Example: Using trig to solve for missing information

WEBVTT
Kind: captions
Language: en

00:00:01.096 --> 00:00:04.407
For acute angle E, so that's this angle

00:00:04.407 --> 00:00:05.143
right over here,

00:00:05.143 --> 00:00:06.352
it tells that sin of E is 5 over

00:00:06.352 --> 00:00:09.601
the square root of 41, and cotangent of E is

00:00:09.601 --> 00:00:11.853
equal to 8/10. Find the values of the

00:00:11.853 --> 00:00:15.137
other four trigonometric ratios. So for

00:00:15.137 --> 00:00:17.272
the trig ratios, I like to use SOH-CAH-TOA

00:00:17.272 --> 00:00:20.263
to remember what the definitions of the trig ratios were

00:00:20.263 --> 00:00:21.100
So let me write this down,

00:00:21.100 --> 00:00:23.845
SOH, CAH

00:00:23.845 --> 00:00:26.069
I'll write CAH in a different color,

00:00:26.069 --> 00:00:28.537
SOH, CAH

00:00:28.537 --> 00:00:29.606
that's not a different color

00:00:29.606 --> 00:00:30.515
I'm trying-

00:00:30.515 --> 00:00:32.604
I'm having trouble changing colors!

00:00:32.604 --> 00:00:35.355
SOH, CAH, TOA.

00:00:35.355 --> 00:00:36.845
SOH, CAH, TOA.

00:00:36.845 --> 00:00:40.404
So, SOH tells us that sine of an angle

00:00:40.404 --> 00:00:41.604
in this case we care about angle E,

00:00:41.604 --> 00:00:44.933
is equal to the opposite over the hypotenuse.

00:00:44.933 --> 00:00:47.076
If we look at angle E, what is the opposite?

00:00:47.076 --> 00:00:48.804
Well, we go across the triangle

00:00:48.804 --> 00:00:50.681
the opposite is right over there.

00:00:50.681 --> 00:00:53.515
That is the opposite so it would be equal to

00:00:53.515 --> 00:00:55.800
5 over the hypotenuse.

00:00:55.800 --> 00:00:58.801
Well, the hypotenuse is opposite the right-angle,

00:00:58.801 --> 00:01:01.867
and it's the longest side of the triangle, it's side DE

00:01:01.867 --> 00:01:04.269
and has length of square-root of 41.

00:01:04.269 --> 00:01:05.853
Square-root of 41.

00:01:05.853 --> 00:01:07.770
So that's consistently the information they gave us

00:01:07.770 --> 00:01:09.869
so this was kind of redundant information,

00:01:09.869 --> 00:01:11.267
we could have figured that out just from

00:01:11.267 --> 00:01:13.432
the diagram.

00:01:13.432 --> 00:01:15.136
But at least we got that out of the way.

00:01:15.136 --> 00:01:16.773
Now let's think about,

00:01:16.773 --> 00:01:20.145
now let's think about the reciprocal of the sine of E,

00:01:20.145 --> 00:01:23.679
which is the co-secant of E.

00:01:23.679 --> 00:01:26.536
Co-secant of E

00:01:26.536 --> 00:01:28.354
which is the reciprocal sine which is

00:01:28.354 --> 00:01:30.351
hypotenuse over the opposite.

00:01:30.351 --> 00:01:31.271
and we could-

00:01:31.271 --> 00:01:32.535
we don't even have to look at the triangle

00:01:32.535 --> 00:01:33.767
we could just say the reciprocal

00:01:33.767 --> 00:01:35.605
which would be the square-root of 41 over 5.

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Or you could look back at this and try to figure that out.

00:01:39.470 --> 00:01:41.934
Now let's think about the cosine.

00:01:41.934 --> 00:01:44.203
What is the cosine?

00:01:44.203 --> 00:01:48.770
What is the cosine of E going to be equal to?

00:01:48.770 --> 00:01:50.736
Well, first think of what the definition is.

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Cosine of E is going to be what?

00:01:53.203 --> 00:01:54.773
Well, CAH tells us that this is going to be

00:01:54.773 --> 00:01:58.688
equal to the adjacent over the hypotenuse.

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Well, if we look over here, we know what the hypotenuse is.

