2 up here can get
12:22
distributed into that X but what this
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means is that exponents have this funny
12:28
property that when you square them it's
12:29
a lot like scaling the input and of
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course the reason I bring that up is
12:32
that you're looking at something very
12:33
similar when it comes to cosine just
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before just by playing around with
12:37
graphs before you even know what they
12:38
mean you can have this tiny little
12:40
instinct in your head that says maybe
12:42
just maybe cosine is somehow related to
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exponents it's not at all obvious how it
12:49
is but it absolutely is and that's what
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one of the things that I'd like to get
12:53
to by the end of the end of the section
12:55
like second lecture we won't get the end
12:57
of there today so with all of that I
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think it's high time that we actually
13:02
talked about what sine and cosine
13:04
actually are I don't want to come in
13:06
assuming that you necessarily know that
13:07
already
13:08
so let's have a little moment to go back
13:09
to the basics I think one of the best
13:12
summaries of trigonometry that I ever
13:14
heard and this was on reddit but I
13:17
cannot for the life of me find it
13:18
remember who originally said this is you
13:20
think that it's about triangles but
13:22
really it's about circles and this is
13:25
absolutely true I think between the time
13:27
when I very first learned about
13:28
trigonometry and
13:30
now I think my mind shifted from
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fundamentally thinking of a triangular
13:34
based definition which we'll talk about
13:36
in a moment two circles and let me just
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show you what I mean with respect to the
13:40
circles because that's the one that
13:41
explains how it comes up in physics and
13:44
often how to get intuitions about where
13:47
the values will be so instead of our
13:50
desmos graph I'm going to go ahead and
13:51
pull up a couple animations here so in
13:55
this one I'll explain what sine of theta
13:57
is where theta is the input and it's
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going to be a measure of how far you've
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walked around a circle so I want you to
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imagine that you've started on the
14:06
rightmost side of a circle that has a
14:08
radius of one okay and then as you walk
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around at a constant rate as a function
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of how far you've walked we're gonna
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graph your height which is to say your
14:18
y-axis on this grid maybe it's think of
14:21
it as the distance between you and the
14:23
x-axis and as you do that you get this
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wave that oscillates and it kind of
14:29
makes sense how it's just going to
14:30
oscillate on and on no matter how far
14:32
you walk which is why when we go back
14:34
over to the desmos graph and you think
14:36
of something like cosine of X not the
14:39
squared version let's just do the plain
14:40
vanila cosine X you know it goes on
14:43
forever continuing to cycle like this
14:44
and I guess here I'm explaining sine of
14:47
X but it works the same way if instead I
14:49
type sine of X we get a similar looking
14:53
shape the only difference is that it
14:54
starts at zero and we can take a look to
14:58
explain why here because if by
15:02
definition if it's giving you a your y
15:04
coordinate when you start off to the
15:05
right it starts at zero before slowly
15:07
increasing okay now by contrast cosine
15:11
it's defined very similarly but it's
15:13
giving you the x coordinate as you walk
15:14
around that circle so the distance to
15:17
the vertical line and in that context it
15:19
starts off at one and then as you walk
15:21
around the circle your distance from
15:24
that y-axis your x coordinate is going
15:27
to get lower it ultimately gets down to
15:29
negative one before it starts increasing
15:31
again okay so this is how you might
15:36
think of sine and cosine with respect to
15:38
circles the input is a kind of it's a
15:42
kind of distance a
15:43
a unit circle but you can think of that
15:46
as an angle and we're going to talk in a
15:47
moment about the difference between
15:48
degrees and radians but when you're
15:50
thinking about walking around the circle
15:52
if it has a radius of 1 you can think of
15:54
that input as being the literal distance
15:56
that you walk so as an example notice
15:58
what happens when we get to the distance
16:00
pi okay
16:01
pi gives you the semi circumference of
16:05
one of these circles if it's got a
16:07
radius of one then the distance that it
16:10
takes to get halfway around is PI which
16:12
is why at that input PI you see the
16:14
output go to negative one at half of
16:17
that if you turned a 90-degree angle
16:18
which means walking around a distance of
16:21
PI halves
16:21
that's why cosine is 0 at that point so
16:25
now I'm going to ask you another
16:26
question where we're going to consider
16:28
the input in terms of this idea of a
16:30
distance around the unit circle okay and
16:33
I'm just gonna have you guess at some of
16:35
the values for sine and cosine so we'll
16:38
go ahead and pull up our quiz again and
16:41
this time let's get it up on here what
16:47
does that question say without looking
16:49
it up or using a calculator which of the
16:52
following is true and it's reminding us
16:55
that that input 3 is going to be
16:57
considered in radians which is to say
16:58
the distance around the unit circle if
17:01
you're just walking along at the arc
17:02
length okay option a the sine of 3 is
17:06
around point 1 4 and the cosine of 3 is
17:09
around point 9 9 be the sine of 3 is
17:13
around point 1 4 and the cosine of 3 is
17:16
around negative 0.99 see sine of 3 is
17:21
around point 9 9 cosine of 3 is around
17:23
point 1 4 and D sine of 3 is around
17:26
negative 0.99 and cosine of 3 is around
17:30
0.14 okay so just to give you a little
17:33
moment to think that through it seems
17:37
like there's not a hundred percent
17:38
consensus on this question and answers
17:40
are still rolling in so I'm gonna go
17:42
ahead and give you a little bit of pause
17:44
and ponder music I'll let you just think
17:48
about it I don't want you to feel rushed
17:49
here
18:20
ok so I think this is probably a pretty
18:22
good time to start grading things so the
18:25
correct answer is B and it looks like
18:30
the majority of you got the correct
18:31
answer which is to say that sign is 0.14
18:35
and cosine is negative 0.99 now the
18:38
specific numbers here don't matter
18:39
what's actually relevant is just kind of
18:41
whether it's positive or negative and
18:43
whether it's very close to 1 or if it's
18:45
closer to 0 so if we think through what
18:47
this might mean in the context of our
18:49
graphs here the cosine which remember is
18:52
measuring the x-coordinate of your
18:54
little dot as you walk around the circle
18:57
what is that going to be when you input
18:59
3 well what does it mean to walk 3 units
19:02
around the circle
19:03
well like I just mentioned walking PI
19:05
units around the circle gets you halfway
19:08
around this is really the way to think
19:09
about pi is it's you've got a unit
19:11
circle it takes you halfway around so to
19:13
walk three units is gonna be something a
