Basic derivative rules: table

WEBVTT
Kind: captions
Language: en

00:00:00.216 --> 00:00:01.323
Voiceover: We've been given

00:00:01.323 --> 00:00:03.211
some interesting information here

00:00:03.211 --> 00:00:05.336
about the functions f, g, and h.

00:00:05.336 --> 00:00:07.874
For f, they tell us for given values of x

00:00:07.874 --> 00:00:09.420
what f of x is equal to

00:00:09.420 --> 00:00:11.725
and what f prime of x is equal to.

00:00:11.725 --> 00:00:14.074
Then they defined g of x for us

00:00:14.074 --> 00:00:16.533
in terms of this kind of
absolute value expression.

00:00:16.533 --> 00:00:18.865
Then they define h of x for us,

00:00:18.865 --> 00:00:21.832
in terms of both f of x and g of x.

00:00:21.832 --> 00:00:27.094
What we're curious about
is what is the derivative

00:00:27.094 --> 00:00:35.018
with respect to x, of h of
x at x is equal to nine.

00:00:35.018 --> 00:00:37.350
I encourage you to pause this video

00:00:37.350 --> 00:00:38.876
and think about it on your own

00:00:38.876 --> 00:00:40.605
before I work through it.

00:00:40.605 --> 00:00:42.739
Let's think about it a little bit.

00:00:42.739 --> 00:00:46.094
Another way just to get
familiar with the notation

00:00:46.094 --> 00:00:48.084
of writing this, the derivative of h of x

00:00:48.084 --> 00:00:50.214
with respect to x at x equals nine.

00:00:50.214 --> 00:00:56.243
This is equivalent to h,
we need that blue color,

00:00:56.243 --> 00:00:59.465
it is equivalent to h prime

00:00:59.465 --> 00:01:02.595
and the prime signifies that
we're taking the derivative.

00:01:02.595 --> 00:01:09.200
H prime of x, when x equals nine

00:01:09.200 --> 00:01:13.664
so h prime of nine is what this really is.

00:01:13.664 --> 00:01:15.566
Actually I'm going to do
this in a different color.

00:01:15.566 --> 00:01:19.659
This is h prime of nine.

00:01:19.659 --> 00:01:21.683
Let's think about what that is.

00:01:21.683 --> 00:01:24.136
Let's take the derivative of both sides

00:01:24.136 --> 00:01:25.048
of this expression

00:01:25.048 --> 00:01:27.300
to figure out what the derivative
with respect to x of h is.

00:01:29.023 --> 00:01:33.198
We get a derivative, I'll
do that same white color.

00:01:33.198 --> 00:01:41.287
A derivative with respect to x of h of x

00:01:41.287 --> 00:01:46.389
is going to be equal to the
derivative with respect to x

00:01:46.389 --> 00:01:47.578
of all of this business.

00:01:47.578 --> 00:01:50.457
I could actually just,
well I'll just rewrite it.

00:01:50.457 --> 00:01:58.130
Three times f of x, plus two times g of x.

00:01:59.253 --> 00:02:01.000
Now this right over here,

00:02:01.000 --> 00:02:03.609
the derivative of the sum of two terms

00:02:03.609 --> 00:02:05.228
that's going to be the same thing

00:02:05.228 --> 00:02:08.869
as the sum of the derivatives
of each of the terms.

00:02:08.869 --> 00:02:10.169
This is going to be the same thing

00:02:10.169 --> 00:02:11.593
as the derivative with respect to x

00:02:11.593 --> 00:02:15.625
of three times, I'll write
that a little bit neater.

00:02:15.625 --> 00:02:22.283
Three times f of x, plus the
derivative with respect to x

00:02:22.283 --> 00:02:27.584
of two times g of x.

00:02:28.122 --> 00:02:30.538
Now the derivative of a number

00:02:30.538 --> 00:02:33.489
or I guess you could say a
scaling factor times a function.

00:02:33.489 --> 00:02:35.514
The derivative of a
scalar times the function

00:02:35.514 --> 00:02:37.219
is the same thing as a scalar times

00:02:37.219 --> 00:02:38.541
the derivative of the function.

00:02:38.541 --> 00:02:39.703
What does that mean?

00:02:39.703 --> 00:02:43.114
Well that just means that this
first term right over here

00:02:43.114 --> 00:02:45.083
that's going to be equivalent to

00:02:45.083 --> 00:02:49.493
three times the derivative
with respect to x

00:02:49.493 --> 00:02:53.803
of f, of our f of x,

00:02:55.079 --> 00:02:59.997
plus this part over here
is the same thing as two.

00:02:59.997 --> 00:03:02.400
Okay, make sure I don't
run out of space here,

00:03:02.400 --> 00:03:06.251
plus two times the
derivative with respect to x.

00:03:07.174 --> 00:03:13.715
The derivative with
respect to x of g of x.

00:03:16.084 --> 00:03:18.440
Derivative of h with respect to x

00:03:18.440 --> 00:03:19.987
is equal to three times the derivative

00:03:19.987 --> 00:03:21.453
of f with respect to x,

00:03:21.453 --> 00:03:24.143
plus two times the derivative
of g with respect to x.

00:03:24.143 --> 00:03:27.427
If we want to write it in this
kind of prime notation here,

00:03:27.427 --> 00:03:33.654
we could rewrite it as
h prime of x is equal to

00:03:33.654 --> 00:03:35.616
three times f prime of x,

00:03:35.616 --> 00:03:39.646
so this part right over here

00:03:39.646 --> 00:03:41.932
that is the same thing as f prime of x.

00:03:41.932 --> 00:03:51.655
It's three times f prime of x,
plus two times g prime of x.

00:03:52.162 --> 00:03:54.726
Once you are more fluent
with this property,

00:03:54.726 --> 00:03:56.210
the derivative of the sum of two things

00:03:56.210 --> 00:03:57.699
is the sum of the derivatives.

00:03:57.699 --> 00:03:59.820
The derivative of a scalar times something

00:03:59.820 --> 00:04:02.006
is the same thing as a
scalar times the derivative

00:04:02.006 --> 00:04:02.898
of that something.

00:04:02.898 --> 00:04:06.543
You really could have
gone straight from here

00:04:06.543 --> 00:04:09.043
to here, pretty quickly.

00:04:09.043 --> 00:04:10.943
Now why is this interesting,

00:04:10.943 --> 00:04:13.029
well now we can evaluate this function

00:04:13.029 --> 00:04:14.792
when x is equal to nine.

00:04:14.792 --> 00:04:18.615
H prime of nine is the same thing

00:04:18.615 --> 00:04:21.020
as three times f prime of nine,

00:04:21.020 --> 00:04:24.493
plus two times g prime of nine.

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Now what is f prime of nine?

00:04:26.839 --> 00:04:28.343
The derivative of our function f

00:04:28.343 --> 00:04:29.711
when x is equal to nine.

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Well they tell us, when
x is equal to nine,

00:04:32.076 --> 00:04:33.610
f of nine is one

00:04:33.610 --> 00:04:36.872
but more importantly f
prime of nine is three.

00:04:36.872 --> 00:04:41.445
This part right over here
evaluates that part's three.

00:04:41.445 --> 00:04:43.797
What's g prime of nine?

00:04:44.151 --> 00:04:47.844
Let's look at this function
a little bit more closely.

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There's a couple of ways
we could think about it.

00:04:51.065 --> 00:04:52.523
Actually let's try to graph it,

00:04:52.523 --> 00:04:53.945
now I think that could be interesting.

00:04:53.945 --> 00:04:56.680
Just to visualize what's going on here.

00:04:56.680 --> 00:04:59.217
Let's say that's our y-axis

00:04:59.217 --> 00:05:03.474
and do this right over here is our x-axis.

00:05:04.643 --> 00:05:07.131
Now when does an absolute
value function like this,

00:05:07.131 --> 00:05:09.048
when is this going to hit a minimum point?

00:05:09.048 --> 00:05:12.154
Well the absolute value of something

00:05:12.154 --> 00:05:13.969
is always going to be non-negative.

00:05:13.969 --> 00:05:18.793
It hits a minimum point when
this thing is equal to zero.

00:05:18.793 --> 00:05:20.514
Well when is this thing equal to zero?

00:05:20.514 --> 00:05:22.816
When x equals one, this
thing is equal to zero.

00:05:22.816 --> 00:05:25.848
We hit a minimum point
when x is equal to one,

00:05:25.848 --> 00:05:28.602
and when x equals one, this term is zero

00:05:28.602 --> 00:05:30.024
absolute value of zero, zero.

00:05:30.024 --> 00:05:31.857
G of one is one.

00:05:31.857 --> 00:05:34.662
We have this point right over there.

00:05:34.662 --> 00:05:36.659
Now what happens after that?

00:05:36.659 --> 00:05:42.473
What happens for x greater than one?

00:05:42.473 --> 00:05:44.338
Actually let me write this down.

00:05:45.425 --> 00:05:47.547
G of x is equal to,

00:05:47.547 --> 00:05:50.178
and in general whenever
you have an absolute value,

00:05:50.178 --> 00:05:52.642
a relatively simple absolute
value function like this

00:05:52.642 --> 00:05:54.442
you could think of it,
you could break it up

00:05:54.442 --> 00:05:55.774
into two function

00:05:55.774 --> 00:05:57.392
or you could think about this function

00:05:57.392 --> 00:05:58.607
over different intervals

00:05:58.607 --> 00:06:00.777
when the absolute value is non-negative

00:06:00.777 --> 00:06:02.530
and when the absolute value is negative.

00:06:02.530 --> 00:06:04.923
When the absolute value is non-negative

00:06:04.923 --> 00:06:09.614
that's when x is greater
than or equal to zero.

00:06:10.198 --> 00:06:12.162
When the absolute value is non-negative,

00:06:12.162 --> 00:06:15.105
if you're taking the absolute
value of a non-negative number

00:06:15.105 --> 00:06:16.649
that is just going to be itself.

00:06:16.649 --> 00:06:17.760
The absolute value of zero, zero.

00:06:17.760 --> 00:06:18.896
Absolute value of one is one.

00:06:18.896 --> 00:06:20.507
The absolute value of
a hundred is a hundred.

00:06:20.507 --> 00:06:23.242
Then you could ignore the absolute value

00:06:23.242 --> 00:06:25.276
for x is greater than or equal to,

00:06:25.276 --> 00:06:26.806
not greater than or equal to zero,

00:06:26.806 --> 00:06:30.810
for x is greater than or equal to one.

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X is greater than or equal to one,

00:06:32.232 --> 00:06:36.056
this thing right over
here is non-negative.

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It will just evaluate to x minus one.

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This is going to be x minus one plus one.

00:06:41.547 --> 00:06:44.715
Which is the same thing as just x,

00:06:44.715 --> 00:06:46.694
minus one plus one, they just cancel out.

00:06:46.694 --> 00:06:52.786
Now when this term right
over here is negative

00:06:52.786 --> 00:06:59.957
and that's going to happen
for x is less than one.

00:06:59.957 --> 00:07:01.822
Well then the absolute value

00:07:01.822 --> 00:07:03.517
is going to be the opposite of it.

00:07:03.517 --> 00:07:05.104
You give me the absolute
value of a negative number

00:07:05.104 --> 00:07:06.553
that's going to be the opposite.

00:07:06.553 --> 00:07:08.794
Absolute value of negative
eight is positive eight.

00:07:08.794 --> 00:07:11.789
It's going to be that the
negative of x minus one

00:07:11.789 --> 00:07:15.696
is one minus x, plus one.

00:07:15.696 --> 00:07:21.300
Or we could say two minus x.

00:07:22.099 --> 00:07:24.041
For x is greater or equal to one,

00:07:24.041 --> 00:07:26.284
we would look at this expression,

00:07:26.284 --> 00:07:27.948
now what's the slope of that?

00:07:27.948 --> 00:07:29.959
Well the slope of that is one.

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We're going to have a
curve that looks like

00:07:32.166 --> 00:07:34.414
or a line I guess we could
say that looks like this.

00:07:36.413 --> 00:07:40.013
For all x is greater than or equal to one.

00:07:40.013 --> 00:07:41.499
The important thing, remember,

00:07:41.499 --> 00:07:44.606
we're going to think about
the slope of the tangent line

00:07:44.606 --> 00:07:46.404
when we think about the derivative of g.

00:07:46.404 --> 00:07:47.779
Slope is equal to one.

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For x less than one or our slope now,

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if we look right over here

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our slope is negative one.

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It's going to look like this.

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It's going to look like that.

00:08:03.682 --> 00:08:05.833
For the pointing question,

00:08:05.833 --> 00:08:07.848
if we're thinking about g prime of nine

00:08:07.848 --> 00:08:09.538
so nine is some place out here,

00:08:09.538 --> 00:08:11.825
so what is g prime of nine?