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The hypotenuse the same.

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The hypotenuse is the square-root of 41.

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What is the adjacent side?

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What length does the adjacent side have?

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Well, for angle E, the adjacent side is side FE

00:02:16.018 --> 00:02:18.203
and we don't know what that is right over here.

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So that is the adjacent side and

00:02:20.203 --> 00:02:21.354
we don't know what that is

00:02:21.354 --> 00:02:22.770
So I'll just write 'a'.

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'a' for how they marked it on this,

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or you could also do this 'a' for 'adjacent.

00:02:26.683 --> 00:02:27.737
So right now we're just gonna

00:02:27.737 --> 00:02:30.069
leave it with the variable 'a' over the square-root of 41

00:02:30.069 --> 00:02:32.199
maybe we can get a little bit more information

00:02:32.199 --> 00:02:34.603
that helps us to figure out what 'a' is

00:02:34.603 --> 00:02:37.535
over the course of this problem.

00:02:37.535 --> 00:02:39.891
If you wanna figure out what the secant of E is,

00:02:39.891 --> 00:02:41.809
well, that's just the inverse of the cosine

00:02:41.809 --> 00:02:44.137
it's hypotenuse over adjacent,

00:02:44.137 --> 00:02:47.473
so it's going to be the square-root of 41 over 'a'.

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Whatever 'a' is. Hopefully we can figure that out.

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Now let's use TOA.

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That tells us that the tangent of E is equal to the opposite over the adjacent.

00:02:57.936 --> 00:03:01.388
Well, what's the opposite side to angle E?

00:03:01.388 --> 00:03:05.224
Well, it's side- it has length 5, it's side DF

00:03:05.224 --> 00:03:06.675
The opposite is 5.

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And we still don't know the length of the adjacent side.

00:03:08.670 --> 00:03:10.804
That's this side of length 'a'.

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So I'll just write 'a', right over there.

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Now what about cotangent?

00:03:15.535 --> 00:03:18.209
Well, cotangent is the reciprocal of tangent.

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So it's adjacent over opposite or in this case,

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'a' over 5.

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Adjacent over opposite; 'a' over 5.

00:03:28.109 --> 00:03:29.358
Well, what do they already tell us?

00:03:29.358 --> 00:03:34.032
And using that, can you figure out what 'a' is?

00:03:34.032 --> 00:03:37.286
Well, they already told us that cotangent of E

00:03:37.286 --> 00:03:40.203
is equal to 8/10.

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There's a little clue here that this is not

00:03:41.604 --> 00:03:43.074
reduced fully; 8/10.

00:03:43.074 --> 00:03:44.473
8 and 10 share common factors.

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So they already tell us that cotangent of E is 8/10.

00:03:47.269 --> 00:03:50.605
So if we use the definition of cotangent of E we get 'a' over 5,

00:03:50.605 --> 00:03:56.014
and they tell us that that is going to be equal to 8/10.

00:03:56.014 --> 00:03:59.939
So we can write, we have an equation now to solve for 'a'.

00:03:59.939 --> 00:04:03.003
And if we solve for 'a' now we can figure out all of the other,

00:04:03.003 --> 00:04:05.088
all of the other ratios.

00:04:05.088 --> 00:04:06.469
So let's do that.

00:04:06.469 --> 00:04:09.843
So we have 'a' over 5 is equal to,

00:04:09.843 --> 00:04:11.268
-we'll simplify this in a little bit-

00:04:11.268 --> 00:04:13.437
what is 8/10 if you simplify it?

00:04:13.437 --> 00:04:16.354
Well, we have 8 by- we have a common factor of 2-

00:04:16.354 --> 00:04:19.140
if you divide 8 by 2 you get 4,

00:04:19.140 --> 00:04:21.933
we divide 10 by two, we get 5.

00:04:21.933 --> 00:04:24.468
So we get 'a' over 5 is equal to 4/5.

00:04:24.468 --> 00:04:26.935
And so this is actually- we could cross-multiply or

00:04:26.935 --> 00:04:29.077
multiply both sides by 5 and you would get

00:04:29.077 --> 00:04:30.203
-either way-

00:04:30.203 --> 00:04:33.204
you would get that 'a' is equal to 4.