19:15
little bit shy of that ok and if you
19:18
notice there's very little change in the
19:20
x direction as you're at that left side
19:23
of the circle so when it approaches one
19:25
it gets really close to one and then it
19:27
doesn't really change that much so maybe
19:29
it shouldn't be too surprising that our
19:30
cosine of three turned out to be around
19:32
0.99 what what excuse me what's relevant
19:35
is that it's negative right it's very
19:37
close to negative one so it's close to
19:39
negative 0.99 now on the other hand if
19:41
we were doing this with sign which same
19:44
game but it's measuring your
19:45
y-coordinate when you walk almost
19:47
halfway around which is to say almost pi
19:51
radians well there's still a little bit
19:54
of height left and importantly it's
19:56
positive because you're still above that
19:57
x-axis so when we turned back to our
19:59
question the only thing differentiating
20:02
our answers was that some of them would
20:05
switch whether sine or cosine was the
20:07
one that's close to negative one and the
20:08
others would just switch the sign s IGN
20:11
whether it's positive or negative so
20:14
congratulations to the 3152
20:16
of you who got that one correct so
20:20
that's loosely how you might think about
20:21
sine and cosine if we're coming at it
20:23
from the direction of circles but this
20:25
is very different from how it's usually
20:26
taught if we're doing things like in a
20:29
typical high school class where you
20:30
start off thinking about right triangles
20:32
one is not better they're just
20:33
very different and worth understanding
20:35
the connection so the classic thing when
20:38
it comes to trig functions is a little
20:42
statement that everyone has ringing
20:44
through their head called so kah TOA
20:49
okay and what you're supposed to
20:52
remember from this is if we have a right
20:54
triangle so let me go ahead and draw a
20:57
right triangle so we know one of the
21:02
angles is 90 degrees and if you know
21:04
another one of the angles theta then
21:07
consider one of the sides to be the
21:10
hypotenuse and then you ask what what of
21:13
the other two legs is adjacent to that
21:16
angle and we label at a for adjacent and
21:18
which of them is opposite so then
21:21
sohcahtoa if that's a phrase that you
21:23
remember running through your head
21:24
tells you that the sine of this angle
21:28
this is the definition of sine for some
21:30
kind of angle that we input is for such
21:33
a right triangle it's the opposite over
21:35
the hypotenuse that's what this is
21:37
telling us opposite over hypotenuse
21:39
similarly the cosine of that angle is
21:44
going to equal the adjacent over the
21:47
hypotenuse so that would be adjacent
21:48
over hypotenuse okay and then the
21:52
tangent of that angle is going to be the
21:59
opposite over the adjacent okay so that
22:01
opposite side o over the adjacent side
22:03
so there's the classic thing to remember
22:05
if students come out of a trig class if
22:08
they know one thing they typically know
22:10
sohcahtoa right and to give us an
22:13
example of how you might apply this I'm
22:16
gonna go ahead and pull up another
22:17
question here for you which is gonna ask
22:20
us about a leaning tower a leaning tower
22:22
of pizza clearing out the last one let's
22:27
read what our new question has to say
22:30
suppose that a tower is 100 meters high
22:33
after several years this tower the tower
22:35
starts to lean so instead of making a 90
22:38
degree angle with the ground it makes an
22:40
80 degree angle when the Sun is directly
22:43
overhead what is the length of the
22:46
tasked by this leaning tower okay the
22:50
length of the shadow cast by this
22:51
leaning tower I would highly encourage
22:53
you to try drying this out yourself I'm
22:56
absolutely gonna draw it for you in a
22:57
moment but it's more fun if you're
22:59
engaged and if you have a pencil and
23:01
paper to noodle it out you could also
23:04
try to imagine it in your head if you
23:05
want but for me at least I always think
23:07
much more clearly
23:08
once I have pencil and paper no matter
23:09
how much I think I can think things
23:10
through in my head so let's give you a
23:14
little pause and ponder time on this one
23:42
looks like we have some very strong
23:44
consensus on this one which is very
23:47
reassuring and I guess one thing that
23:53
should maybe go without saying
23:54
especially given that I've turned the
23:56
super chat or the live chat into the
23:58
very aggressive slow mode this is most
24:00
fun if you don't know what the majority
24:03
of people are already thinking right the
24:04
whole idea of the statistics is right
24:06
now you can see there's a bandwagon
24:07
right there's twenty six hundred people
24:09
who are voting for one of the answers
24:12
who think that's it but we don't know
24:14
what one of those answers is so if you
24:17
have the answer try not to just blurt it
24:19
out in the live chat that's kind of the
24:21
equivalent of a student in a classroom
24:22
where when the teacher asks something
24:24
rather than raising their hand to ask it
24:26
or submitting it quietly just shouting
24:28
okay I get it you're excited you really
24:30
want to do it but in this case it's a
24:32
little bit more fun if we have this low
24:33
reveal all right so that's probably
24:36
enough time for people to have rolled
24:38
their answers in now let's go ahead and
24:40
see what they see what the majority has
24:43
submitted and the correct answer is B
24:49
which is 100 times the cosine of 80 and
24:52
it looks like the second most common
24:53
answer was a which is a hundred times
24:55
the sine of eighty so it looks like you
24:58
guys correctly knew that you were
25:00
multiplying it by one of these
25:01
trigonometric functions but there might
25:03
have just been a little swap up for
25:04
which one was which so let's think this
25:06
through okay let's go back to our paper
25:08
and our question had us imagine a tower
25:12
which is a hundred meters tall and
25:15
specifically it said it was a hundred
25:17
meters tall and then it started tilting
25:19
so we can think of the the length of the
25:21
towers 100 meters that's not necessarily
25:23
how far its new top is off of the ground
25:26
and then it has this imagine that
25:28
there's the Sun directly overhead okay
25:31
so it's going to be casting a shadow
25:33
perpendicular to the ground that's what
25:35
gives us our nice right triangle and it
25:38
further specified that the angle here
25:40
was 80 degrees okay so the thing we want
25:43
to know is this side here the length of
25:46
the shadow which if we look at our
25:47
sohcahtoa that's the adjacent side so we
25:50
might write for ourselves cosine of this
25:53
angle which is
25:54
80 degrees is equal to that adjacent
25:58
side the thing that we want to know I'll
26:00
just call it X or maybe I'll call it s
26:02
foreshadow tried to give our variables
26:04
readable meanings divided by the
26:07
hypotenuse which is a hundred so that
26:10
means that the shadow once we rearrange
26:13
is 100 times the cosine of 80 degrees
26:18
now already this feels very different
26:21
from the thought of trigonometry via
26:23
circles like I said you start to think
26:25
about it in terms of triangles later it
26:26
comes to circles
26:27
one of the main differences here is that
26:30
we're dealing with a different unit
26:32
right here I was saying degrees and
26:34
that's very natural when we're in the
26:36
context of triangles we could also talk
26:37
about angles in terms of radians like
26:39
I'll say in a moment but we humans
26:42
assign numbers a numerical system to
26:44
things we really like whole numbers
26:46
between 0 and 100 that we just love that
26:48
instead of thinking of proportions we