00:08:11.825 --> 00:08:14.704
G prime of nine, let me make it clear,

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this graph right over here,

00:08:15.954 --> 00:08:17.521
this is the graph of g of x

00:08:17.521 --> 00:08:21.145
or we could say y, this is
the graph y equals g of x.

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Y is equal to g of x.

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What is g prime of nine?

00:08:26.266 --> 00:08:29.847
Well that's the slope
when x is equal to nine.

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The slope is going to be equal to one.

00:08:32.202 --> 00:08:37.780
G prime of nine is one.

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What does this evaluate to?

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This is going to be three times three,

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so this part right over here is nine

00:08:44.052 --> 00:08:51.084
plus two times one, plus
two, which is equal to 11.

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The slope of the tangent line of h

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when x is equal to nine is 11.


Midpoint and distance on complex plane

WEBVTT
Kind: captions
Language: en

00:00:00.776 --> 00:00:02.953
Voiceover:So we have two
complex numbers here.

00:00:02.953 --> 00:00:06.309
The complex number z is
equal to two plus three i

00:00:06.309 --> 00:00:10.157
and the complex number w is
equal to negative five minus i.

00:00:11.430 --> 00:00:12.786
What I want to do in
this video is to first

00:00:12.786 --> 00:00:16.658
plot these two complex
numbers on the complex plane

00:00:16.658 --> 00:00:19.600
and then think about what
the distance is between

00:00:19.600 --> 00:00:22.326
these two numbers on the
plane and what complex

00:00:22.326 --> 00:00:26.139
number is exactly halfway
between these two numbers

00:00:26.139 --> 00:00:27.877
or another way of thinking
about it, what complex

00:00:27.877 --> 00:00:31.291
number is the midpoint
between these two numbers.

00:00:31.486 --> 00:00:34.510
So I encourage you to
pause this video and think

00:00:34.510 --> 00:00:36.879
about it on your own
before I work through it.

00:00:38.226 --> 00:00:41.640
So let's first try to plot
these on the complex planes.

00:00:41.640 --> 00:00:43.310
So let me draw, so right over here,

00:00:43.310 --> 00:00:46.238
let me draw our imaginary axis.

00:00:46.821 --> 00:00:49.119
So our imaginary axis, and over

00:00:49.119 --> 00:00:51.402
here let me draw our real axis.

00:00:52.452 --> 00:00:56.384
Real axis right over
there, and let's first,

00:00:56.384 --> 00:00:58.319
let's see, we're gonna
have it go as high as

00:00:58.319 --> 00:01:01.442
positive two in the real axis
and as low as negative five

00:01:01.442 --> 00:01:07.632
along the real axis so let's
go one, two, three, four, five.

00:01:07.632 --> 00:01:11.542
One, two, three, four, five.

00:01:11.592 --> 00:01:13.985
Along the imaginary axis
we go as high as positive

00:01:13.985 --> 00:01:16.891
three and as low as negative one.

00:01:16.891 --> 00:01:19.699
So we could do one, two,
three and we could do

00:01:19.699 --> 00:01:22.452
one, two, three and of
course I could keep going

00:01:22.452 --> 00:01:26.140
up here just to have nice
markers there although we

00:01:26.140 --> 00:01:28.197
won't use that part of the plane.

00:01:28.733 --> 00:01:30.399
Now let's plot these two points.

00:01:30.579 --> 00:01:36.659
So the real part of z
is two and then we have

00:01:36.659 --> 00:01:40.290
three times i so the
imaginary part is three.

00:01:40.494 --> 00:01:43.939
So we would go right over here.

00:01:43.939 --> 00:01:47.845
So this is two and this
is three right over here.

00:01:47.845 --> 00:01:52.583
Two plus three i, so that
right over there is z.

00:01:52.597 --> 00:01:55.772
Now let's plot w, w is negative five.

00:01:55.772 --> 00:01:58.598
One, two, three, four, five, negative five

00:01:58.598 --> 00:02:02.944
minus i, so this is negative
one right over here.

00:02:02.944 --> 00:02:05.271
So minus i, that is w.

00:02:05.282 --> 00:02:08.452
So first we can think about
the distance between these two

00:02:08.452 --> 00:02:11.398
complex numbers; the distance
on the complex plane.

00:02:11.899 --> 00:02:13.310
So one way of thinking
about it, that's really

00:02:13.310 --> 00:02:17.128
just the distance of this
line right over here.

00:02:17.604 --> 00:02:20.834
And to figure that out
we can really just think

00:02:20.834 --> 00:02:22.870
about the Pythagorean theorem.

00:02:22.955 --> 00:02:27.825
If you hear about the Distance
Formula in two dimensions,

00:02:27.825 --> 00:02:29.770
well that's really just
an application of the

00:02:29.770 --> 00:02:32.693
Pythagorean theorem, so let's
think about that a little bit.

00:02:32.906 --> 00:02:34.582
So we can think about
how much have we changed

00:02:34.582 --> 00:02:39.436
along the real axis which is
this distance right over here.

00:02:39.650 --> 00:02:42.580
This is how much we've
changed along the real axis.

00:02:42.580 --> 00:02:46.929
And if we're going from
w to z, we're going from

00:02:46.929 --> 00:02:49.805
negative 5 along the real axis to two.

00:02:49.805 --> 00:02:51.409
What is two minus negative 5?

00:02:51.409 --> 00:02:57.375
Well it's seven, if we
go five to get to zero

00:02:57.375 --> 00:02:59.957
along the real axis and then
we go two more to get to two,

00:02:59.957 --> 00:03:02.120
so the length of this
right over here is seven.

00:03:02.259 --> 00:03:09.712
And what is the length of
this side right over here?

00:03:09.712 --> 00:03:11.582
Well along the imaginary
axis we're going from

00:03:11.582 --> 00:03:15.572
negative one to three so
the distance there is four.

00:03:15.867 --> 00:03:18.037
So now we can apply the
Pythagorean theorem.

00:03:18.037 --> 00:03:20.617
This is a right triangle, so the distance

00:03:20.617 --> 00:03:24.171
is going to be equal to the distance.

00:03:24.171 --> 00:03:26.501
Let's just say that this
is x right over here.

00:03:26.501 --> 00:03:30.894
x squared is going to be
equal to seven squared,

00:03:30.894 --> 00:03:33.625
this is just the Pythagorean
theorem, plus four squared.

00:03:34.609 --> 00:03:38.740
Plus four squared or we
can say that x is equal

00:03:38.740 --> 00:03:42.613
to the square root of 49 plus 16.

00:03:42.672 --> 00:03:44.959
I'll just write it out so
I don't skip any steps.

00:03:44.985 --> 00:03:49.376
49 plus 16, now what is
that going to be equal to?

00:03:49.376 --> 00:03:56.890
That is 65 so x, that's right,
59 plus another 6 is 65.

00:03:56.890 --> 00:04:01.668
x is equal to the square root of 65.

00:04:01.668 --> 00:04:04.357
Now let's see, 65 you can't factor this.

00:04:04.357 --> 00:04:06.282
There's no factors that
are perfect squares here,

00:04:06.282 --> 00:04:09.990
this is just 13 times five so

00:04:09.990 --> 00:04:10.604
we can just leave it like that.

00:04:10.604 --> 00:04:13.427
x is equal to the square
root of 65 so the distance

00:04:13.427 --> 00:04:16.281
in the complex plane between
these two complex numbers,

00:04:16.281 --> 00:04:21.826
square root of 65 which is I
guess a little bit over eight.

00:04:21.924 --> 00:04:23.739
Now what about the complex number that is

00:04:23.739 --> 00:04:26.371
exactly halfway between these two?

00:04:27.295 --> 00:04:29.381
Well to figure that out, we just have to

00:04:29.381 --> 00:04:31.218
figure out what number
has a real part that

00:04:31.218 --> 00:04:33.126
is halfway between these two real parts

00:04:33.126 --> 00:04:34.870
and what number has an imaginary part

00:04:34.870 --> 00:04:37.133
that's halfway between
these two imaginary parts.

00:04:37.659 --> 00:04:40.477
So if we had some, let's say
that some complex number,

00:04:40.477 --> 00:04:43.295
let's just call it a, is
the midpoint, it's real part

00:04:43.295 --> 00:04:45.873
is going to be the mean
of these two numbers.

00:04:45.873 --> 00:04:48.925
So it's going to be
two plus negative five.

00:04:49.619 --> 00:04:54.387
Two plus negative five over two, over two,

00:04:54.387 --> 00:04:56.323
and it's imaginary part
is going to be the mean

00:04:56.323 --> 00:05:02.247
of these two numbers so
plus, plus three minus one.

00:05:03.340 --> 00:05:13.960
Three minus one, minus
one, over two times i

00:05:13.960 --> 00:05:15.975
and this is equal to, let's
see, two plus negative five

00:05:15.975 --> 00:05:21.707
is negative three so
this is negative 3/2 plus

00:05:21.707 --> 00:05:27.989
this is three minus 1 is
negative, is negative two

00:05:27.989 --> 00:05:33.972
over two is let's see three,
make sure I'm doing this right.

00:05:33.972 --> 00:05:36.643
Three, something in the
mean, three minus one is two

00:05:36.643 --> 00:05:42.169
divided by two is one,
so three plus three.

00:05:42.169 --> 00:05:46.936
Negative 3/2 plus i is the
midpoint between those two

00:05:46.936 --> 00:05:49.934
and if we plot it we can verify
that actually makes sense.

00:05:49.934 --> 00:05:52.957
So real part negative 3/2,
so that's negative one,

00:05:52.957 --> 00:05:56.664
negative one and a half so
it'll be right over there

00:05:56.664 --> 00:06:02.549
and then plus i so it's
going to be right over there.

00:06:02.549 --> 00:06:04.556
And I'll just have to
draw it perfectly to scale

00:06:04.556 --> 00:06:06.401
but this makes sense, that this right

00:06:06.401 --> 00:06:09.684
over here would be the midpoint.


Identity and zero matrix equation

WEBVTT
Kind: captions
Language: en

00:00:00.754 --> 00:00:02.998
So we have a matrix equation

00:00:02.998 --> 00:00:04.762
set up right over here,

00:00:04.762 --> 00:00:07.244
we have matrix A times this

00:00:07.244 --> 00:00:09.278
plus matrix B times this

00:00:09.278 --> 00:00:10.903
plus matrix C times this

00:00:10.903 --> 00:00:13.919
is equal to this two by two matrix.

00:00:13.919 --> 00:00:14.731
And the way it's set up,

00:00:14.731 --> 00:00:16.311
it's a little bit of a puzzle.

00:00:16.311 --> 00:00:19.151
And I'll give you one clue as to what type

00:00:19.151 --> 00:00:22.049
of matrices A, B, and C could be.

00:00:22.049 --> 00:00:24.434
They're each either going to be

00:00:24.434 --> 00:00:28.058
an Identity Matrix,

00:00:28.058 --> 00:00:32.397
or a Zero Matrix.

00:00:32.397 --> 00:00:34.378
So given that clue,

00:00:34.378 --> 00:00:35.874
that each of these are either an

00:00:35.874 --> 00:00:38.364
identity matrix or a zero matrix,

00:00:38.364 --> 00:00:40.786
can you pause this video,

00:00:40.786 --> 00:00:42.093
and essentially solve this puzzle?

00:00:42.093 --> 00:00:43.541
Which of these are identity matrices,

00:00:43.541 --> 00:00:47.737
and which of these are zero matrices?

00:00:47.737 --> 00:00:49.305
I'm assuming you've given a go at it.

00:00:49.305 --> 00:00:51.203
So let's go entry by entry.

00:00:51.203 --> 00:00:53.204
So if you look at this first entry

00:00:53.204 --> 00:00:54.477
right over here,

00:00:54.477 --> 00:00:58.082
how can we get a two
for this top left entry?

00:00:58.082 --> 00:00:59.381
Well, let's see.

00:00:59.381 --> 00:01:02.861
When we multiply, if any
of these are identities,

00:01:02.861 --> 00:01:05.127
and essentially you just
get the value of the matrix,

00:01:05.127 --> 00:01:06.372
if any of these are zero,

00:01:06.372 --> 00:01:08.007
then essentially that
product doesn't get added.

00:01:08.007 --> 00:01:10.778
It's one way of thinking about it,

00:01:10.778 --> 00:01:13.245
is that matrix won't matter anymore,

00:01:13.245 --> 00:01:15.734
because it's going to
be multiplied by zero.

00:01:15.734 --> 00:01:18.246
So if A was a zero matrix

00:01:18.246 --> 00:01:20.613
and B and C were identity matrices,

00:01:20.613 --> 00:01:23.425
you would add one plus one to get to two.