00:04:33.204 --> 00:04:35.771
Let's do that, just to show you you can do it systematically.

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And you're left with 'a' is equal to 4 which is great

00:04:39.203 --> 00:04:42.803
because we can now say that the cotangent of E,

00:04:42.803 --> 00:04:46.101
yes it's 8/10 which is the same thing as 4/5.

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We could say that the tangent of E

00:04:47.472 --> 00:04:48.768
-instead of saying it's 5 over 'a'-

00:04:48.768 --> 00:04:51.936
we can now say that it's 5 over 4.

00:04:51.936 --> 00:04:54.670
And now what would the cosine of E be?

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Well, it was 'a' over (square-root of) 41,

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now it is -let me do that in a different color-

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now it is 4 over the square-root of 41.

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And what is the secant of E now?

00:05:05.431 --> 00:05:08.184
Well, it was square-root of 41 over 'a',

00:05:08.184 --> 00:05:12.871
now it's square-root of 41 over 4.

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Because now we know the value of 'a'.

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And we can verify that 'a' is equal to 4

00:05:18.373 --> 00:05:20.019
by using the Pythagorean theorem.

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In fact, we could've solved it that way.

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But the whole point, I'm suspecting, of this problem,

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is to actually establish this (a = 4)

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using this (cotangent of E= 8/10) information

00:05:28.767 --> 00:05:30.813
even although we could have done it using the Pythagorean theorem.

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But let's just verify that this 'a=4' satisfies

00:05:33.068 --> 00:05:34.936
the Pythagorean theorem.

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So if we take this side right over here,

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we got 4 squared plus 5 squared

00:05:40.603 --> 00:05:44.471
should be equal to the square-root of 41 squared.

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Should be equal to the hypotenuse squared.

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Square-root of 41 squared.

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So 4 squared is 16,

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5 squared is 25,

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so does this actually meet-

00:05:54.135 --> 00:05:56.939
does this actually satisfy the Pythagorean theorem?

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So 16 plus 25,

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square-root of 41 squared is 41,

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16 plus 25 is indeed 41.

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So at least it's consistent with the Pythagorean theorem,

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and we are done!


Secant (sec), cosecant (csc) and cotangent (cot) example

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:04.230
Determine the six
trigonometric ratios

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for angle A in the
right triangle below.

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So this right over here is
angle A, it's at vertex A.

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And to help me remember
the definitions of the trig

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ratios-- and these are human
constructed definitions that

00:00:16.620 --> 00:00:19.790
have ended up being very,
very useful for analyzing

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a whole series of
things in the world.

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To help me remember them, I
use the words soh cah toa.

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Let me write that down.

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Soh cah toa.

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Sometimes you can think
of it as one word,

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but it's really the three parts
that define at least three

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of the trig functions for you.

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And then we can
get the other three

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by looking at the first three.

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So soh tells us that sine of an
angle-- in this case it's sine

00:00:48.940 --> 00:00:57.030
of A-- so sine of A is equal
to the opposite, that's the O,

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over the hypotenuse.

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Well in this context, what is
the opposite side to angle A?

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Well, we go across the triangle,
it opens up onto side BC.

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It has length 12.

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So that is the opposite side.

00:01:12.130 --> 00:01:14.260
So, this is going
to be equal to 12.

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And what's the hypotenuse?

00:01:15.830 --> 00:01:19.950
Well, the hypotenuse is the
longest side of the triangle.

00:01:19.950 --> 00:01:22.390
It's opposite the
90 degree angle.

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And so we go opposite
the 90 degree angle,

00:01:24.580 --> 00:01:26.270
longest side is side AB.

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It has length 13.

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So this right over
here is the hypotenuse.

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So, the sine of A is 12/13.

00:01:34.870 --> 00:01:36.270
Now let's go to cah.

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Cah defines cosine for us.

00:01:38.950 --> 00:01:42.340
It tells us that cosine of
an angle-- in this case,

00:01:42.340 --> 00:01:47.600
cosine of A-- is equal
to the adjacent side

00:01:47.600 --> 00:01:52.490
to the angle over
the hypotenuse.

00:01:52.490 --> 00:01:55.970
So, what's the adjacent
side to angle A?