26:50
think in terms of percentages which is
26:52
the same thing but it just turns it to
26:53
whole numbers between 0 and 100 when we
26:56
think about like sound with decibels
26:58
which scales a whole bunch and it's got
27:00
this crazy exponential pattern to it we
27:02
logarithmic really change it and we have
27:04
this very awkward system that's really
27:05
twisting the math behind its back so
27:07
that we can think of whole numbers
27:08
between 0 and 100 same deal with what's
27:10
going on with degrees but mathematicians
27:12
don't really like that they don't really
27:14
like to think in terms of degrees
27:15
because it's so unnatural instead they
27:19
want some sort of natural unit something
27:20
where you imagine if you talk to an
27:22
alien civilization about math they would
27:24
have the same conventions so how does
27:26
any of this connect to the unit circle
27:28
and the idea of the distance around that
27:30
unit circle that we were walking like we
27:31
saw earlier well let me just pull up a
27:35
pre-printed unit circle because trust me
27:37
you don't want to see me try to draw a
27:38
circle what's neat here is we can for
27:42
each of these triangles imagine
27:44
rescaling it so that that hypotenuse is
27:47
really one because all we care about our
27:49
ratios of lengths of sides so we might
27:51
as well say okay let's scale this so
27:53
that the hypotenuse is 1 meaning we
27:55
divide everything by H and what that's
27:58
going to get us is the opposite side is
28:01
now o over H whatever that was and then
28:05
the adjacent side is a over H
28:07
meaning that the wykel component here is
28:12
sine of theta where theta was our angle
28:15
down here okay and then that bottom part
28:19
is cosine of theta okay and if we want
28:25
to think about all possible right
28:26
triangles who have a hypotenuse of one
28:28
well you might think of is taking a unit
28:30
circle because that's saying every point
28:33
is a distance of one away so think of
28:35
that line as the hypotenuse of a
28:37
triangle if you have a triangle with a
28:39
bigger angle it shows up there if you
28:41
have a triangle with a smaller angle it
28:43
shows up there and it says if you take
28:45
all possible right triangles and you fix
28:49
them so that their points are all in
28:51
common and search that their hypotenuse
28:53
is are all one you scaled them down and
28:55
the other tip of the triangle is going
28:58
to trace out a unit circle like this and
29:00
each one of these points for the
29:02
corresponding angle is going to have
29:04
coordinates cosine of theta as the
29:08
x-coordinate and then sine of theta for
29:12
the y-coordinate and I'm just writing it
29:14
vertically so I can fit it all on here
29:16
okay now typically like I said
29:19
mathematicians like to think in terms of
29:21
radians which is really a way of saying
29:23
if we know that the radius of our circle
29:25
is 1 what is the distance that you've
29:29
walked along the outside here yeah so
29:33
for example if you were to say 180
29:35
degrees excuse-me degrees which is
29:39
walking halfway around the circle that's
29:41
the same as pi radians and if you were
29:47
to take something like 60 degrees okay
29:50
which would maybe end up oh I don't know
29:52
around here that might be 60 degrees
29:56
well that's a third of that you're
29:58
walking a third of the half turn around
30:00
the circle everyone who's an enthusiast
30:02
about tau is sort of yelling right now
30:04
because it would make the conventions
30:05
easier but pi is the standard so we're
30:07
working with it it's PI thirds radians
30:14
so from this point forward and really
30:16
for a lot of your math career I would
30:17
encourage you to in every time you want
30:19
to think in terms of degrees where that
30:21
makes the angle intuitive try to
30:22
translate it to radians because that's
30:24
going to be the more natural way to
30:25
understand it and the farther you get
30:27
down the road when it comes to how sine
30:29
and cosine play into calculus how you
30:31
actually find values of them almost all
30:34
of the time it's easier to think of that
30:35
input in in terms of radians so I think
30:39
it's high time I I asked you a couple
30:40
questions about these and you might
30:44
wonder you know how do we actually
30:45
compute some of the values right how to
30:48
compute and the almost answer is that
30:53
it's hard if there was an easy answer to
30:55
say oh well this is the formula that
30:57
shows you what sine of theta is we
30:59
wouldn't have given sine of theta a
31:01
fancy name we would just always work in
31:02
terms of that formula and we would call
31:04
it the circle y-coordinate formula and
31:06
it might get a name but it wouldn't be a
31:08
special function it wouldn't have a
31:09
special button on your calculator
31:10
because effectively what you have to do
31:13
is before you know fancy techniques
31:16
associated with calculus or associated
31:18
with even fancier things like what's
31:20
known as the quartic algorithm you could
31:23
just have to have a handful of values
31:24
that you do know that come from
31:25
geometric settings where there's enough
31:27
symmetry so for example let's say I try
31:30
to draw an equilateral triangle okay an
31:35
equilateral triangle will be nice and
31:36
symmetric and we're going to be able to
31:39
leverage that symmetry to figure out a
31:42
concrete value of sine and cosine
31:48
this looks roughly equilateral wouldn't
31:51
you say and let's say each of those side
31:55
links is one because you know if we get
31:58
to choose our side lengths why not one
32:01
because remember sine and cosine have
32:03
everything to do with just ratios so on
32:06
an equilateral triangle this angle here
32:09
is 1/3 of the total the total is 180
32:12
degrees so you might think of this as 60
32:14
degrees but like I said earlier I would
32:16
encourage you to try to get in the habit
32:18
of thinking of that instead of 60
32:19
degrees say okay the full angle is PI so
32:22
a third of that is going to be PI thirds
32:26
and then similarly this one up here is
32:28
half that angle here's where we're
32:30
leveraging that symmetry to some extent
32:32
that one is 30 degrees which in terms of
32:35
radians is now a sixth of a a sixth of a
32:39
half turn all the Tallinn theists get
32:41
very mad at this point but again hey
32:43
it's the convention this is the thing
32:44
that you should come out of a trig class
32:46
knowing in your head
32:47
how to think in terms of radians
32:48
measured in PI because that's how we
32:50
that's how we often write it down in
32:52
tests and such so that one will be PI
32:55
sixths so one thing I might ask you now
32:58
is if you can leverage the symmetry of
33:00
the set up the fact that equilateral
33:02
triangles are going to be very nice and
33:03
that the right triangle coming about
33:04
from it is very nice to answer something
33:07
like a very concrete calculation what is
33:11
the sine of PI sixths okay what do you
33:15
think it is and in fact let's do this as
33:17
another live question which as you can
33:19
see I'm like a kid in a candy shop I'm
33:20
just very enthusiastic with my new toy
33:22
so let's go ahead and use it pull up
33:26
another question here and what is our
33:28
question asked what is the sine of pi/6
33:34
of course so again if you want if you're
33:37
just joining now you can hop over to
33:39
three b-1b cows live you can answer that
33:42
I imagine the best way to do this if
33:44
you're doing it live is you might be
33:45
answering on your phone watching on
33:47
another screen or if you want to switch
33:48
between tabs that's also good now while
33:52
those answers are rolling in let's take
33:55
a look to see if we have anything from
33:56
the audience and we do when you do maths