00:01:23.425 --> 00:01:25.200
Just like oh, maybe that's the case.

00:01:25.200 --> 00:01:26.983
But it could be the other way around.

00:01:26.983 --> 00:01:30.118
It could be that A is identity matrix,

00:01:30.118 --> 00:01:31.540
B is a zero matrix,

00:01:31.540 --> 00:01:33.245
and C is an identity matrix,

00:01:33.245 --> 00:01:35.015
and you add one plus one

00:01:35.015 --> 00:01:36.502
over there to get two.

00:01:36.502 --> 00:01:39.539
Or you could say that
maybe C is the zero matrix,

00:01:39.539 --> 00:01:41.072
and B is the identity matrix,

00:01:41.072 --> 00:01:42.997
and you add one plus one here.

00:01:42.997 --> 00:01:45.714
So really, all this is telling us is

00:01:45.714 --> 00:01:47.415
two of these matrix,

00:01:47.415 --> 00:01:50.763
two of A, B and C are going
to be identity matrices,

00:01:50.763 --> 00:01:52.210
and one is going to be a zero matrix,

00:01:52.210 --> 00:01:54.918
but we don't know which
one is which just yet.

00:01:54.918 --> 00:01:56.441
Any of them can be the zero matrix,

00:01:56.441 --> 00:01:59.034
at least based on looking
at that first entry.

00:01:59.034 --> 00:02:00.968
Now let's look at, I don't know,

00:02:00.968 --> 00:02:03.257
let's look at this entry right over here.

00:02:03.257 --> 00:02:05.740
How can we add up to four?

00:02:05.740 --> 00:02:06.418
Well, let's see.

00:02:06.418 --> 00:02:08.787
Three, if this was an identity matrix

00:02:08.787 --> 00:02:10.871
and this was an identity matrix,

00:02:10.871 --> 00:02:11.762
then you're going to,

00:02:11.762 --> 00:02:13.298
then essentially you'll just be left with

00:02:13.298 --> 00:02:14.595
this matrix plus this matrix.

00:02:14.595 --> 00:02:17.033
This is going to be the zero.

00:02:17.033 --> 00:02:20.769
So three plus negative five.

00:02:20.769 --> 00:02:24.980
That is not equal to four.

00:02:24.980 --> 00:02:26.013
So both of these.

00:02:26.013 --> 00:02:28.499
You can't have A and
B be identity matrices

00:02:28.499 --> 00:02:30.834
and C be the zero matrix.

00:02:30.834 --> 00:02:33.899
So let's think about
other combinations here.

00:02:33.899 --> 00:02:38.154
So what about B and C being
the identity matrices,

00:02:38.154 --> 00:02:40.307
and A being the zero matrix?

00:02:40.307 --> 00:02:41.463
In that situation,

00:02:41.463 --> 00:02:44.262
so A is the zero matrix,
that's not going to matter,

00:02:44.262 --> 00:02:45.489
and we're essentially, when you multiply

00:02:45.489 --> 00:02:47.392
B times this you're just
going to get this matrix,

00:02:47.392 --> 00:02:49.744
and C times this, you're
going to get that matrix,

00:02:49.744 --> 00:02:51.784
if B and C are identity matrices.

00:02:51.784 --> 00:02:54.080
And so you have negative five plus one

00:02:54.080 --> 00:02:55.823
would be negative four.

00:02:55.823 --> 00:02:58.682
So that doesn't work either.

00:02:58.682 --> 00:03:02.681
So our last scenario is going to be

00:03:02.681 --> 00:03:04.190
so we can essentially rule out B

00:03:04.190 --> 00:03:05.608
as an identity matrix.

00:03:05.608 --> 00:03:07.805
Because when we had B

00:03:07.805 --> 00:03:09.866
be one of the identity matrices,

00:03:09.866 --> 00:03:11.364
and we picked the other two options,

00:03:11.364 --> 00:03:12.948
we still couldn't get to four here.

00:03:12.948 --> 00:03:16.918
So B is going to be a zero matrix.

00:03:16.918 --> 00:03:18.491
And let's see if that works out.

00:03:18.491 --> 00:03:21.615
A is identity, then A would be an identity

00:03:21.615 --> 00:03:24.098
and C would be an idenitty matrix as well.

00:03:24.098 --> 00:03:27.726
So let's see if that actually makes sense.

00:03:27.726 --> 00:03:30.917
So if this right over
here is identity matrix,

00:03:30.917 --> 00:03:32.984
and that over there is an identity matrix,

00:03:32.984 --> 00:03:35.163
this whole thing simplifies to

00:03:35.163 --> 00:03:39.778
one three four comma negative two

00:03:39.778 --> 00:03:44.242
plus one one three two,

00:03:44.242 --> 00:03:46.762
and it does indeed equal

00:03:46.762 --> 00:03:48.042
one one plus is two,

00:03:48.042 --> 00:03:50.062
three plus one is four,

00:03:50.062 --> 00:03:52.942
four plus three is seven,

00:03:52.942 --> 00:03:56.942
negative two plus two is indeed zero.


Zero matrix

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.252
Voiceover:We've been drawing analogies

00:00:02.252 --> 00:00:03.755
between I guess we could say

00:00:03.755 --> 00:00:06.688
traditional multiplication,
or scalar multiplication,

00:00:06.688 --> 00:00:07.840
and the first one we drew is

00:00:07.840 --> 00:00:10.268
when you have traditional multiplication,

00:00:10.268 --> 00:00:13.364
you multiply 1 times any number

00:00:13.364 --> 00:00:15.139
and you get that number again.

00:00:15.139 --> 00:00:17.998
And you could view 1 as
essentially the identity.

00:00:17.998 --> 00:00:19.177
The identity number,

00:00:19.177 --> 00:00:21.535
or this is the identity
property of multiplication.

00:00:21.535 --> 00:00:22.813
You multiply 1 times any number,

00:00:22.813 --> 00:00:24.063
you get that number again.

00:00:24.063 --> 00:00:26.222
And that essentially inspired our thinking

00:00:26.222 --> 00:00:29.020
behind having identity matrices.

00:00:29.020 --> 00:00:30.748
Said hey, maybe there are some matrices

00:00:30.748 --> 00:00:34.546
that if I multiply
times some other matrix,

00:00:34.546 --> 00:00:38.101
I'm going to get that matrix again.

00:00:38.101 --> 00:00:40.071
And you've probably shown for yourselves

00:00:40.071 --> 00:00:42.069
that you can do it in either way.

00:00:42.069 --> 00:00:47.470
You could have some matrix
times an identity matrix

00:00:47.470 --> 00:00:50.931
and get that matrix again.

00:00:50.931 --> 00:00:54.964
Now if matrix A right over
here is a square matrix,

00:00:54.964 --> 00:00:58.580
then in either situation,
this identity matrix

00:00:58.580 --> 00:01:00.344
is going to be the same identity matrix.

00:01:00.344 --> 00:01:02.298
But if matrix A is not a square matrix,

00:01:02.298 --> 00:01:03.116
then these are going to be

00:01:03.116 --> 00:01:05.178
two different identity matrices,

00:01:05.178 --> 00:01:08.722
depending on the appropriate dimensions.

00:01:08.722 --> 00:01:11.385
Now, let's see if we
can extend this analogy

00:01:11.385 --> 00:01:13.616
between traditional multiplication

00:01:13.616 --> 00:01:16.109
and matrix multiplication.

00:01:16.109 --> 00:01:17.873
We know that there's
another spacial number

00:01:17.873 --> 00:01:20.898
in traditional multiplication,
and that's a 0.

00:01:20.898 --> 00:01:27.394
So, we know that 0 times
anything is equal to 0.

00:01:27.394 --> 00:01:31.840
Or, anything times 0 is equal to 0.

00:01:31.840 --> 00:01:33.233
So what would be the analogy

00:01:33.233 --> 00:01:35.122
if we're thinking about
matrix multiplication?

00:01:35.122 --> 00:01:37.535
Well, it would be some matrix that

00:01:37.535 --> 00:01:40.747
if I were to multiply
it times another matrix,

00:01:40.747 --> 00:01:45.777
I get, I guess you could say
that same 0 matrix again.

00:01:45.777 --> 00:01:47.007
And that is what we call it.

00:01:47.007 --> 00:01:49.122
We call it a 0 matrix.

00:01:49.122 --> 00:01:53.954
So if I take some matrix
A, and essentially,

00:01:53.954 --> 00:01:57.324
if I multiply it times
one of these 0 matrices,

00:01:57.324 --> 00:02:01.473
or I multiply one of
the 0 matrices times A,

00:02:01.473 --> 00:02:07.249
I should get another 0 matrix.

00:02:07.249 --> 00:02:08.825
And it depends on the dimensions.

00:02:08.825 --> 00:02:11.461
You might not get a 0 matrix

00:02:11.461 --> 00:02:13.458
with the same dimensions.

00:02:13.458 --> 00:02:14.897
It depends what the dimensions

00:02:14.897 --> 00:02:17.328
of A are going to be, but you could image

00:02:17.328 --> 00:02:20.161
what a 0 matrix might look like.

00:02:20.161 --> 00:02:26.138
For example, if A is 1,2,3,4,

00:02:26.138 --> 00:02:28.657
what's a 0 matrix that I could multiply

00:02:28.657 --> 00:02:31.815
this by to get another 0 matrix?

00:02:31.815 --> 00:02:33.642
Well, it might be pretty straight forward,

00:02:33.642 --> 00:02:35.855
if you just had a ton of zeros here,

00:02:35.855 --> 00:02:37.746
when you multiply this out,

00:02:37.746 --> 00:02:39.977
you're going to get this -

00:02:39.977 --> 00:02:42.703
you date the dot product of
this row and this column.

00:02:42.703 --> 00:02:45.713
0 times 1 plus 0 times 3 is going to be 0.

00:02:45.713 --> 00:02:49.654
You keep going, 0,0,0,0.

00:02:49.669 --> 00:02:54.761
If we had a - just to
make the point clear -

00:02:54.761 --> 00:03:03.205
let's say we had a matrix 1,2,3,4,5,6.

00:03:03.205 --> 00:03:05.916
So over here, we want
to multiply this times -

00:03:05.916 --> 00:03:08.165
let's see, in order for
the matrix multiplication

00:03:08.165 --> 00:03:12.078
to work, my 0 matrix has got to have

00:03:12.078 --> 00:03:19.213
the same number of columns
as this one has rows,

00:03:19.213 --> 00:03:25.630
so it's got to have 2 columns,

00:03:25.630 --> 00:03:28.276
but I could make it have 3 rows.

00:03:28.276 --> 00:03:31.792
So it could look like this, 0,0,0,0

00:03:31.792 --> 00:03:34.646
and I encourage you to multiply these two.

00:03:34.646 --> 00:03:36.610
Pause the video right
now, and see what you get.

00:03:37.453 --> 00:03:40.079
Well when you multiply
them, let's think about it.

00:03:40.079 --> 00:03:41.807
So the top left entry -

00:03:41.807 --> 00:03:43.606
so let me just write the dimensions.

00:03:43.606 --> 00:03:48.135
This is a 3x2 matrix, this is a 2x3matrix.

00:03:48.135 --> 00:03:51.678
So, we know that we have
valid matrix multiplication

00:03:51.678 --> 00:03:53.116
going on right over same here.

00:03:53.116 --> 00:03:55.016
The number of columns in the first matrix

00:03:55.016 --> 00:03:56.914
is equal to the number of
rows in the second one.

00:03:56.914 --> 00:03:58.938
And we also know that
the resulting product

00:03:58.938 --> 00:04:01.862
is going to be a 3x3 matrix.

00:04:01.862 --> 00:04:04.960
So it's going to be a 3x3 matrix,

00:04:04.960 --> 00:04:07.236
and I'll leave it up to you to verify

00:04:07.236 --> 00:04:13.903
that all the entries
here are going to be 0.

00:04:13.903 --> 00:04:16.351
And it makes sense, you
could go through the math,

00:04:16.351 --> 00:04:17.550
but you can see, well
you're just everytime,

00:04:17.550 --> 00:04:22.293
you're multiplying say
this row by this column

00:04:22.293 --> 00:04:23.851
to get that entry, we'll just have

00:04:23.851 --> 00:04:28.007
0 times 1 plus 0 times
4 to get that 0 there.

00:04:28.007 --> 00:04:30.041
But the whole point of
showing you this example is,

00:04:30.041 --> 00:04:32.240
we have one 0 matrix multiplying

00:04:32.240 --> 00:04:35.474
by this matrix right over here,

00:04:35.474 --> 00:04:38.130
and then we get another 0 matrix

00:04:38.130 --> 00:04:40.544
but it has different dimensions.