00:01:55.970 --> 00:01:58.074
Well, if we look
at angle A, there

00:01:58.074 --> 00:01:59.490
are two sides that
are next to it.

00:01:59.490 --> 00:02:00.930
One of them is the hypotenuse.

00:02:00.930 --> 00:02:03.410
The other one has length 5.

00:02:03.410 --> 00:02:05.800
The adjacent one is side CA.

00:02:05.800 --> 00:02:07.200
So it's 5.

00:02:07.200 --> 00:02:09.729
And what is the hypotenuse?

00:02:09.729 --> 00:02:11.320
Well, we've already
figure that out.

00:02:11.320 --> 00:02:12.736
The hypotenuse is
right over here,

00:02:12.736 --> 00:02:14.190
it's opposite the
90 degree angle.

00:02:14.190 --> 00:02:16.023
It's the longest side
of the right triangle.

00:02:16.023 --> 00:02:17.480
It has length 13.

00:02:17.480 --> 00:02:19.565
So the cosine of A is 5/13.

00:02:19.565 --> 00:02:21.000
And let me label this.

00:02:21.000 --> 00:02:24.120
This right over here
is the adjacent side.

00:02:24.120 --> 00:02:25.720
And this is all
specific to angle A.

00:02:25.720 --> 00:02:28.150
The hypotenuse would be the
same regardless of what angle

00:02:28.150 --> 00:02:30.120
you pick, but the
opposite and the adjacent

00:02:30.120 --> 00:02:31.710
is dependent on
the angle that we

00:02:31.710 --> 00:02:33.620
choose in the right triangle.

00:02:33.620 --> 00:02:35.560
Now let's go to toa.

00:02:35.560 --> 00:02:37.590
Toa defines tangent for us.

00:02:37.590 --> 00:02:42.050
It tells us that the
tangent of an angle

00:02:42.050 --> 00:02:50.020
is equal to the opposite
side over the adjacent side.

00:02:50.020 --> 00:02:54.560
So given this definition,
what is the tangent of A?

00:02:54.560 --> 00:02:57.350
Well, the opposite side,
we already figured out,

00:02:57.350 --> 00:03:00.380
has length 12.

00:03:00.380 --> 00:03:02.560
And the adjacent side,
we already figure out,

00:03:02.560 --> 00:03:03.940
has length 5.

00:03:03.940 --> 00:03:06.720
So the tangent of A, which
is opposite over adjacent,

00:03:06.720 --> 00:03:08.250
is 12/5.

00:03:08.250 --> 00:03:12.850
Now, we'll go the to the
other three trig ratios, which

00:03:12.850 --> 00:03:14.479
you could think of
as the reciprocals

00:03:14.479 --> 00:03:15.520
of these right over here.

00:03:15.520 --> 00:03:16.760
But I'll define it.

00:03:16.760 --> 00:03:18.320
So first you have cosecant.

00:03:18.320 --> 00:03:21.440
And cosecant, it's always
a little bit unintuitive

00:03:21.440 --> 00:03:23.840
why cosecant is the
reciprocal of sine of A,

00:03:23.840 --> 00:03:26.320
even though it starts
with a co like cosine.

00:03:26.320 --> 00:03:29.000
But, cosecant is the
reciprocal of the sine of A.

00:03:29.000 --> 00:03:31.190
So sine of A is opposite
over hypotenuse.

00:03:31.190 --> 00:03:35.570
Cosecant of A is
hypotenuse over opposite.

00:03:35.570 --> 00:03:38.690
And so what's the hypotenuse
over the opposite?

00:03:38.690 --> 00:03:44.230
Well, the hypotenuse is 13
and the opposite side is 12.

00:03:44.230 --> 00:03:47.920
And notice that 13/12 is
the reciprocal of 12/13.

00:03:47.920 --> 00:03:51.006
Now, secant of A
is the reciprocal.

00:03:51.006 --> 00:03:52.880
So instead of being
adjacent over hypotenuse,

00:03:52.880 --> 00:03:54.950
which we got from the
cah part of soh cah toa,

00:03:54.950 --> 00:03:59.880
it's hypotenuse over adjacent.

00:03:59.880 --> 00:04:01.880
So what is the secant of A?