33:59
problems do you use pen or pencil for
34:02
sure I use pencil one I'm very error
34:05
prone so I'm gonna make mistakes all the
34:07
time the only reason I'm writing in pen
34:09
now is because it shows up better on the
34:10
camera and let me just tell you with
34:12
every stroke that I write I'm deeply
34:14
terrified that there's about to be an
34:15
error because I know there will be so if
34:17
that terror doesn't show in the shaking
34:19
of my hands then you'll start to notice
34:22
it from this point forward back to our
34:25
question though it looks like we've got
34:27
pretty strong consensus now around two
34:30
thousand ninety-one people agreeing on
34:32
one of the answers and then a pretty
34:34
even split among the other ones oh no no
34:37
I guess the length of the bar is a
34:38
little whoever worked on this game
34:40
you may be a little bit misleading it
34:43
looks like one of the answers is much
34:44
less popular than the other two but I
34:47
guess the its considering them to start
34:49
from somewhere like that maybe we should
34:52
have some axes on that are indicating
34:53
that so there's one very unpopular
34:55
answer and then one very popular answer
34:57
so let's go ahead and see what that
34:58
turns out to be correct answer is
35:04
one-half
35:05
congratulations to the twenty eight
35:08
hundred no no twenty 915 of you coming
35:11
into the last minute there to poll in a
35:13
correct answer okay so why is that the
35:15
case right why is he like necessarily
35:17
gonna be one-half well remember sine of
35:20
pi/6 we're asking sohcahtoa to kind of
35:24
remember in your head sine is opposite
35:25
over adjacent opposite over hypotenuse
35:28
excuse me
35:29
so with respect to this smaller angle
35:32
and they all change colors with respect
35:36
to this smaller angle the opposite side
35:38
is this segment here and because of the
35:42
symmetry of our triangle that's going to
35:44
be one half because each side length of
35:47
the equilateral triangle is one so half
35:48
of it should be one half and divided by
35:51
the hypotenuse but by definition the
35:53
hypotenuse was one here so one half
35:55
divided by one it's just the same as one
35:57
half now slightly trickier is if I'd ask
36:01
you the cosine of pi/6 and let's do this
36:05
one again as a live quiz but maybe a
36:06
little bit more quickly this time I'm
36:08
gonna hop over get to our next question
36:11
and as you might expect the next
36:13
question is going to ask you to find the
36:16
cosine of pi divided by six there we go
36:18
and your options are a 1/2 be square
36:23
root of 2/2 see square root of 3/2 or d
36:28
square root of 5/2 if you can I would
36:32
HIGHLY encourage you to be keeping your
36:34
own notes right so if you wanted to kind
36:36
of check back on your own piece of paper
36:37
you could take a look through it as
36:39
you're working out these questions but
36:41
for right now I'm just going to give you
36:42
maybe 20 seconds or so to kind of noodle
36:45
to it think of what it might be get a
36:47
little pause and ponder music going
37:10
okay once again it looks like we have
37:15
some pretty kind of strong consensus
37:17
from about the same number of people
37:18
that we had before so let's go ahead and
37:20
grade the answers and the correct one is
37:24
C square root of three divided by two so
37:28
congratulations to the 3183 of you who
37:31
knew that it was square root of three
37:33
divided by two
37:33
now how do you figure something like
37:36
that out well the trick is basically if
37:39
you have one of these right triangles if
37:41
you know one of their sides then you
37:43
know both of them so let me draw a
37:44
little copy for ourselves here we know
37:48
that the hypotenuse is one one of the
37:50
side lengths is one half and the only
37:52
thing that's missing if we're thinking
37:54
the cosine of 30 degrees is this
37:56
adjacent side so because we were doing
37:58
it with the angle up there adjacent side
38:00
is here now the Pythagorean theorem
38:02
tells us that that adjacent value
38:05
squared plus the other leg length
38:07
squared is going to equal one squared
38:12
our hypotenuse squared which means that
38:16
value is equal to the square root of one
38:20
minus well what's 1/2 squared it's 1/4
38:23
which means it is the square root of 1
38:27
minus 1/4 is 3/4 so we can go ahead and
38:31
take the square root of that bottom and
38:32
just leave the radical on the top and
38:34
what we get is the square root of 3 all
38:36
divided by 2 now this is a value that
38:39
any anybody coming out of a trig class
38:41
becomes very familiar with they're like
38:43
okay if the answer isn't 1/2 it's going
38:45
to be square root of 3 over 2 or root 2
38:47
over 2 also comes up in another
38:49
circumstance but this is one of those
38:51
that kind of becomes burned in your
38:52
memory square root of 3/2
38:54
and one very important fact that's come
38:57
about here that's worth highlighting is
38:59
that how does the Pythagorean
39:02
Pythagorean theorem play into our
39:06
trigonometry because if we have a right
39:08
triangle whose hypotenuse is 1 and we
39:11
know that one of the side lengths let's
39:12
let me switch to thinking with respect
39:14
to this lower left angle again if one of
39:16
the side lengths is cosine of theta and
39:19
one of them is sine of theta
39:24
well the Pythagorean theorem tells us
39:26
that cosine of x squared plus sine of x
39:33
squared is always equal to 1 squared
39:36
okay now this is a very important
39:40
identity it basically tells you if you
39:41
know cosine you know sine if you know
39:43
sine you know cosine and it also is
39:45
giving you a little reminder that it's
39:47
all really about the Pythagorean theorem
39:48
in fact this is equivalent to the
39:51
Pythagorean theorem because if you think
39:52
of how cosine and sine are defined it's
39:54
just defined in terms of the leg lengths
39:56
of a right triangle and then everything
39:59
scaled down as needed
40:00
now there's one very silly convention
40:02
with how we write these things so we've
40:04
already started to make friends with the
40:06
cosine of x squared function and the
40:07
fact that is a little more interesting
40:09
than you might think at first sight but
40:11
you never see it written down as open
40:13
parenthesis cosine of X close
40:15
parenthesis squared because people don't
40:17
want to write too many parenthesis
40:19
instead they write cosine squared of X
40:23
plus sine squared of X it's all very
40:28
nice how lovely
40:31
we don't have to write too many symbols
40:32
what a wonderful convention except for
40:35
the fact that literally every other time
40:37
in math you see something written as f
40:39
squared of X what that means is not F of
40:44
x squared that's almost never what it
40:46
means instead what it's supposed to mean
40:49
for most of math is taking a function
40:51
and applying it to itself
40:54
you're like layering it over itself
40:57
that's what that too usually refers to
40:58
sometimes later on it refers to the
41:01
second derivative especially if it has
41:02
some parentheses around it but the idea
41:05
is that it's very different from the
41:07
convention in trigonometry trigonometry
41:08
is just flying by its own rules and
41:11
saying yeah yeah we've got our own
41:12
conventions for what it means when we
41:14
feel a little too up there I understand
41:16
some students will be confused but I'm
41:17
trigonometry I don't care I'm moving
41:19
forward my conventions and that's fine
41:22
right you get used to it this is the
41:24
standard way that you'll end up seeing
41:25
it what I was writing earlier when we
41:27
were thinking about the cosine squared
41:29
function which again is very interesting
41:32
you would almost always see it written
41:34