Dimensions of identity matrix

WEBVTT
Kind: captions
Language: en

00:00:00.389 --> 00:00:04.054
Voiceover:Let's say that
you've got some matrix C,

00:00:04.054 --> 00:00:05.950
trying my best to bold it,

00:00:05.950 --> 00:00:08.814
to make sure you realize
that this is a matrix.

00:00:08.814 --> 00:00:14.068
Let's say that we know that it has a rows

00:00:14.068 --> 00:00:16.108
and b columns.

00:00:16.108 --> 00:00:20.355
It's an a by b matrix.

00:00:20.355 --> 00:00:23.452
Let's say that we are going to multiply it

00:00:23.452 --> 00:00:27.765
by some identity matrix.

00:00:27.765 --> 00:00:30.854
Once again let me do my
best to attempt to bold this

00:00:30.854 --> 00:00:31.962
right over here.

00:00:31.962 --> 00:00:35.426
We're going to multiply the
identity matrix I times C

00:00:35.426 --> 00:00:37.738
and of course we are going to get C again

00:00:37.738 --> 00:00:40.043
because that's the identity matrix,

00:00:40.043 --> 00:00:42.086
that's the property of
the identity matrix.

00:00:42.086 --> 00:00:46.503
Of course C, we already
know is an a by b matrix.

00:00:46.503 --> 00:00:51.978
A rows and b columns.

00:00:54.193 --> 00:00:57.312
Based on this, what
are the dimensions of I

00:00:57.312 --> 00:00:58.608
going to be?

00:00:58.608 --> 00:01:02.802
I encourage you to pause
this video and think about it

00:01:02.802 --> 00:01:03.963
on your own.

00:01:03.963 --> 00:01:06.680
We've already done this
exercise a little bit,

00:01:06.680 --> 00:01:09.037
where we first looked at identity matrices

00:01:09.037 --> 00:01:10.522
but now we're doing it with a very ...

00:01:10.522 --> 00:01:11.773
We're multiplying the identity matrix

00:01:11.773 --> 00:01:13.594
times a very general matrix.

00:01:13.594 --> 00:01:15.834
I'm just even speaking in generalities

00:01:15.834 --> 00:01:17.157
about these dimensions.

00:01:17.157 --> 00:01:19.874
Well one thing we know is
that matrix multiplication

00:01:19.874 --> 00:01:20.991
is only defined

00:01:20.991 --> 00:01:23.016
is if the column, the number of columns

00:01:23.016 --> 00:01:25.932
of the first matrix is
equal to the number of rows

00:01:25.932 --> 00:01:27.228
of the second matrix.

00:01:27.228 --> 00:01:35.449
This one has a rows, so this
one's going to have a columns.

00:01:35.449 --> 00:01:38.696
Now how many rows is
this one going to have?

00:01:38.696 --> 00:01:41.477
We already know that matrix
multiplication is only defined

00:01:41.477 --> 00:01:43.474
if the number of columns
on the first matrix

00:01:43.474 --> 00:01:45.620
is equal to the number of
rows on the second one.

00:01:45.620 --> 00:01:54.615
We know that the product
gets its number of rows

00:01:54.615 --> 00:01:56.365
from the number of rows

00:01:56.365 --> 00:01:58.794
of the first matrix being multiplied.

00:01:58.794 --> 00:02:01.593
The product has a rows

00:02:01.593 --> 00:02:03.690
then the identity matrix right over here

00:02:03.690 --> 00:02:05.975
has to have a rows.

00:02:05.975 --> 00:02:08.610
What's interesting about this?

00:02:08.610 --> 00:02:12.058
When we first got introduced
to identity matrices,

00:02:12.058 --> 00:02:13.651
we were multiplying,

00:02:13.651 --> 00:02:15.837
we picked out a three by three example

00:02:15.837 --> 00:02:17.870
and we got a three by
three identity matrix.

00:02:17.870 --> 00:02:21.353
What's interesting about what
we've just proven to ourselves

00:02:21.353 --> 00:02:24.321
is the identity matrix for any matrix,

00:02:24.321 --> 00:02:25.889
even a non square matrix,

00:02:25.889 --> 00:02:27.888
a and b could be two different values.

00:02:27.888 --> 00:02:30.741
The identity matrix for any matrix

00:02:30.741 --> 00:02:33.288
is going to be a square matrix.

00:02:33.288 --> 00:02:37.345
It's going to have the same number of rows

00:02:37.345 --> 00:02:39.208
and the same number of columns.

00:02:39.208 --> 00:02:41.603
When we think about identity matrices,

00:02:41.603 --> 00:02:45.515
we can really just say, well
is this the identity matrix

00:02:45.515 --> 00:02:46.886
that is a four by four?

00:02:46.886 --> 00:02:47.745
Is it a three by three?

00:02:47.745 --> 00:02:48.707
Is it a two by two?

00:02:48.707 --> 00:02:50.724
Or I guess one by one?

00:02:52.846 --> 00:02:55.240
The convention is, it isn't
even to write identity

00:02:55.240 --> 00:03:00.166
two by two is equal to
one, zero, zero, one.

00:03:00.166 --> 00:03:02.911
The convention is actually just write I2

00:03:02.911 --> 00:03:04.997
because you know it's
going to be a two by two.

00:03:06.720 --> 00:03:08.157
It's going to be a two by two matrix,

00:03:08.157 --> 00:03:10.320
it's going to be one, zero, zero, one.

00:03:10.320 --> 00:03:14.099
Identity five is going to
be a five by five matrix.

00:03:14.099 --> 00:03:16.394
It's going to be one,
one, two, three, four.

00:03:16.394 --> 00:03:19.121
Zero, one, two, one, three.

00:03:19.121 --> 00:03:22.206
Zero, zero, one, zero, zero.

00:03:22.206 --> 00:03:25.006
Zero ... you get the idea,

00:03:25.006 --> 00:03:27.795
zero, zero, zero, one, zero.

00:03:27.795 --> 00:03:30.266
Zero, zero, zero, zero, one.

00:03:31.004 --> 00:03:32.130
Just like that.

00:03:32.130 --> 00:03:33.671
The whole point here is just to realize

00:03:33.671 --> 00:03:34.857
that your identity matrix

00:03:34.857 --> 00:03:37.152
is always going to be a square matrix

00:03:37.152 --> 00:03:39.356
and it works even when you're multiplying

00:03:39.356 --> 00:03:42.227
non square other matrices.


Identity matrix

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.389
Voiceover:When you first
learned multiplication

00:00:02.389 --> 00:00:04.900
many, many, many years ago,

00:00:04.900 --> 00:00:08.767
you got exposed to the
idea that 1 times ...

00:00:08.767 --> 00:00:10.102
I shouldn't use that symbol ...

00:00:10.102 --> 00:00:13.966
1 times some number is
equal to that number again,

00:00:13.966 --> 00:00:15.701
and that makes intuitive sense.

00:00:15.701 --> 00:00:17.766
You're just literally
saying one of this thing

00:00:17.766 --> 00:00:20.114
is just going to be that
thing right over there.

00:00:20.114 --> 00:00:21.900
And you could view it as 1,

00:00:21.900 --> 00:00:23.833
when you're thinking about
regular multiplication

00:00:23.833 --> 00:00:25.233
or scalar multiplication,

00:00:25.233 --> 00:00:27.833
it has this identity property.

00:00:27.833 --> 00:00:32.166
It has the identity
property of multiplication.

00:00:32.166 --> 00:00:35.680
1 times some number is equal
to that some number again.

00:00:35.680 --> 00:00:37.766
Since we're now exploring matrices

00:00:37.766 --> 00:00:40.033
and matrix multiplication,

00:00:40.033 --> 00:00:42.306
the question arises is there some matrix

00:00:42.306 --> 00:00:46.168
that has the same property
for matrix multiplication?

00:00:46.168 --> 00:00:48.566
To make that a little bit more concrete,

00:00:48.566 --> 00:00:51.034
is there some matrix I,

00:00:51.034 --> 00:00:54.366
and let me bold it as best
as I can in my handwriting,

00:00:54.366 --> 00:00:58.032
is there some matrix I that
if I were to multiply it

00:00:58.032 --> 00:01:00.100
times any other ...

00:01:00.100 --> 00:01:01.766
I think I over-bolded that one,

00:01:01.766 --> 00:01:03.105
but I'll just go with it.

00:01:03.105 --> 00:01:07.233
If I were to multiply it
times any other matrix, A,

00:01:07.233 --> 00:01:13.300
that the resulting product
is going to be matrix A again

00:01:13.300 --> 00:01:17.166
by the standard conventions
of matrix multiplication.

00:01:17.166 --> 00:01:20.235
To make that a little bit
concrete, let's just imagine.

00:01:20.235 --> 00:01:22.101
Let's just take an example for A.

00:01:22.101 --> 00:01:24.900
Let's say that our matrix A,

00:01:26.622 --> 00:01:28.400
let's go 3 by 3.

00:01:28.700 --> 00:01:37.201
Let's say it is 1, 2, 3, 4, 5, 6, 7, 8, 9.

00:01:37.368 --> 00:01:40.100
What I encourage you
to is pause this video

00:01:40.100 --> 00:01:42.235
and try to think about
whether you can construct

00:01:42.235 --> 00:01:44.233
some matrix I,

00:01:44.233 --> 00:01:45.567
and first think about
even what the dimensions

00:01:45.567 --> 00:01:49.566
of matrix I have to be in order to,

00:01:49.566 --> 00:01:51.033
when you multiply the two this way,

00:01:51.033 --> 00:01:52.633
when you multiply I times A,

00:01:52.633 --> 00:01:54.333
you get A again.

00:01:55.500 --> 00:01:57.633
I'm assuming you've given a go at it,

00:01:57.633 --> 00:01:58.904
so let's think this through.

00:01:58.904 --> 00:02:00.767
Let's throw matrix A down there.

00:02:00.767 --> 00:02:04.867
Let's say copy and paste.

00:02:04.968 --> 00:02:06.367
Let's first think about
what the dimensions

00:02:06.367 --> 00:02:07.900
are going to have to be.

00:02:07.900 --> 00:02:12.500
When I multiply my matrix I,

00:02:12.500 --> 00:02:16.967
when I multiply my matrix
I times A right over here,

00:02:16.967 --> 00:02:20.535
I get A again.

00:02:20.535 --> 00:02:25.501
I'm multiplying something times a 3 by 3,

00:02:25.501 --> 00:02:28.073
3 by 3 matrix,

00:02:28.073 --> 00:02:32.148
and I'm getting another 3 by 3 matrix.

00:02:32.294 --> 00:02:34.633
There's a few things that we know.

00:02:34.633 --> 00:02:37.366
First of all, in order for
this matrix multiplication

00:02:37.366 --> 00:02:39.434
to even be defined,

00:02:39.434 --> 00:02:41.166
this matrix, the identity matrix,

00:02:41.166 --> 00:02:46.013
has to have the same number
of columns as A has rows.

00:02:46.034 --> 00:02:49.100
We already see that A has 3 rows,

00:02:49.100 --> 00:02:51.435
so this character, the identity matrix,

00:02:51.435 --> 00:02:53.600
is going to have to have 3 columns.

00:02:53.766 --> 00:02:55.967
It's going to have to have 3 columns.

00:02:55.967 --> 00:03:01.366
We also know that the
dimensions of the product,

00:03:01.366 --> 00:03:05.566
the rows of the product are defined

00:03:05.566 --> 00:03:07.666
by the rows of the first matrix,

00:03:07.666 --> 00:03:10.500
so this has to be also a 3 by 3,

00:03:10.500 --> 00:03:12.569
and of course, the columns of the product

00:03:12.569 --> 00:03:15.467
are defined by the columns
of the second matrix.

00:03:16.653 --> 00:03:18.925
This is what defines this.

00:03:18.925 --> 00:03:20.566
These middle two have to match,

00:03:20.566 --> 00:03:22.501
and then the rows of the first matrix

00:03:22.501 --> 00:03:24.366
define the rows of the product,

00:03:24.366 --> 00:03:26.233
and then the columns of the second matrix

00:03:26.233 --> 00:03:27.634
define the columns of the product.

00:03:27.634 --> 00:03:29.867
We know this has to be a 3 by 3 matrix.

00:03:30.166 --> 00:03:32.366
Now what else do we know?

00:03:32.366 --> 00:03:34.100
We know what the product needs to be.

00:03:34.100 --> 00:03:42.934
It also needs to be 1,
2, 3, 4, 5, 6, 7, 8, 9.

00:03:43.566 --> 00:03:44.566
Let's think about it.