00:04:01.880 --> 00:04:05.470
Well, the hypotenuse, we've
figured out multiple times

00:04:05.470 --> 00:04:07.220
already, is 13.

00:04:07.220 --> 00:04:09.415
And what is the adjacent side?

00:04:09.415 --> 00:04:10.070
It's 5.

00:04:10.070 --> 00:04:12.270
So it's 13/5, which
is, once again,

00:04:12.270 --> 00:04:15.130
the reciprocal of the
cosine of A, 5/13.

00:04:15.130 --> 00:04:17.790
Finally, let's
get the cotangent.

00:04:17.790 --> 00:04:20.880
And the cotangent is the
reciprocal of tangent of A.

00:04:20.880 --> 00:04:22.720
Instead of being
opposite over adjacent,

00:04:22.720 --> 00:04:26.560
it is adjacent over opposite.

00:04:26.560 --> 00:04:29.180
So what is the cotangent of A?

00:04:29.180 --> 00:04:31.510
Well, we've figured
out the adjacent side

00:04:31.510 --> 00:04:34.060
multiple times for
angle A. It's length 5.

00:04:34.060 --> 00:04:37.690
And the opposite side
to angle A is 12.

00:04:37.690 --> 00:04:40.480
So it's 5/12, which is,
once again, the reciprocal

00:04:40.480 --> 00:04:43.420
of the tangent of
A, which is 12/5.


Example: Trig to solve the sides and angles of a right triangle

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.470
We're asked to solve the
right triangle shown below.

00:00:04.470 --> 00:00:07.140
Give the lengths to
the nearest tenth.

00:00:07.140 --> 00:00:08.920
So when they say solve
the right triangle,

00:00:08.920 --> 00:00:09.810
we can assume that
they're saying,

00:00:09.810 --> 00:00:11.680
hey figure out the
lengths of all the sides.

00:00:11.680 --> 00:00:14.500
So whatever a is equal to,
whatever b is equal to.

00:00:14.500 --> 00:00:17.180
And also what are all the
angles of the right triangle?

00:00:17.180 --> 00:00:18.410
They've given two of them.

00:00:18.410 --> 00:00:21.550
We might have to figure out
this third right over here.

00:00:21.550 --> 00:00:24.070
So there's multiple
ways to tackle this,

00:00:24.070 --> 00:00:28.460
but we'll just try to
tackle side XW first,

00:00:28.460 --> 00:00:30.920
try to figure out what a is.

00:00:30.920 --> 00:00:32.229
And I'll give you a hint.

00:00:32.229 --> 00:00:34.600
You can use a calculator,
and using a calculator,

00:00:34.600 --> 00:00:36.540
you can use your
trigonometric functions

00:00:36.540 --> 00:00:38.735
that we've looked
at a good bit now.

00:00:38.735 --> 00:00:40.360
So I'll give you a
few seconds to think

00:00:40.360 --> 00:00:45.629
about how to figure
out what a is.

00:00:45.630 --> 00:00:47.090
Well, what do we know?

00:00:47.090 --> 00:00:49.610
We know this angle
y right over here.

00:00:49.610 --> 00:00:52.839
We know the side
adjacent to angle y.

00:00:52.840 --> 00:00:55.410
And length a, this
is the side that's

00:00:55.410 --> 00:01:00.699
the length of the side that
is opposite to angle Y.

00:01:00.700 --> 00:01:04.239
So what trigonometric
ratio deals

00:01:04.239 --> 00:01:06.680
with the opposite
and the adjacent?

00:01:06.680 --> 00:01:08.990
So if we're looking at angle
Y, relative to angle Y,

00:01:08.990 --> 00:01:10.910
this is the opposite.

00:01:10.910 --> 00:01:14.024
And this right over
here is the adjacent.

00:01:14.024 --> 00:01:16.315
Well if we don't remember,
we can go back to SohCahToa.

00:01:21.240 --> 00:01:23.660
Sine deals with
opposite and hypotenuse.

00:01:23.660 --> 00:01:26.420
Cosine deals with
adjacent and hypotenuse.

00:01:26.420 --> 00:01:29.320
Tangent deals with
opposite over adjacent.