as Coast squared of X
41:37
now very interesting is that this is not
41:41
just a consequence of the Pythagorean
41:43
theorem that gives us a relation between
41:45
our trig functions at the end of the
41:47
lecture I want to show you how we can
41:48
use trig functions to prove the
41:50
Pythagorean theorem in a sneaky way and
41:53
it's pretty sneaky like it's it's not
41:54
one of the common proofs that you'll see
41:56
that's associated with you know
41:57
rearranging triangles or anything like
41:59
that but it is very visually satisfying
42:01
so that's something that we'll get to by
42:04
the end of the lecture but in the
42:06
meantime I want to think through a
42:08
couple more of our of our trig function
42:11
questions okay so if we kind of come
42:13
back up here we just found cosine of PI
42:15
sixths which is square root of 3 over 2
42:19
but let's answer it for a couple more
42:21
complicated ones right so for I'm
42:23
dealing only with angles that were
42:24
between 0 and 90 degrees or I should say
42:27
between zero and PI halves but let's try
42:29
a question that's a little bit different
42:31
from that and what if I instead said
42:34
cosine of negative 2 PI over 3 ok now
42:43
that's one that is hard to think of in
42:45
terms of triangles because what does it
42:47
mean to have a negative angle so this is
42:49
a good reminder that cosine and sine
42:52
even if you think they're about
42:53
triangles are really about circles and
42:55
remember if you walk all the way around
42:58
the circle and if it has a radius of one
43:00
walking all the way around the circle is
43:02
going to take you a distance of 2 pi and
43:04
so if you want to think about 2 pi
43:06
thirds what you might start by doing is
43:08
chopping the triangle into thirds and
43:12
then using that to help guide the
43:14
intuitions but again I kind of want to
43:16
do this as a live quiz so let's go ahead
43:18
and pull up the next question and have
43:25
you guys answer this for me
43:26
what do you think the cosine of two pi
43:29
divided by three now let's go ahead and
43:31
pull it up cosine of negative two pi
43:34
divided by three yeah so again if you're
43:38
keeping notes that can be very helpful
43:39
you reference your notes now look just
43:42
to make sure that you have enough time
43:43
to think this through I'm gonna go ahead
43:45
and give you a little bit more of that
43:46
pause and ponder music
43:59
it looks like one of you figured out how
44:02
to submit an answer that is not one of
44:06
the four multiple choices so other than
44:09
that we've got a very strong lead with
44:11
eleven hundred and fifty two of you as I
44:13
say this agreeing on one of the answers
44:15
and then second place by about a factor
44:20
of four is gonna sit on another one I'm
44:22
curious about actually what what all the
44:24
answer distribution here is I'm very
44:26
curious about oh now we have a sixth
44:28
okay so two of you have now gone outside
44:32
the realm of the typical multiple choice
44:34
universe and have entered your own
44:36
things two of you sit there and say I
44:39
don't want one of your choices I figured
44:41
out my own answer and you just didn't
44:42
put it in your form so I'm gonna figure
44:44
out a way to submit it differently if
44:46
only you could do that on the SAT what
44:49
is the answer a through e sir I think
44:51
the answer is J 1729 we got Rama
44:56
Newton's constant oh if I pause it now
44:58
though it's not going to stay there for
45:00
a brief glorious moment we had a
45:02
wonderful number they make composite
45:04
around the year though cuz it's 1999
45:07
20 31 okay well I've posited in the
45:10
future so the correct answer in this
45:12
context is negative one-half okay and it
45:15
looks like the second most common answer
45:17
was D which is negative root three over
45:20
two okay so that's very good because
45:22
often is kind of this back and forth
45:23
between whether you think it's a half or
45:26
whether you think it's root three over
45:28
two because it's sort of the difference
45:30
between things that feel like 30 degree
45:34
angles and things that feel like 60
45:35
degree angles so let's take a look back
45:37
at our drawing and see how we might
45:39
think about this so like I said negative
45:42
2 pi divided by 3 that's taking us a
45:45
negative distance over here and this is
45:48
why cosine will be a negative number
45:49
because the x-coordinate is pointing in
45:53
the negative direction and if you want
45:56
to think of this in terms of triangles I
45:58
might draw this triangle here and ask
46:01
what is this angle okay we know that
46:04
it's 90 degrees plus something or PI
46:07
halves plus something and negative 2 PI
46:10
over 3 if we wanted to think
46:12
in terms of degrees if that's something
46:13
we're more comfortable with that's
46:15
negative 360 divided by three you know
46:19
degrees which is negative 120 degrees so
46:24
if it takes 120 degrees to get around
46:26
here then that minus the 90 degree angle
46:28
between the X and y axis gives us 30
46:31
degrees okay so then the question is
46:35
what is the cosine of this value what is
46:37
the x-coordinate there with your
46:39
30-60-90 triangles it's just kind of a
46:41
nice thing to remember that the shorter
46:44
side is 1/2 and the longer side is root
46:48
3 over 2 and if you forget why it's
46:50
exactly the diagram that we just drew
46:52
before associated with equilateral
46:55
triangles right because if you imagine
46:57
taking an equilateral triangle and
46:59
chopping it in half then the shorter
47:01
side has length 1/2 and then the longer
47:03
side comes from the Pythagorean theorem
47:06
so congratulations to those of you who
47:10
correctly found that now I want to bring
47:13
back that identity that you guys found
47:14
at the very beginning ok associated with
47:17
cosine squared and cosine of 2 theta and
47:20
show how that extends our capability for
47:22
making some of these computations and
47:24
figuring out certain values so let's
47:26
remind ourselves what that actually was
47:27
okay I'm gonna go back here I'm gonna
47:30
pull up again the desmos graph and
47:32
remember how what we said is that cosine
47:34
squared which looks like this little
47:37
scaled-down version of cosine
47:39
oscillating more quickly is the same as
47:42
cosine of 2x that quicker oscillation
47:45
but we had to kind of shift it and
47:47
rescale it to make them equal so now
47:49
we're going to use that fact that you
47:51
can discover just by playing around with
47:52
graphs to make a computation that
47:55
otherwise would be pretty tricky to make
47:57
and I think that when you're able to
47:59
make concrete computations it's often a
48:01
reflection of the fact that you have a
48:02
deeper conceptual understanding of
48:04
what's going on so even if it's always
48:06
going to be your calculator computing
48:07
this you're not sitting here working out
48:08
cosine values by hand the fact that you
48:11
can is an illustrate that you understand
48:13
that cosine function a little bit better
48:14
so over on our paper let me write down
48:17
again what that identity was let's go
48:20
back to black good old friendly black
48:24
sine squared of X using the dum-dum
48:27
convention is 1 plus cosine of 2x this
48:34
is kind of oscillating more quickly all
48:36
divided by 2 okay so now I'm going to
48:40
ask you a trickier question which is
48:42
going to be what is the cosine of PI
48:47
twelfths okay and for those of you
48:51
keeping track PI twelfths is the same as
48:53
15 degrees so if we were doing this with
48:57
a unit circle what you might think is
48:59
okay a 30 degree angle so it's looking
49:04
like this that's 30 degrees I'm now
49:09
going to be taking half of that a half
49:12
of that and I want to know the cosine of
49:15