00:03:44.566 --> 00:03:46.774
To get this first entry right over here,

00:03:46.774 --> 00:03:50.366
we're going to have to multiply this row,

00:03:50.366 --> 00:03:53.100
this row times this column,

00:03:53.100 --> 00:03:55.235
since you take the dot product of it.

00:03:55.235 --> 00:03:59.435
I'm going to have to
multiply something times 1

00:03:59.435 --> 00:04:01.434
plus something else times 4

00:04:01.434 --> 00:04:04.200
plus something else times 7 to get 1.

00:04:05.100 --> 00:04:06.039
Let's just think about it in the most,

00:04:06.039 --> 00:04:08.438
I guess we could say, naive possible way.

00:04:08.438 --> 00:04:12.904
What happens if we just
multiply 1 times this 1 to get 1

00:04:12.904 --> 00:04:17.568
and then 0 times 4 and add to it

00:04:17.568 --> 00:04:19.234
and then 0 times 7.

00:04:19.234 --> 00:04:20.634
I think that works out.

00:04:20.634 --> 00:04:22.367
When you take this product,

00:04:22.367 --> 00:04:25.601
this entry right over here
is going to be 1 times 1,

00:04:25.700 --> 00:04:28.734
1 times 1 plus 0 times 4,

00:04:29.100 --> 00:04:32.534
0 times 4 plus 0 times 7,

00:04:32.700 --> 00:04:35.534
plus 0 times 7.

00:04:35.567 --> 00:04:37.700
That worked out quite well,

00:04:37.700 --> 00:04:39.166
but let's just make sure
that that still holds.

00:04:39.166 --> 00:04:43.902
What happens when we multiply
this row times this column

00:04:43.902 --> 00:04:47.234
or times this column to get
this entry right over here?

00:04:47.234 --> 00:04:47.767
It works out.

00:04:47.767 --> 00:04:51.833
It's 1 times 2 plus 0
times 5 plus 0 times 8,

00:04:51.833 --> 00:04:52.633
so it makes sense.

00:04:52.633 --> 00:04:54.267
You get 2 again.

00:04:54.635 --> 00:04:56.700
Same thing when you do
it for this 3rd column.

00:04:56.700 --> 00:05:01.666
1 times 3 plus 0 times 6 plus
0 times 9 is going to be 3.

00:05:02.167 --> 00:05:03.700
Now what do we do in the second row?

00:05:03.700 --> 00:05:05.300
Let's think about it a little bit.

00:05:05.300 --> 00:05:07.033
The second row right over here

00:05:07.033 --> 00:05:09.700
is going to determine what
values we get over here.

00:05:09.700 --> 00:05:13.638
For example, to get this
entry right over there,

00:05:13.638 --> 00:05:16.600
we're going to multiply this row,

00:05:16.773 --> 00:05:21.066
we're going to multiply
this row times this column,

00:05:21.166 --> 00:05:22.268
times this column.

00:05:22.902 --> 00:05:23.901
We want it to have the 4,

00:05:23.901 --> 00:05:24.834
so one way to think about it,

00:05:24.834 --> 00:05:26.833
we just want this middle entry here,

00:05:26.833 --> 00:05:34.101
so let's multiply 0 times 1
plus 1 times 4 plus 0 times 7,

00:05:34.101 --> 00:05:35.433
and then we're going to get 4.

00:05:35.433 --> 00:05:40.501
That works out for this
next entry right over here.

00:05:40.501 --> 00:05:46.100
0 times 2 plus 1 times 5 plus 0 times 8.

00:05:46.100 --> 00:05:47.566
We get 5.

00:05:47.566 --> 00:05:50.141
It will work out the same
for this entry over there.

00:05:50.301 --> 00:05:52.505
Now, for this last entry,

00:05:52.505 --> 00:05:57.036
for this bottom row right
over here of our product,

00:05:57.036 --> 00:05:58.766
to do that, we're going
to have to multiply

00:05:58.766 --> 00:06:01.386
this row times these columns,

00:06:01.386 --> 00:06:03.640
or take, I guess you could
say, the dot product.

00:06:03.640 --> 00:06:05.166
To get the 7,

00:06:05.166 --> 00:06:08.766
we want to multiply this
row times this column,

00:06:08.766 --> 00:06:11.901
or take the dot product of
this row and that column.

00:06:11.901 --> 00:06:15.017
If we want the 7, let's
multiply 0 times a 1

00:06:15.017 --> 00:06:19.602
plus 0 times a 4 plus a 1 times the 7.

00:06:19.987 --> 00:06:21.900
Just like that, you'll
see that that works.

00:06:21.900 --> 00:06:23.566
That gives us a 7 for this entry.

00:06:23.566 --> 00:06:25.835
It gives us, when you take
the dot of this and that,

00:06:25.835 --> 00:06:27.833
it gives you an 8 for this entry.

00:06:27.833 --> 00:06:29.633
You take the dot product of that and that.

00:06:29.633 --> 00:06:31.500
It gives you the 9,

00:06:31.500 --> 00:06:33.500
the 9 for that entry.

00:06:33.500 --> 00:06:35.366
Just like that, we have constructed

00:06:35.366 --> 00:06:38.433
a 3 by 3 identity matrix.

00:06:38.433 --> 00:06:45.433
The 3 by 3 identity matrix
is equal to 1, 0, 0, 0, 1, 0,

00:06:45.433 --> 00:06:49.266
and 0, 0, 1.

00:06:50.166 --> 00:06:53.976
As you will see, whenever you construct

00:06:53.976 --> 00:06:54.900
an identity matrix,

00:06:54.900 --> 00:06:57.833
if you're constructing a
2 by 2 identity matrix,

00:06:57.833 --> 00:07:00.767
so I can say identity matrix 2 by 2,

00:07:00.767 --> 00:07:02.233
it's going to have a very similar pattern.

00:07:02.233 --> 00:07:04.966
It's going to be 1, 0, 0, 1.

00:07:04.966 --> 00:07:12.233
If you have a 4 by 4 identity matrix,

00:07:12.233 --> 00:07:14.306
it is going to be, you could guess it,

00:07:14.306 --> 00:07:23.500
1, 0, 0, 0, 0, 1, 0, 0,
0, 0, 1, 0, 0, 0, 0, 1.

00:07:23.500 --> 00:07:25.703
You essentially just
have 1s down the diagonal

00:07:25.703 --> 00:07:28.139
going from the top left
to the bottom right.

00:07:28.166 --> 00:07:30.303
What's neat about identity matrices,

00:07:30.303 --> 00:07:32.033
you multiply it times any matrix,

00:07:32.033 --> 00:07:34.266
and you're going to get that matrix again.

00:07:34.602 --> 00:07:37.366
Now another thing I encourage you to do

00:07:37.366 --> 00:07:40.967
is we've just shown that
I times A is equal to A,

00:07:40.967 --> 00:07:43.500
but I'll let you do this after this video,

00:07:43.500 --> 00:07:49.433
what about A times I?

00:07:49.433 --> 00:07:51.308
We've seen that matrix multiplication,

00:07:51.308 --> 00:07:52.700
the order matters,

00:07:52.700 --> 00:07:53.767
so what happens here?

00:07:53.767 --> 00:07:58.267
If you take A times I, do you still get A?


Matrix multiplication introduction

WEBVTT
Kind: captions
Language: en

00:00:00.634 --> 00:00:03.330
Voiceover:Let's say that
we have got 2 matrices,

00:00:03.330 --> 00:00:04.301
and I'll just, for simplicity,

00:00:04.301 --> 00:00:06.967
I'll start with two 2 by 2 matrices.

00:00:06.967 --> 00:00:09.566
Let's say that this
first one right over here

00:00:09.566 --> 00:00:19.233
is 2, negative 2, 5,
and let's say 5 and 3,

00:00:19.233 --> 00:00:24.330
and then I have this
matrix right over here that

00:00:24.330 --> 00:00:26.100
it's also going to be 2 by 2.

00:00:26.100 --> 00:00:30.300
Let's say it's negative 1, 4,

00:00:30.300 --> 00:00:35.100
and let's say 7 and negative 6.

00:00:35.100 --> 00:00:36.766
What I want to go through in this video,

00:00:36.766 --> 00:00:40.188
what I want to introduce
you to is the convention,

00:00:40.188 --> 00:00:41.890
the mathematical convention

00:00:41.890 --> 00:00:44.766
for multiplying two matrices like these.

00:00:44.766 --> 00:00:47.500
I want to stress that
because mathematicians

00:00:47.500 --> 00:00:49.300
could have come up with
a bunch of different ways

00:00:49.300 --> 00:00:51.700
to define matrix multiplication.

00:00:51.700 --> 00:00:53.566
But the convention that
I'm going to show you

00:00:53.566 --> 00:00:55.572
is the way that it is done,

00:00:55.572 --> 00:00:57.366
and it's done this way
especially as you go into

00:00:57.366 --> 00:00:58.966
deeper linear algebra classes

00:00:58.966 --> 00:01:00.900
or you start doing computer graphics

00:01:00.900 --> 00:01:04.330
or even modeling different
types of phenomena,

00:01:04.330 --> 00:01:07.234
you'll see why this type
of matrix multiplication,

00:01:07.234 --> 00:01:08.566
which I'm about to show you,

00:01:08.566 --> 00:01:11.366
why it has the most applications.

00:01:11.366 --> 00:01:13.766
But I really want to stress
this is a human construct.

00:01:13.766 --> 00:01:16.500
Humans have found defining
matrix multiplication

00:01:16.500 --> 00:01:19.766
the way I'm about to
show you to be useful.

00:01:19.766 --> 00:01:22.166
Let's just think about how this could be.

00:01:22.166 --> 00:01:24.566
Once again, I want to stress
it's a human construct.

00:01:24.566 --> 00:01:26.635
There's several ways that
you could have thought about

00:01:26.635 --> 00:01:29.330
multiplying two 2 by 2 matrices.

00:01:29.330 --> 00:01:31.633
You could have done it the
same way that you add matrices.

00:01:31.633 --> 00:01:32.634
When you add matrices,

00:01:32.634 --> 00:01:35.566
both matrices have to
have the same dimensions,

00:01:35.566 --> 00:01:39.563
and you just add the corresponding
entries in the matrices.

00:01:40.334 --> 00:01:41.967
One convention could have been

00:01:41.967 --> 00:01:45.700
why don't we just, for our
product right over here,

00:01:45.700 --> 00:01:49.966
why don't we just multiply
corresponding entries?

00:01:49.966 --> 00:01:52.368
2 times negative 1 would
put a negative 2 here.

00:01:52.368 --> 00:01:54.966
Negative 2 times 4, put a negative 8 here.

00:01:54.966 --> 00:01:55.900
That's how we did addition.

00:01:55.900 --> 00:01:57.500
We added corresponding entries,

00:01:57.500 --> 00:02:01.366
but that is not the convention
for multiplying matrices.

00:02:01.366 --> 00:02:03.566
That is not the standard convention.

00:02:03.566 --> 00:02:06.634
The standard convention
for multiplying matrices

00:02:06.634 --> 00:02:07.900
is we're essentially going to take ...

00:02:07.900 --> 00:02:11.700
To get this top left
entry right over here,

00:02:11.700 --> 00:02:16.166
we're going to take the
product of this row,

00:02:16.166 --> 00:02:19.768
of that row with this
column right over here.

00:02:19.768 --> 00:02:20.567
Now what does it mean

00:02:20.567 --> 00:02:23.433
to take the product of a row and a column?

00:02:23.433 --> 00:02:25.778
If you are familiar with
vector dot products,

00:02:25.778 --> 00:02:26.633
this might ring a bell,

00:02:26.633 --> 00:02:29.300
where you take the product
of the corresponding terms,

00:02:29.300 --> 00:02:30.366
the product of the first terms,

00:02:30.366 --> 00:02:31.700
products of the second terms,

00:02:31.700 --> 00:02:33.330
and then add those together.

00:02:33.330 --> 00:02:34.833
That's essentially what
we're going to be doing.

00:02:34.833 --> 00:02:37.433
We're going to be taking the dot product

00:02:37.433 --> 00:02:39.900
of this first row and this first column

00:02:39.900 --> 00:02:42.700
to get this top left
entry right over here.

00:02:42.700 --> 00:02:45.500
If the word "dot product"
makes no sense to you,

00:02:45.500 --> 00:02:47.768
I will show you what that actually means.

00:02:48.491 --> 00:02:52.503
Actually, let's get some
more real estate here

00:02:52.503 --> 00:02:54.300
just so I think it will be useful,

00:02:54.300 --> 00:02:56.167
especially this very first time

00:02:56.167 --> 00:03:00.568
that we attempt to multiply matrices.

00:03:00.568 --> 00:03:05.966
This top left entry, it's
going to be 2 times negative 1,

00:03:05.966 --> 00:03:09.233
so 2 times negative 1,

00:03:09.233 --> 00:03:11.433
plus negative 2,

00:03:11.433 --> 00:03:14.854
plus negative 2 times 7,

00:03:15.501 --> 00:03:17.300
plus negative 2 times 7.

00:03:17.300 --> 00:03:18.700
Notice, I took the product,

00:03:18.700 --> 00:03:20.851
first entry in the row,
first entry in the column,

00:03:20.851 --> 00:03:22.330
those two products,

00:03:22.330 --> 00:03:23.566
then the product of
second entry in the row,

00:03:23.566 --> 00:03:25.433
second entry in the column
that's right over there,

00:03:25.433 --> 00:03:26.967
and then I added them together.

00:03:26.967 --> 00:03:28.434
That's essentially taking the dot product

00:03:28.434 --> 00:03:30.778
of this row vector and this column vector.

00:03:30.778 --> 00:03:32.101
If that doesn't make sense to you,

00:03:32.101 --> 00:03:34.340
if you're not familiar with
vectors and dot products,

00:03:34.340 --> 00:03:35.233
don't worry about it.

00:03:35.233 --> 00:03:37.300
We just took the first end
product of the first entry,

00:03:37.300 --> 00:03:40.367
product of the second entry,
added them together to get ...

00:03:40.367 --> 00:03:42.330
This is going to give us some number,

00:03:42.330 --> 00:03:44.245
and we'll calculate that in a few seconds.

00:03:44.245 --> 00:03:45.900
But let's think about the other ones.

00:03:45.900 --> 00:03:47.968
To get this,

00:03:47.968 --> 00:03:50.901
to get this entry right over here,

00:03:50.901 --> 00:03:54.300
we're going to take the
first row from this matrix

00:03:54.300 --> 00:03:59.104
and the second column from this matrix.

00:03:59.104 --> 00:04:01.833
That makes sense because
we're still in the first row

00:04:01.833 --> 00:04:04.966
but we're in the second column
of the first row right here.

00:04:04.966 --> 00:04:06.767
First row, second column.

00:04:06.767 --> 00:04:08.833
It's going to be 2 times 4,

00:04:08.833 --> 00:04:12.833
2 times 4 plus negative 2,

00:04:12.833 --> 00:04:16.633
plus negative 2 times negative 6.

00:04:16.633 --> 00:04:18.330
At this point, I encourage
you to pause the video.

00:04:18.330 --> 00:04:20.300
Seeing what you just saw, see
if you could complete this.

00:04:20.300 --> 00:04:22.234
See if you can figure
out the bottom left entry

00:04:22.234 --> 00:04:23.502
and the bottom right entry.

00:04:23.502 --> 00:04:24.833
I'll give you a clue.

00:04:24.833 --> 00:04:27.933
It has something to do
with this second row here.

00:04:28.166 --> 00:04:29.701
I'm assuming you've given a go at it.

00:04:29.701 --> 00:04:32.101
Now let's just power through it together.

00:04:32.101 --> 00:04:33.186
Sometimes matrix multiplication

00:04:33.186 --> 00:04:35.301
can get a little bit intense.

00:04:35.670 --> 00:04:37.301
We're now in the second row,

00:04:37.301 --> 00:04:39.568
so we're going to use the
second row of this first matrix,

00:04:39.568 --> 00:04:42.166
and for this entry,
second row, first column,

00:04:42.166 --> 00:04:43.633
second row, first column.

00:04:43.633 --> 00:04:45.790
5 times negative 1,

00:04:45.790 --> 00:04:49.133
5 times negative 1 plus 3 times 7,

00:04:49.300 --> 00:04:51.868
plus 3 times 7.

00:04:52.330 --> 00:04:54.966
Then finally, we're in
the home stretch now,

00:04:54.966 --> 00:05:00.633
to get this bottom row second column,

00:05:00.633 --> 00:05:01.900
or second row, second column,

00:05:01.900 --> 00:05:04.367
we multiply this row essentially

00:05:04.367 --> 00:05:07.166
by this column right over here.

00:05:07.166 --> 00:05:11.135
It is going to be 5 times 4,

00:05:11.167 --> 00:05:16.330
5 times 4 plus 3 times negative 6,

00:05:16.330 --> 00:05:18.835
plus 3 times negative 6.

00:05:18.835 --> 00:05:23.233
Now what does all of this simplify to?

00:05:23.233 --> 00:05:27.233
This is going to be equal to, let's see,

00:05:27.233 --> 00:05:32.566
so negative 2 plus negative 14,

00:05:32.566 --> 00:05:34.835
that's going to be negative 16.

00:05:34.835 --> 00:05:37.567
That, right over there, is negative 16.

00:05:37.567 --> 00:05:42.766
Then we have 8 plus 12,
so that's going to be 20.

00:05:42.766 --> 00:05:48.100
Then we have negative 5 plus 21,

00:05:48.100 --> 00:05:51.900
which is going to be 16, positive 16.

00:05:51.900 --> 00:05:52.767
Did I do that right?

00:05:52.767 --> 00:05:54.700
Yup, positive 16.

00:05:54.700 --> 00:05:58.701
Then finally, you're
going to have 20 minus 18,

00:05:58.701 --> 00:06:00.576
so that's just going to be 2.

00:06:00.576 --> 00:06:02.500
The product of these 2 matrices,

00:06:02.500 --> 00:06:04.900
we deserve a little bit of
a drum roll at this point,

00:06:04.900 --> 00:06:06.966
when we multiply this 2 by 2 matrix

00:06:06.966 --> 00:06:08.900
times this 2 by 2 matrix,

00:06:08.900 --> 00:06:13.878
we are going to get negative 16, 20,

00:06:14.263 --> 00:06:16.900
20, 16,

00:06:17.623 --> 00:06:21.933
and 16 and 2,

00:06:22.100 --> 00:06:24.932
and we are done.


Periodicity of tan example

WEBVTT
Kind: captions
Language: en

00:00:00.649 --> 00:00:03.656
Voiecover:One angle whose tangent is half

00:00:03.656 --> 00:00:05.544
is 0.46 radians.

00:00:05.544 --> 00:00:09.113
So we're saying that the
tangent right over here is...

00:00:09.113 --> 00:00:10.896
So the tangent...

00:00:10.896 --> 00:00:12.169
So we're gonna write this down.

00:00:12.169 --> 00:00:16.945
So we're saying that the
tangent of 0.46 radians

00:00:16.945 --> 00:00:18.417
is equal to half.

00:00:18.417 --> 00:00:20.217
And another way of
thinking about the tangent

00:00:20.217 --> 00:00:22.104
of an angle is that's the slope of

00:00:22.104 --> 00:00:23.520
that angle's terminal ray.

00:00:23.520 --> 00:00:25.409
So it's the slope of
this ray right over here.

00:00:25.409 --> 00:00:26.800
Yeah that makes sense

00:00:26.800 --> 00:00:28.879
that that slope is about half.

00:00:28.879 --> 00:00:32.663
Now what other angles
have a tangent of 1 half?

00:00:32.663 --> 00:00:33.759
So let's look at these choices.

00:00:33.759 --> 00:00:37.448
So this is our original
angle, 0.46 radians,

00:00:37.448 --> 00:00:39.001
plus pi over 2.

00:00:39.001 --> 00:00:41.414
If you think in degrees, pi is 180.

00:00:41.414 --> 00:00:43.568
pi over 2 is 90 degrees.

00:00:43.568 --> 00:00:45.782
So this one...

00:00:45.782 --> 00:00:47.374
Actually let me do in a color

00:00:47.374 --> 00:00:48.246
you're more likely to see.

00:00:48.246 --> 00:00:53.670
This one is gonna look like this.

00:00:53.670 --> 00:00:58.982
Where this is an angle of pi over 2.

00:00:58.982 --> 00:01:00.254
And just eyeballing it,

00:01:00.254 --> 00:01:02.214
you immediately see that
the slope of this ray

00:01:02.214 --> 00:01:04.845
is very different than
the slope of this ray

00:01:04.845 --> 00:01:06.349
right over here.

00:01:06.349 --> 00:01:08.022
In fact they look like they are.

00:01:08.022 --> 00:01:09.813
They are perpendicular
because they have an angle

00:01:09.813 --> 00:01:11.541
of pi over 2 between them.

00:01:11.541 --> 00:01:12.285
But they're definitely not

00:01:12.285 --> 00:01:15.876
going to have the same tangent.

00:01:15.876 --> 00:01:17.626
They don't have the same slope.

00:01:17.626 --> 00:01:21.449
Let's think about pi minus 0.46.

00:01:21.449 --> 00:01:23.064
So that's essentially pi is going

00:01:23.064 --> 00:01:24.721
along the positive x axis.

00:01:24.721 --> 00:01:26.201
You go all the way around.

00:01:26.201 --> 00:01:28.573
Or half way around to your pi radians.

00:01:28.573 --> 00:01:30.819
But then we're gonna subtract 0.46.

00:01:30.819 --> 00:01:33.963
So it's gonna look something like this.

00:01:33.963 --> 00:01:36.219
It's gonna look something like that

00:01:36.219 --> 00:01:40.347
where this is 0.46 that
we have subtracted.

00:01:40.347 --> 00:01:41.299
Another way to think about it,

00:01:41.299 --> 00:01:43.370
if we take our original terminal ray

00:01:43.370 --> 00:01:46.139
and we flip it over the y axis,

00:01:46.139 --> 00:01:49.147
we get to this terminal
ray right over here.

00:01:49.147 --> 00:01:51.106
And you could immediately
see that the slope

00:01:51.106 --> 00:01:53.110
of the terminal ray is not the same

00:01:53.110 --> 00:01:54.406
as the slope of this one,

00:01:54.406 --> 00:01:55.746
of our first one, of our original,

00:01:55.746 --> 00:01:58.998
in fact they look like the
negatives of each other.

00:01:58.998 --> 00:02:01.984
So we can rule that one out as well.

00:02:01.984 --> 00:02:05.936
0.46 plus pi or pi plus 0.46.

00:02:05.936 --> 00:02:08.624
So that's going to take us...

00:02:08.624 --> 00:02:11.025
If you add pi to this you're essentially

00:02:11.025 --> 00:02:12.592
going half way around the unit circle

00:02:12.592 --> 00:02:14.752
and you're getting to a point that is...

00:02:14.752 --> 00:02:17.097
Or you're forming a ray that is collinear

00:02:17.097 --> 00:02:19.096
with the original ray.

00:02:19.096 --> 00:02:22.112
So that's that angle right over here.

00:02:22.112 --> 00:02:28.456
So pi plus 0.46 is this
entire angle right over there.

00:02:28.456 --> 00:02:30.273
And when you just look at this ray,

00:02:30.273 --> 00:02:31.896
you see its collinear is going to have the

00:02:31.896 --> 00:02:35.459
exact same slope as the terminal ray for

00:02:35.459 --> 00:02:37.665
the 0.46 radion.

00:02:37.665 --> 00:02:40.184
So just that tells you that the tangent

00:02:40.184 --> 00:02:41.528
is going to be the same.

00:02:41.528 --> 00:02:43.057
So I could check that there.

00:02:43.057 --> 00:02:45.281
And in previous videos when we explore

00:02:45.281 --> 00:02:47.089
the symmetries of the tangent function,

00:02:47.089 --> 00:02:48.465
we in fact saw that.

00:02:48.465 --> 00:02:50.849
That if you took an angle
and you add pi to it,

00:02:50.849 --> 00:02:52.633
you're going to have the same tangent.

00:02:52.633 --> 00:02:53.841
And if you wanna dig a little bit deeper,

00:02:53.841 --> 00:02:55.097
I encourage you to look at that video

00:02:55.097 --> 00:02:58.249
on the symmetries of
unit circle symmetries

00:02:58.249 --> 00:03:00.349
for the tangent function.

00:03:00.349 --> 00:03:01.969
So let's look at these other choices.

00:03:01.969 --> 00:03:04.664
2 pi minus 0.46.

00:03:04.664 --> 00:03:07.073
So 2 pi...

00:03:07.073 --> 00:03:09.033
If this is 0 degrees,

00:03:09.033 --> 00:03:13.665
2 pi gets you back to the positive x axis

00:03:13.665 --> 00:03:16.185
and then you're going to subtract 0.46.

00:03:16.185 --> 00:03:19.992
So that's going to be this
angle right over here.

00:03:19.992 --> 00:03:22.529
And that looks like it
has the negative slope

00:03:22.529 --> 00:03:24.976
of this original ray right up here.

00:03:24.976 --> 00:03:28.745
So these aren't going to
have the same tangent.

00:03:28.745 --> 00:03:29.663
Now this one,

00:03:29.663 --> 00:03:32.991
you're taking 0.46 and
you're adding 2 pi.