this value which is going to be the
49:16
x-coordinate of that value and right
49:18
away you can probably make a guess that
49:20
it's very close to one right because
49:22
it's a small angle and the small angle
49:26
means that our we haven't walked that
49:28
far around a circle so we haven't gotten
49:30
that far away from one especially given
49:32
the fact that as you start to walk
49:33
almost all of your velocity is in the up
49:35
and down direction alright so cosine of
49:37
PI 12 15 degrees you know what's about
49:39
to happen you know that this is a live
49:41
quiz dynamic so let's pull it up
49:50
all right what is the cosine of pi
49:55
divided by 12 okay lots of answers
50:00
rolling in very fun very exciting again
50:02
if you just happen to be tuning in now
50:04
you haven't been following along before
50:07
go to 3 B 1 B coast live your answers
50:10
will become part of what you're seeing
50:12
on screen it affects the overall
50:13
distribution this one unlike what we had
50:17
before much less overall consensus and
50:20
it definitely seems to be taking a bit
50:22
more time so I'm going to just stop
50:23
talking and give you some some pause and
50:25
ponder moments here because this one is
50:28
tricky this one takes time to think
50:30
through
51:15
now it looks like there's a total of
51:17
five of you who have extended beyond the
51:19
realm of usual multiple-choice by the
51:24
way as you guys are entering answers
51:27
here let me just say a huge thanks to
51:29
Ben eater and cam Christensen they're
51:32
the ones who actually worked on the item
51:34
pool website that is making all of this
51:37
possible either many of you might
51:40
recognize from his own youtube channel
51:41
named Ben eater about computer
51:43
engineering if you haven't checked it
51:45
out you absolutely should and then item
51:48
pool is a pretty cool thing that will be
51:50
unfolding over the next coming months
51:53
and year so you might want to keep your
51:55
eye on that it looks like some more
51:58
consensus is starting to form so I'm
51:59
gonna go ahead and grade this question
52:05
it looks like I'm getting a phone call
52:09
awesome so the correct answer is B which
52:12
is square root of it's a very
52:14
complicated thing right square root of 1
52:16
plus the square root of 3 over 2 divided
52:18
by 2 so congratulations to the two 2 to
52:22
9 of you who - 2 - 4 - 2 4 3 2243 of you
52:28
how are the answers still changing gonna
52:30
have to talk to caman eater about that
52:32
my hands are not on this who got that
52:34
correct now it looks like the second
52:36
most common answer was a which is of a
52:39
very similar form there's just a minus
52:41
sign in there so it's probably the case
52:43
that while doing the arithmetic
52:44
something got flipped we're going to
52:45
walk through it in a moment so you can
52:46
see and a special congratulations to the
52:50
two of you who figured out how to submit
52:52
an @ sign on this who the three of you
52:55
who submitted an @ sign that's not the
52:57
correct answer but well done well done
53:00
four of you very nice alright
53:02
so clearly this is a complicated answer
53:05
but where does it come from
53:07
well if we look at our identity here the
53:11
key is that it's relating two times an
53:13
angle to the angle itself
53:15
so if we just plug this in for 15
53:18
degrees or 4pi 12th what we would get is
53:21
that the cosine squared of PI twelfths
53:26
okay if we enter that as X is equal to
53:30
one plus the cosine of two times that
53:34
value which is PI sixths PI sixths all
53:40
divided by two now the cosine of PI
53:42
sixths we know right that's the same as
53:45
our 30-degree angle here again when you
53:48
have a 30-60-90 triangle the shorter
53:51
side of that which is going to be the
53:55
sine in this context the shorter side is
53:57
one half and then the longer side is
54:00
square root of three over two assuming
54:03
that the hypotenuse is one which in the
54:05
context of the unit circle we always do
54:07
assume so that means that we now have
54:09
one plus the square root of three over
54:12
two that's what the cosine of PI sixths
54:14
was all divided by two which means our
54:17
final answer when we just want to say
54:21
cosine of PI twelfths no squared about
54:23
it is going to be the square root of
54:26
this value okay very interesting so this
54:30
is showing us that cosine and sine can
54:33
get complicated right we've got a
54:34
handful of values like PI sixths and PI
54:37
thirds where we can figure them out we
54:39
can have even more I won't call them
54:41
trivial values but easier to compute
54:43
values like cosine of PI where you would
54:45
look at your unit circle and say oh PI
54:47
that means I've walked half way around
54:48
so my x-coordinate is negative one right
54:52
but outside of just a handful of values
54:54
like that they're very hard to compute
54:57
by hand and in the olden days the only
54:59
way that people could compute them by
55:00
hand before something like the existence
55:02
of calculus would be measuring it right
55:05
you like draw out with a protractor and
55:06
you try to carefully see what the
55:08
coordinates are or layering on usage of
55:11
identities like this over and over in
55:13
ways that you can get approximate values
55:16
like this and just to check ourselves we
55:18
can actually go and plug this into a
55:19
calculator if we want it to something
55:21
like desmos and just see one how big the
55:25
value is and kind of confirm for
55:26
ourselves that they turn out to be the
55:28
same so if we pop over here and I just
55:30
typed something like cosine of PI over
55:36
12 point nine six five okay so like I
55:41
said very close to one and that's kind
55:42
of what we would expect and the answer
55:44
we just got was the square root of 1
55:45
plus the square root of 3 I gotta make
55:49
sure I do the divided by two correctly /
55:52
- right that looks correct but then all
55:56
of that was divided by 2 all that stuff
55:57
on the inside was divided by 2 and
55:59
indeed they're the same value and so
56:01
even just something like PI twelfths
56:03
we end up getting a very convoluted
56:04
image now like I said this is a
56:08
reflection of the fact that our cosine
56:10
identity that we found at the very
56:11
beginning just by playing through with
56:13
graphs
56:14
it's definitely non-obvious and what
56:16
we'll find next time is that this is
56:17
actually a shadow of the fact that
56:20
trigonometric functions are related to
56:22
complex numbers and the idea of doubling
56:24
the angle here corresponding to
56:26
multiplying by something twice not at
56:28
all a coincidence I think it's very fun
56:30
I'm excited to show you guys all about
56:32
that now what I wanted to do just to
56:34
finish things off here though was to
56:36
show two things that are not commonly
56:38
shown about the unit circle okay so we
56:40
see where a sine and cosine are but
56:42
there might be a couple questions where
56:44
is tan okay where in our picture do we
56:49
see tangent of theta and then also where
56:54
is cosine of X or cosine of theta
56:59
squared or with the dumb convention
57:01
cosine squared of theta so pulling up
57:04
another one of my infinite supply of
57:05
unit circles here actually you know what
57:10
this one's not going to have enough
57:11
y-axis so I'm going to do something that
57:13
I really shouldn't do which is to try to
57:16
use a compass live on camera I'm sort of
57:19
terrible at using compasses because it
57:20
always slips out for me but we're just
57:22
going to do this I'm so nervous that
57:24
this is gonna somehow slip out oh yeah
57:28
I'm actually didn't turn out too badly
57:30
hopefully the pencil shows up okay on
57:32
screen the reason I'm doing this is the
57:34
tangent is going to demand that we think
57:36
outside of the box that we think outside
57:39
of the circle in order
57:40
- geometrically interpreted you could
57:44
just ask about the the opposite divided
57:47
by the adjacent which is where that
57:48
sohcahtoa TOA comes from but what I
57:51
really want is a single length in our
57:54
diagram that corresponds to the tangent
57:56
so we can kind of get that nice
57:57
geometric intuitive feel for it and the
58:01
idea is we take that radius which is 1
58:03
and if we think of the fact that tangent
58:07
of theta
58:08
sohcahtoa is supposed to be the opposite
58:10
over the adjacent
58:12
I want a triangle where that adjacent is
58:14
equal to 1 okay so a right triangle such
58:17
that this one is the adjacent side to
58:20
our angle theta well in order to do that
58:24
let's just draw a right-angled
58:26
perpendicular line off of our radial
58:28
line and some of you might know that if
58:31
you draw a perpendicular line to the
58:34
radius of a circle it actually lies
58:36
tangent to the circle so you see tangent
58:38
this is where the word comes from then
58:42
what we have is a triangle where the
58:43
adjacent side to theta is 1 and then the
58:46
opposite side is this length here so
58:49
that length of there is the tangent of
58:52
theta right and so right now you can
58:55
kind of intuitively understand that as
58:57
theta is very small when the opposite
59:00
over adjacent is very close to 0 because
59:03
that opposite is close to 0 and the
59:05
adjacent is close to 1 well indeed this
59:07
length also is getting very close to 0
59:09
and then you can also think through what
59:11
happens as that angle approaches 90
59:13
degrees this is something I actually
59:15
Illustrated once before in a previous
59:16
video a a lesser-known video entitled
59:21
tattoos on math that fun fact includes
59:25
someone who is behind the scenes as we
59:27
say all of this right now but the idea
59:30
is that when you draw that tangent line
59:32
tangent is this length that I just
59:35
pointed out to you and that context you
59:37
could also think about cotangent there's
59:38
this whole other triad of trigonometric
59:41
functions that aren't worth teaching but
59:43
are kind of historical quirks that's
59:44
actually what that whole video was about
59:46
but from here you can just play around
59:48
with moving the angle theta and seeing
59:50
what that does to this tangent value to
59:52
that
59:53
length okay and using that intuition
59:56
actually I want you to place a guess on
59:59
what the graph of tangent of theta is
60:01
going to look like so once again let's
60:03
go over to our live question dynamic
60:05
again congratulations to those of you
60:06
who got the last one right and as our
60:09
final question before the end of this
60:10
particular stream which is a rather tall
60:13
question so it might actually pop up
60:17
over over below the values we have
60:19
available here is the final question I
60:21
just want you to guess what the tangent
60:23
of theta looks like when we graph it
60:25
yeah and it should be pulled up now and
60:31
oh there we go
60:32
it is rather tall though okay okay so
60:36
let me go ahead and pull up a separate
60:38
image for you here so that you can see
60:39
all of the options let's pop over here
60:42
and take a look okay so again if you go
60:46
to 3 B 1 B Co /live where the link is in
60:49
the description you can submit your
60:50
answers to this basically which one of
60:53
these graphs shows you the tangent of
60:56
theta okay unfortunately we're gonna
61:01
have to kind of switch back and forth to
61:02
see the statistics it looks like there's
61:04
pretty strong consensus in one direction
61:06
but I'm gonna give you I'm gonna give
61:09
you a little bit more time to think
61:11
about it if you want to don't feel
61:12
rushed don't feel like you have to jump
61:14
right into it
61:33
I went a little ambient pause and ponder
61:40
music while we're at it I'm mildly
61:44
addicted to it
62:05
all right so while you're answering so
62:09
as we can see some people are clearly
62:11
messing with the website and having fun
62:12
or there's a bug in the website to
62:14
answer various other things other fun
62:16
thing going on it seems like we've got a
62:19
couple comments in the in the JSON
62:22
associated with the answers so to the
62:25
person who decided to skip over my very
62:28
draconian limits on the live chat
62:30
dynamic and instead entered in JSON
62:33
comment through the answer forum forum I
62:36
really enjoyed this class thank you I
62:39
appreciate that
62:40
and I appreciate your your hockey
62:42
instincts you're gonna go far in life my
62:44
friend you're gonna go very very far so
62:47
let's go ahead and grade this question
62:49
here um it looks like we have very
62:50
strong consensus hopefully around the
62:53
correct answer and the correct answer is
62:57
a again maybe a little bit hard to see
62:59
on our on our indication of the question
63:02
over here but a is the graph that looks
63:05
like this and what you'll notice is it
63:07
starts at zero and then as you approach
63:09
PI halves it blows up to infinity okay
63:12
now in terms of the animations and the
63:16
the geometric interpretation of tangent
63:19
that should actually make sense because
63:21
if we go back to what we were looking at
63:22
as the angle blows up the angle doesn't
63:27
blow up the angle does a very modest
63:28
thing of simply going to PI halves a
63:30
fine number but the tangent of that
63:32
angle is going to just get bigger and
63:34
bigger and ultimately it's it's not
63:36
defined it's the same as dividing by
63:37
zero once you get to PI halves and then
63:40
after that if you play the same game
63:42
it's going to be it's going to be
63:44
considered a negative value which
63:45
basically comes down to the fact that
63:47
the opposite and adjacent sides of our
63:51
triangle are going to be considered to
63:52
have separate signs so the very last
63:57
thing that I want to talk about like I
63:59
said is the question of where does
64:01
cosine squared of theta show up in our
64:04
image because this is the very thing
64:05
that we opened with is trying to
64:06
understand cosine squared and you might
64:09
think oh if we can see that in our image
64:11
somewhere maybe it really reveals why it
64:13
looks like a miniature version of the
64:14
cosine wave now one of the naive things
64:17
naive but one of the first things you
64:18
might try to do is pull up the original
64:21
cosine animation that we had or
64:23
something like it and say oh well I'm
64:24
gonna take that cosine length and
64:26
literally square it I'm just gonna draw
64:28
the square based on that and if I were
64:30
to do this where I'm gonna plot the
64:31
length of that horizontal line and I'm
64:34
also going to plot the area of that
64:35
square you know it's a little bit
64:37
complicated it certainly shows us what
64:43
we already saw before which is that the
64:45
square of that value looks like another
64:47
cosine wave but this is something where
64:50
just by staring at the image it actually
64:53
doesn't yield much insight I think
64:55
oftentimes in geometry or math when you
64:57
simply write down the definition of
64:59
things the answer pops out this is a
65:01
sign that again it's not trivial it's
65:03
very unclear why the way that that
65:05
square that literal square would
65:07
oscillate is going to be related to
65:09
cosine much less cosine of two theta so
65:12
it's going to come from somewhere else
65:13
inside our image okay and for that let
65:18
me go ahead and draw a much bigger
65:20
triangle for ourselves let's say we've
65:24
got one of these guys that was living
65:26
inside the unit circle this by the way I
65:34
think is the most exciting part of the
65:35
lecture so if there was just one thing