00:03:32.991 --> 00:03:38.440
So you're taking 0.46 and
then you're adding 2 pi

00:03:38.440 --> 00:03:39.463
which essentially is just going

00:03:39.463 --> 00:03:40.751
around the unit circle once

00:03:40.751 --> 00:03:42.679
and you get to the exact same point.

00:03:42.679 --> 00:03:45.232
So you add 2 pi to any angle measure,

00:03:45.232 --> 00:03:46.151
you're going to not only have the

00:03:46.151 --> 00:03:47.743
same tangent value,

00:03:47.743 --> 00:03:49.952
you're gonna have the same
sine value, cosine value

00:03:49.952 --> 00:03:51.847
because you're essentially
going back around

00:03:51.847 --> 00:03:54.705
to the exact same angle
when you add 2 pi.

00:03:54.705 --> 00:03:58.705
So this is definitely
also going to be true.


Symmetry of trig values

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.166
Voiceover:Let's explore the unit circle

00:00:03.166 --> 00:00:04.433
a little bit more in depth.

00:00:04.433 --> 00:00:07.242
Let's just start with some angle theta,

00:00:07.242 --> 00:00:08.768
and for the sake of this video,

00:00:08.768 --> 00:00:11.000
we'll assume everything is in radians.

00:00:13.433 --> 00:00:15.303
This angle right over here,

00:00:15.303 --> 00:00:17.633
we would call this theta.

00:00:17.633 --> 00:00:19.769
Now let's flip this, I guess we could say,

00:00:19.769 --> 00:00:21.648
the terminal ray of this angle.

00:00:21.648 --> 00:00:24.566
Let's flip it over the X and Y-axis.

00:00:24.566 --> 00:00:27.766
Let's just make sure we
have labeled our axes.

00:00:27.766 --> 00:00:30.971
Let's flip it over the positive X-axis.

00:00:30.971 --> 00:00:33.300
If you flip it over the positive X-axis,

00:00:33.300 --> 00:00:34.413
you just go straight down,

00:00:34.413 --> 00:00:36.186
and then you go the same
distance on the other side.

00:00:36.186 --> 00:00:38.438
You get to that point right over there,

00:00:38.438 --> 00:00:41.000
and so you would get this ray.

00:00:41.633 --> 00:00:45.001
You would get this ray that
I'm attempting to draw in blue.

00:00:45.367 --> 00:00:47.501
You would get that ray right over there.

00:00:47.501 --> 00:00:49.100
Now what is the angle between this ray

00:00:49.100 --> 00:00:52.933
and the positive X-axis if you
start at the positive X-axis?

00:00:53.253 --> 00:00:56.084
Well, just using our conventions
that counterclockwise

00:00:56.084 --> 00:00:59.024
from the X-axis is a positive angle,

00:00:59.024 --> 00:01:00.433
this is clockwise.

00:01:00.433 --> 00:01:02.901
Instead of going theta above the X-axis,

00:01:02.901 --> 00:01:03.900
we're going theta below,

00:01:03.900 --> 00:01:06.033
so we would call this, by our convention,

00:01:06.033 --> 00:01:08.733
an angle of negative theta.

00:01:09.167 --> 00:01:11.700
Now let's flip our original green ray.

00:01:11.700 --> 00:01:14.066
Let's flip it over the positive Y-axis.

00:01:14.366 --> 00:01:16.434
If you flip it over the positive Y-axis,

00:01:16.434 --> 00:01:21.435
we're going to go from there
all the way to right over there

00:01:21.435 --> 00:01:26.600
then we can draw ourselves a ray.

00:01:27.702 --> 00:01:30.601
My best attempt at that
is right over there.

00:01:30.640 --> 00:01:34.436
What would be the measure of
this angle right over here?

00:01:34.436 --> 00:01:38.800
What was the measure of
that angle in radians?

00:01:39.036 --> 00:01:41.281
We know if we were to go all the way

00:01:41.619 --> 00:01:44.433
from the positive X-axis
to the negative X-axis,

00:01:44.433 --> 00:01:46.101
that would be pi radians

00:01:46.101 --> 00:01:48.667
because that's halfway around the circle.

00:01:49.517 --> 00:01:53.100
This angle, since we
know that that's theta,

00:01:53.100 --> 00:01:54.767
this is theta right over here,

00:01:54.767 --> 00:01:56.367
the angle that we want to figure out,

00:01:56.367 --> 00:01:58.300
this is going to be all the way around.

00:01:58.300 --> 00:02:01.100
It's going to be pi minus,

00:02:01.100 --> 00:02:04.402
it's going to be pi minus theta.

00:02:04.402 --> 00:02:06.767
Notice, pi minus theta plus theta,

00:02:06.767 --> 00:02:09.566
these two are supplementary,

00:02:09.566 --> 00:02:14.133
and they add up to pi
radians or 180 degrees.

00:02:14.433 --> 00:02:18.000
Now let's flip this one
over the negative X-axis.

00:02:18.855 --> 00:02:21.967
If we flip this one over
the negative X-axis,

00:02:21.967 --> 00:02:23.567
you're going to get right over there,

00:02:23.567 --> 00:02:26.600
and so you're going to get an
angle that looks like this,

00:02:27.034 --> 00:02:28.533
that looks like this.

00:02:28.966 --> 00:02:31.700
Now what is going to be
the measure of this angle?

00:02:31.700 --> 00:02:35.833
If we go all the way around like that,

00:02:35.833 --> 00:02:37.233
what is the measure of that angle?

00:02:37.233 --> 00:02:39.900
To go this far is pi,

00:02:39.900 --> 00:02:42.233
and then you're going another theta.

00:02:42.233 --> 00:02:43.700
This angle right over here is theta,

00:02:43.700 --> 00:02:46.104
so you're going pi plus another theta.

00:02:46.104 --> 00:02:47.967
This whole angle right over here,

00:02:47.967 --> 00:02:49.703
this whole thing,

00:02:49.703 --> 00:02:53.202
this whole thing is pi plus theta radians.

00:02:53.434 --> 00:02:55.033
Pi plus theta,

00:02:55.033 --> 00:02:56.167
let me just write that down.

00:02:56.167 --> 00:02:59.267
This is pi plus theta.

00:02:59.767 --> 00:03:02.459
Now that we've figured out

00:03:02.658 --> 00:03:04.769
these have different
symmetries about them,

00:03:04.769 --> 00:03:06.700
let's think about how
the sines and cosines

00:03:06.700 --> 00:03:08.866
of these different angles
relate to each other.

00:03:09.233 --> 00:03:12.833
We already know that this
coordinate right over here,

00:03:12.833 --> 00:03:15.233
that is sine of theta,

00:03:15.233 --> 00:03:18.374
sorry, the X-coordinate
is cosine of theta.

00:03:18.374 --> 00:03:20.100
The X-coordinate is cosine of theta,

00:03:20.100 --> 00:03:22.566
and the Y-coordinate is sine of theta.

00:03:22.566 --> 00:03:23.968
Or another way of thinking about it

00:03:23.968 --> 00:03:27.100
is this value on the
X-axis is cosine of theta,

00:03:27.100 --> 00:03:29.566
and this value right
over here on the Y-axis

00:03:29.566 --> 00:03:31.050
is sine of theta.

00:03:31.633 --> 00:03:33.740
Now let's think about
this one down over here.

00:03:33.740 --> 00:03:36.505
By the same convention, this point,

00:03:36.505 --> 00:03:38.368
this is really the unit circle definition

00:03:38.368 --> 00:03:40.166
of our trig functions.

00:03:40.166 --> 00:03:42.903
This point, since our angle
is negative theta now,

00:03:42.903 --> 00:03:47.252
this point would be cosine
of negative theta, comma,

00:03:47.252 --> 00:03:52.267
sine of negative theta.

00:03:52.674 --> 00:03:54.982
And we can apply the same thing over here.

00:03:54.982 --> 00:03:56.766
This point right over here,

00:03:56.766 --> 00:04:02.833
the X-coordinate is
cosine of pi minus theta.

00:04:02.833 --> 00:04:04.366
That's what this angle is when we go

00:04:04.366 --> 00:04:06.300
from the positive X-axis.

00:04:06.300 --> 00:04:10.233
This is cosine of pi minus theta.

00:04:10.233 --> 00:04:16.633
And the Y-coordinate is
the sine of pi minus theta.

00:04:16.633 --> 00:04:19.800
Then we could go all the
way around to this point.

00:04:20.167 --> 00:04:22.268
I think you see where this is going.

00:04:22.367 --> 00:04:26.233
This is cosine of, I guess
we could say theta plus pi

00:04:26.233 --> 00:04:27.833
or pi plus theta.

00:04:27.833 --> 00:04:35.600
Let's write pi plus data
and sine of pi plus theta.

00:04:35.767 --> 00:04:38.233
Now how do these all relate to each other?

00:04:38.233 --> 00:04:41.900
Notice, over here, out here
on the right-hand side,

00:04:41.900 --> 00:04:45.233
our X-coordinates are
the exact same value.

00:04:45.233 --> 00:04:47.169
It's this value right over here.

00:04:47.169 --> 00:04:50.033
So we know that cosine
of theta must be equal

00:04:50.033 --> 00:04:52.500
to the cosine of negative theta.

00:04:52.500 --> 00:04:53.700
That's pretty interesting.

00:04:53.700 --> 00:04:55.167
Let's write that down.

00:04:55.167 --> 00:05:00.767
Cosine of theta is equal to ...

00:05:00.767 --> 00:05:02.367
let me do it in this blue color,

00:05:02.367 --> 00:05:05.300
is equal to the cosine of negative theta.

00:05:05.300 --> 00:05:07.200
That's a pretty interesting result.

00:05:07.233 --> 00:05:09.367
But what about their sines?

00:05:09.367 --> 00:05:11.833
Well, here, the sine of
theta is this distance

00:05:11.833 --> 00:05:13.769
above the X-axis,

00:05:13.769 --> 00:05:15.433
and here, the sine of negative theta

00:05:15.433 --> 00:05:17.900
is the same distance below the X-axis,

00:05:17.900 --> 00:05:20.900
so they're going to be the
negatives of each other.

00:05:20.900 --> 00:05:26.100
We could say that sine of negative theta,

00:05:26.100 --> 00:05:30.900
sine of negative theta is equal to,

00:05:30.900 --> 00:05:34.433
is equal to the negative sine of theta,

00:05:34.433 --> 00:05:37.500
equal to the negative sine of theta.

00:05:37.500 --> 00:05:38.433
It's the opposite.

00:05:38.433 --> 00:05:41.900
If you go the same amount
above or below the X-axis,

00:05:41.900 --> 00:05:45.835
you're going to get the
negative value for the sine.

00:05:45.835 --> 00:05:47.100
We could do the same thing over here.

00:05:47.100 --> 00:05:48.767
How does this one relate to that?

00:05:48.767 --> 00:05:52.900
These two are going to
have the same sine values.

00:05:52.900 --> 00:05:54.300
The sine of this, the Y-coordinate,

00:05:54.300 --> 00:05:56.433
is the same as the sine of that.

00:05:56.433 --> 00:05:59.768
We see that this must be equal to that.

00:05:59.768 --> 00:06:01.100
Let's write that down.

00:06:01.100 --> 00:06:14.067
We get sine of theta is equal
to sine of pi minus theta.

00:06:14.967 --> 00:06:16.917
Now let's think about how
do the cosines relate.

00:06:17.054 --> 00:06:18.119
The same argument, they're going to be

00:06:18.119 --> 00:06:19.433
the opposites of each other,

00:06:19.433 --> 00:06:22.366
where the X-coordinates
are the same distance

00:06:22.366 --> 00:06:24.800
but on opposite sides of the origin.

00:06:24.833 --> 00:06:29.900
We get cosine of theta is equal to

00:06:29.900 --> 00:06:33.369
the negative of the cosine of ...

00:06:33.369 --> 00:06:34.700
let me do that in same color.

00:06:34.700 --> 00:06:36.734
Actually, let me make
sure my colors are right.

00:06:37.166 --> 00:06:41.166
We get cosine of theta is equal to

00:06:41.166 --> 00:06:46.200
the negative of the
cosine of pi minus theta.

00:06:46.700 --> 00:06:49.933
Now finally, let's think
about how this one relates.

00:06:50.166 --> 00:06:54.967
Here, our cosine value, our
X-coordinate is the negative,

00:06:54.967 --> 00:06:58.366
and our sine value is also the negative.

00:06:58.366 --> 00:07:00.533
We've flipped over both axes.

00:07:00.900 --> 00:07:01.966
Let's write that down.

00:07:01.966 --> 00:07:09.366
Over here, we have sine of theta plus pi,

00:07:09.366 --> 00:07:11.441
which is the same thing as pi plus theta,

00:07:11.441 --> 00:07:17.768
is equal to the negative
of the sine of theta,

00:07:17.768 --> 00:07:19.367
and we see that this is sine of theta,

00:07:19.367 --> 00:07:21.643
this is sine of pi plus theta,

00:07:21.643 --> 00:07:23.438
or sine of theta plus pi,

00:07:23.438 --> 00:07:29.100
and we get the cosine of theta plus pi.

00:07:29.100 --> 00:07:31.243
Cosine of theta plus pi is going to be

00:07:31.243 --> 00:07:33.766
the negative of cosine of theta,

00:07:33.766 --> 00:07:38.067
is equal to the negative
of cosine of theta.

00:07:38.100 --> 00:07:39.151
Even here, and you could see,

00:07:39.151 --> 00:07:39.900
you could keep going.

00:07:39.900 --> 00:07:41.900
You could try to relate
this one to that one

00:07:41.900 --> 00:07:43.433
or that one to that one.

00:07:43.433 --> 00:07:45.701
You can get all sorts
of interesting results.

00:07:45.701 --> 00:07:48.101
I encourage you to really
try to think this through

00:07:48.101 --> 00:07:51.034
on your own and think
about how all of these

00:07:51.034 --> 00:07:53.567
are related to each other
based on essentially

00:07:53.567 --> 00:07:57.969
symmetries or reflections
around the X or Y-axis.


Unit circle symmetries for tan

WEBVTT
Kind: captions
Language: en

00:00:00.767 --> 00:00:02.563
Voiceover:The previous video we explored

00:00:02.563 --> 00:00:05.373
how the cosine and sines of angles relate.

00:00:05.373 --> 00:00:07.301
We essentially take the
terminal ray of the angle and

00:00:07.301 --> 00:00:12.600
we reflect it about the X
or the Y axis, or both axes.

00:00:12.835 --> 00:00:14.238
What I want to do in this
video is think a little bit

00:00:14.238 --> 00:00:17.509
about the tangent of
these different angles.

00:00:17.509 --> 00:00:19.381
So just as a little bit of a reminder,

00:00:19.381 --> 00:00:23.506
we know that the tangent
of theta is equal to

00:00:23.506 --> 00:00:28.313
the sine of an angle over
the cosine of an angle,

00:00:28.313 --> 00:00:30.382
and by the Unit Circle Definition,

00:00:30.382 --> 00:00:32.172
it's essentially saying,
"What is the slope

00:00:32.172 --> 00:00:34.641
"of the terminal ray right over here?"

00:00:34.641 --> 00:00:37.841
We remind ourselves
slope is rise over run.

00:00:37.841 --> 00:00:40.311
It is our change in the vertical axis

00:00:40.311 --> 00:00:44.107
over our change in the horizontal axis.

00:00:44.107 --> 00:00:46.575
If we're starting at the
origin, what is our change

00:00:46.575 --> 00:00:49.978
in the vertical axis if we
go from zero to sine theta?

00:00:49.978 --> 00:00:53.376
Well, our change in the
vertical axis is sine theta.

00:00:53.376 --> 00:00:55.166
What is our change in the horizontal axis?

00:00:55.166 --> 00:00:56.979
It's cosine of theta.

00:00:56.979 --> 00:00:58.642
So this is change in Y over

00:00:58.642 --> 00:01:02.138
change in X for the terminal ray.

00:01:07.581 --> 00:01:09.709
So the tangent of theta
is the sine of theta

00:01:09.709 --> 00:01:11.105
over cosine of theta, or you could view it

00:01:11.105 --> 00:01:15.075
as the slope of this ray right over here.

00:01:15.511 --> 00:01:18.506
Lets think about what other angles are

00:01:18.506 --> 00:01:22.074
going to have the exact
same tangent of theta?

00:01:22.307 --> 00:01:25.240
This ray is collinear with
this ray right over here.

00:01:25.240 --> 00:01:28.444
In fact if you put them
together you get a line.

00:01:28.444 --> 00:01:32.042
So the tangent of this
angle right over here,

00:01:32.042 --> 00:01:34.382
this pink angle going all the way around,

00:01:34.382 --> 00:01:36.910
the tangent of pi plus theta,

00:01:36.910 --> 00:01:40.579
or the tangent of theta plus pi,

00:01:40.579 --> 00:01:42.645
obviously you could write theta

00:01:42.645 --> 00:01:45.141
plus pi instead of pi plus theta.

00:01:45.177 --> 00:01:48.968
This should be, just based
on this slope argument,

00:01:48.968 --> 00:01:52.068
this should be equal to
the tangent of theta.

00:01:55.637 --> 00:01:58.205
Lets see if this actually is the case.

00:01:58.713 --> 00:02:01.167
So these two things should be equal

00:02:01.167 --> 00:02:04.435
if we agree that the
tangent of an angle is

00:02:04.435 --> 00:02:08.545
equal to the slope of the terminal ray.

00:02:08.714 --> 00:02:10.108
Of course the other side of the angle is

00:02:10.108 --> 00:02:11.900
going to be the positive X axis based

00:02:11.900 --> 00:02:13.934
on the conventions that we've set up.

00:02:14.370 --> 00:02:17.110
Lets think about what it
is when the tangent of

00:02:17.110 --> 00:02:22.345
theta plus pi is in
terms of sine and cosine.

00:02:25.511 --> 00:02:27.473
Let me write this down in the pink color.

00:02:27.711 --> 00:02:29.043
The tangent.

00:02:29.043 --> 00:02:30.544
That's not pink.

00:02:30.775 --> 00:02:37.309
The tangent of pi plus theta,
that's going to be equal to,

00:02:37.309 --> 00:02:39.245
put the parentheses to avoid ambiguity,

00:02:39.245 --> 00:02:41.945
that's equal to the sine of pi plus theta,

00:02:43.314 --> 00:02:48.473
or theta plus pi, over the
cosine of theta plus pi.

00:02:49.111 --> 00:02:50.437
And in the previous video we established

00:02:50.437 --> 00:02:54.112
that the sine of theta plus pi,

00:02:54.112 --> 00:02:57.160
that's the same thing
as negative sine theta.

00:02:57.316 --> 00:03:01.407
So this is equal to negative sine theta.

00:03:01.970 --> 00:03:03.938
And the cosine of theta plus pi,

00:03:03.970 --> 00:03:04.902
we already established that's the

00:03:04.902 --> 00:03:07.734
same thing as negative cosine of theta.

00:03:10.383 --> 00:03:11.580
We have a negative divided by a negative

00:03:11.580 --> 00:03:13.311
that you could say the
negatives cancel out,

00:03:13.311 --> 00:03:16.433
and we're left with sine
theta over cosine theta,

00:03:16.433 --> 00:03:19.167
which is indeed tangent of theta,

00:03:19.167 --> 00:03:21.571
so we can feel pretty good about that.

00:03:21.571 --> 00:03:25.700
Now what about the points, or the

00:03:25.700 --> 00:03:28.605
terminal rays right over here?

00:03:32.101 --> 00:03:33.177
Lets think about this point.

00:03:33.177 --> 00:03:38.603
What is the tangent of
negative theta going to be?

00:03:39.380 --> 00:03:41.839
We know that the tangent of negative theta

00:03:41.839 --> 00:03:45.980
is the same thing as the
sine of negative theta

00:03:45.980 --> 00:03:49.367
over the cosine of negative theta,

00:03:49.367 --> 00:03:54.034
and we already established
the sine of negative theta,

00:03:54.034 --> 00:03:57.207
that's negative sine theta.

00:03:58.300 --> 00:03:59.111
We see that right over here,

00:03:59.111 --> 00:04:00.845
sine of negative theta.

00:04:00.845 --> 00:04:03.313
That's the negative, that's the opposite,

00:04:03.313 --> 00:04:04.976
of the sine of theta,
so we have that there,

00:04:04.976 --> 00:04:07.702
but the cosine of negative theta is

00:04:07.702 --> 00:04:10.379
the same thing as the cosine of theta,

00:04:10.379 --> 00:04:12.879
so these things are the same.

00:04:13.046 --> 00:04:17.048
So we're left with
negative sine theta over

00:04:17.048 --> 00:04:19.773
cosine of theta, which is the same thing,

00:04:19.773 --> 00:04:25.407
equal to, negative tangent theta.

00:04:25.709 --> 00:04:30.513
So we see here if you take
the negative of the angle,

00:04:30.513 --> 00:04:32.300
you're going to get the
negative of the tangent,

00:04:32.300 --> 00:04:34.714
and that's because the
sine, the numerator,

00:04:34.714 --> 00:04:36.233
in our definition of tangent,

00:04:36.233 --> 00:04:41.668
changes signs, but the
denominator does not.

00:04:42.102 --> 00:04:43.907
So the tangent of negative theta is

00:04:43.907 --> 00:04:46.668
the same thing as
negative tangent of theta.

00:04:46.913 --> 00:04:51.011
Now, what about this
point right over here?

00:04:51.040 --> 00:04:53.708
Well over here, relative to theta,

00:04:53.708 --> 00:04:56.833
when we're looking at pi minus theta,

00:04:56.833 --> 00:04:59.872
so when we're looking at
tangent of pi minus theta,

00:05:05.710 --> 00:05:13.603
that's sine of pi minus theta
over cosine of pi minus theta.

00:05:13.901 --> 00:05:16.946
and we already established
in the previous video,

00:05:17.042 --> 00:05:20.543
that sine of pi minus theta
is equal to sine of theta,

00:05:20.647 --> 00:05:21.902
and we see that right over here,

00:05:21.902 --> 00:05:24.036
they have the exact same sines,

00:05:24.036 --> 00:05:27.710
so this is equal to sine of theta,

00:05:27.710 --> 00:05:30.102
while cosine of pi minus theta,

00:05:30.102 --> 00:05:32.910
well, it's the opposite
of cosine of theta,

00:05:32.910 --> 00:05:34.681
it's the negative of cosine of theta.

00:05:37.569 --> 00:05:38.643
And so this once again is going

00:05:38.643 --> 00:05:41.175
to be equal to the
negative sine over cosine,

00:05:41.175 --> 00:05:45.380
or the negative tangent of
theta, which makes sense.

00:05:45.380 --> 00:05:48.243
This ray should have the same slope

00:05:48.243 --> 00:05:52.905
as this ray right over here.

00:05:52.905 --> 00:05:54.842
And we see that slope, we could view

00:05:54.842 --> 00:05:59.447
this as negative tangent of theta.

00:05:59.447 --> 00:06:03.310
And we see just looking at these two,

00:06:03.310 --> 00:06:06.836
if you combine the rays, that these two

00:06:06.836 --> 00:06:09.370
intersecting lines have the negative slope

00:06:09.370 --> 00:06:13.543
of each other, they're mirror
images across the X axis.

00:06:13.912 --> 00:06:16.833
And so, we've just seen,
if you take an angle

00:06:16.833 --> 00:06:19.301
and you add pi to the
angle, you're tangent

00:06:19.301 --> 00:06:21.578
won't change because you're going to

00:06:21.578 --> 00:06:23.441
essentially be sitting on the same line.

00:06:23.441 --> 00:06:26.511
pi, everything in degrees,
you're going 180 degrees around.

00:06:26.511 --> 00:06:27.908
You're going the opposite direction but

00:06:27.908 --> 00:06:30.314
the slope of your ray has not changed.

00:06:30.314 --> 00:06:32.640
So tangent of theta is the same thing

00:06:32.640 --> 00:06:36.362
as the tangent of theta plus pi,

00:06:36.377 --> 00:06:38.444
but if you take the
negative of your angle,

00:06:38.444 --> 00:06:42.177
then you're going to get
the negative of your tangent

00:06:42.177 --> 00:06:46.177
or, if you were to go
over here, and if you were

00:06:46.177 --> 00:06:52.376
to take pi minus your
angle, then you're also

00:06:52.376 --> 00:06:55.303
going to get the negative of your tangent.

00:06:55.303 --> 00:06:57.234
Hopefully this makes you a little bit,

00:06:57.234 --> 00:06:58.842
this is very useful when you're trying

00:06:58.842 --> 00:07:00.968
to work though trigonometric problems

00:07:00.968 --> 00:07:02.642
or try to find relationships
or even when we're

00:07:02.642 --> 00:07:05.504
trying to use our identities
or prove our identities,

00:07:05.504 --> 00:07:06.578
and essentially what we've done here is

00:07:06.578 --> 00:07:08.845
we have proven some identities,

00:07:08.845 --> 00:07:10.302
but it's really helpful
to think about these

00:07:10.302 --> 00:07:14.342
symmetrys that we have
within the unit circle.