65:36
that you were going to stick around for
65:37
I would say that this is the thing that
65:39
you should because it gives a very
65:41
sneaky proof of the Pythagorean theorem
65:43
it might feel like a circular argument
65:45
at first but I assure you that it's not
65:47
and I also assure you that the word
65:49
circular was not meant as a pun there so
65:51
remember if that's our angle theta then
65:53
one of our side lengths is sine of theta
65:55
and the other one is cosine of theta let
66:00
me ask you if I draw well okay so you
66:04
know the example we had earlier was this
66:06
leaning tower thing where we kind of
66:08
imagined casting a shadow from a leaning
66:11
tower onto the ground this is sort of
66:13
gonna flip it on its head and say well
66:15
this angle theta between them who's to
66:17
say who's the tower and who's the ground
66:18
what if I wanted to have a light source
66:20
over here and I wanted to cast a shadow
66:22
from the ground onto the tower right a
66:24
nice perpendicular line in this
66:26
direction that way my angle
66:30
is going to do double duty for me right
66:32
it's describing angle between these two
66:34
lines but in one context this ground
66:37
line was the adjacent side it was the
66:39
short side but now if I ask you a
66:41
question like what is this side length
66:44
in the shadow that the ground lays down
66:47
on the tower if that's not more
66:49
confusing than I want it to be what are
66:52
we going to get well in terms of this
66:56
smaller right triangle that we've just
66:58
created by definition the cosine of that
67:02
angle is going to be the adjacent side
67:04
the thing we want to know maybe I'll
67:06
call it s for shadow again I'm going to
67:10
write down what it equals it's going to
67:12
be the adjacent side divided by the
67:14
hypotenuse but for this little triangle
67:16
the hypotenuse is cosine so it's divided
67:19
by the cosine of theta which means
67:21
cosine squared of theta dumb convention
67:24
is equal to the answer that we want let
67:29
me just write it down in a different
67:29
color to really emphasize cosine squared
67:33
of theta and let me emphasize this is
67:37
not using the Pythagorean theorem in any
67:39
way this is not having the cosine
67:41
squared arrived because we were assuming
67:43
that this sits on a circle where x
67:45
squared plus y squared equals 1 we're
67:46
just using the definition of cosine
67:48
twice to cast one shadow and then to
67:50
cast another now you can probably see
67:52
where this is going do the same thing
67:54
for the sine of theta so another place
67:57
where theta shows up in our diagram is
67:59
right here and a way that you can see
68:01
that is that this angle plus something
68:04
must equal 90 degrees right but this
68:07
other something plus R theta had to be
68:11
the remaining angles of the little right
68:13
triangle that we had what you might
68:15
think up here I can just give it a
68:17
different name if I called this other
68:18
angle alpha then we know that alpha plus
68:21
theta alpha plus theta plus 90 degrees
68:25
or PI halves if you want to get used to
68:29
the better conventions has to equal PI
68:32
that's got to equal 180 degrees this is
68:35
maybe confusing just think alpha plus
68:37
theta plus PI halves and the back of
68:40
your mind you can go to 90 degrees if
68:41
that's something you want for
68:43
for comfort but that also implies that
68:46
when we have a new right triangle
68:47
sitting here this little angle here has
68:49
got to be theta because it's alpha plus
68:51
90 plus something so that something's
68:53
got to be theta now the reason I have
68:56
stated that so much is because we're
68:57
gonna play our same shadow casting game
68:59
to figure out what this side length is
69:01
now with respect to that we have sine of
69:05
theta of this theta in particular is the
69:09
opposite the new shadow we don't know
69:11
I'll call it s prime is equal to s prime
69:14
divided by the hypotenuse well on this
69:17
smaller right triangle what was the long
69:20
side has become the hypotenuse so that
69:22
sine of theta sine of theta which means
69:27
sine squared of theta is our other
69:31
shadow this here is sine squared of
69:34
theta so that's kind of cool the side
69:37
length which was by definition 1 because
69:41
we've always been rescaling our
69:42
triangles such that the hypotenuse is 1
69:43
can be broken down into one part that's
69:47
cosine squared and another part that
69:49
sine squared and the way to break that
69:51
down is to just draw a single line the
69:53
perpendicular projection from our corner
69:56
point on to onto that hypotenuse so that
70:00
means is every time you see a unit
70:02
circle out in the wild which I'm sure
70:03
you do pretty regularly and you draw the
70:06
standard triangle to understand sine and
70:08
cosine so that height is going to be
70:11
sine the x-coordinate is going to be
70:14
cosine and our hypotenuse is always 1 so
70:21
that hypotenuse is 1 if this is Theta
70:24
and you ask yourself where is cosine
70:27
squared in this diagram where is sine
70:29
squared you can just draw a little
70:31
perpendicular line here and the part on
70:38
the bottom is cosine squared so Coast
70:41
squared of theta is this length this
70:46
length here and what we'll do in a
70:50
future lecture actually I want to talk
70:52
about geometry and geometry proofs and
70:54
in particular one that comes up all the
70:56
I've used it in many past videos it's
70:58
high time we just sit down and say how
70:59
would you prove it is the inscribed
71:01
angle theorem so after showing you how
71:04
complex numbers play into things next
71:06
time I want to turn back to this
71:08
particular diagram to give another very
71:10
intuitive understanding for the identity
71:13
that you found simply by playing around
71:14
with graphs at the very beginning here
71:16
which again is a very non-trivial
71:18
identity that the cosine squared of some
71:20
value is 1 plus the cosine of 2 times
71:23
that value divided by 2 the more
71:25
standard way you see it by the way if
71:27
you're to look this up in a textbook
71:29
instead of writing this as X this is
71:32
commonly written as something like alpha
71:33
halves for some angle and then 2x would
71:36
be alpha and the whole thing ends up
71:39
looking like cosine of alpha divided by
71:43
2 is equal to the square root of 1 plus
71:48
cosine of alpha divided by 2 I for one I
71:55
have such trouble memorizing the trig
71:56
identities and I always would just sort
71:58
of read arrive them on the fly and this
72:00
is one where I think the easiest way to
72:02
read arrive is to practice kind of
72:03
smashing around graphs in your head when
72:05
you have the 1 key insight which is that
72:07
squaring cosine gets you a smaller
72:09
cosine graph and the next time I'll show
72:11
you how complex numbers help you kind of
72:13
read arrive a bunch of other identities
72:14
on the fly but this one in particular I
72:16
just always have trouble remembering
72:18
it's called the half angle identity and
72:20
I just think it's delightful that before
72:22
you even know any trigonometry simply
72:24
hopping over to something like desmos
72:26
and being a little bit playful that I
72:28
mean that lands you on a highly
72:30
non-trivial fact and once again it's one
72:33
that indicates that cosine is related to
72:35
Exponential's that's not obvious and
72:38
hopefully next time we can start to get
72:40
a glimpse of why thanks for sticking
72:42
around this long thank you once again to
72:44
cam and Ben eater who have been making
72:46
the item pool magic behind the scenes to
72:48
get us our live statistics and I'll see
72:50
you go at the very end I had almost
72:53
everything down into the very end I will
72:55
see you all on Friday
73:07
[Music]