00:01:31.890 --> 00:01:38.150
So we can say that the
tangent of 65 degrees,

00:01:38.150 --> 00:01:42.250
of that angle of 65 degrees,
is equal to the opposite,

00:01:42.250 --> 00:01:44.050
the length of the
opposite side, which

00:01:44.050 --> 00:01:48.750
we know has length a over the
length of the adjacent side,

00:01:48.750 --> 00:01:53.170
which they gave us in the
diagram, which has length five.

00:01:53.170 --> 00:01:55.010
And you might say,
how do I figure out a?

00:01:55.010 --> 00:01:57.350
Well we can use our
calculator to evaluate

00:01:57.350 --> 00:01:59.734
what the tangent
of 65 degrees are.

00:01:59.734 --> 00:02:00.899
And then we can solve for a.

00:02:00.900 --> 00:02:04.320
And actually if we just want to
get the expression explicitly

00:02:04.320 --> 00:02:05.960
solving for a, we
could just multiply

00:02:05.960 --> 00:02:08.500
both sides of this
equation times 5.

00:02:08.500 --> 00:02:09.930
So let's do that.

00:02:09.930 --> 00:02:13.120
5 times, times 5.

00:02:13.120 --> 00:02:15.410
These cancel out,
and we are left with,

00:02:15.410 --> 00:02:18.230
if we flip the
equal around, we're

00:02:18.230 --> 00:02:26.489
left with a is equal to 5 times
the tangent of 65 degrees.

00:02:26.490 --> 00:02:28.780
So now we can get our
calculator out and figure out

00:02:28.780 --> 00:02:31.760
what this is to
the nearest tenth.

00:02:31.760 --> 00:02:41.733
That's my handy TI-85 out and
I have 5 times the tangent--

00:02:41.734 --> 00:02:43.900
I didn't need to press that
second right over there,

00:02:43.900 --> 00:02:48.890
just a regular
tangent-- of 65 degrees.

00:02:48.890 --> 00:02:52.649
And I will get, if I
round to the nearest tenth

00:02:52.650 --> 00:02:55.800
like they ask me to, I get 10.7.

00:02:55.800 --> 00:03:01.290
So a is approximately
equal to 10.7.

00:03:01.290 --> 00:03:05.730
I say approximately
because I rounded it down.

00:03:05.730 --> 00:03:07.519
This is not the exact number.

00:03:07.520 --> 00:03:09.870
But a is equal to 10.7.

00:03:09.870 --> 00:03:14.070
So we now know that this has
length 10.7, approximately.

00:03:14.070 --> 00:03:17.450
There are several ways that
we can try to tackle b.

00:03:17.450 --> 00:03:19.290
And I'll let you pick
the way you want to.

00:03:19.290 --> 00:03:21.420
But then I'll just do it
the way I would like to.

00:03:21.420 --> 00:03:23.890
So my next question
to you is, what

00:03:23.890 --> 00:03:28.829
is the length of the side YW?

00:03:28.830 --> 00:03:31.560
Or what is the value of b?

00:03:34.229 --> 00:03:35.770
Well there are
several ways to do it.

00:03:35.770 --> 00:03:36.860
This is the hypotenuse.

00:03:36.860 --> 00:03:38.900
So we could use
trigonometric functions

00:03:38.900 --> 00:03:41.510
that deal with adjacent
over hypotenuse or opposite

00:03:41.510 --> 00:03:42.840
over hypotenuse.

00:03:42.840 --> 00:03:45.750
Or we could just use
the Pythagorean theorem.

00:03:45.750 --> 00:03:47.770
We know two sides
of a right triangle.

00:03:47.770 --> 00:03:50.320
We can come up with
the third side.

00:03:50.320 --> 00:03:54.799
I will go with using
trigonometric ratios

00:03:54.800 --> 00:03:57.280
since that's what we've
been working on a good bit.

00:03:57.280 --> 00:04:01.710
So this length of b, that's
the length of the hypotenuse.

00:04:01.710 --> 00:04:07.734
So this side WY
is the hypotenuse.

00:04:07.734 --> 00:04:09.400
And so what trigonometric
ratios-- or we

00:04:09.400 --> 00:04:10.691
can decide what we want to use.

00:04:10.691 --> 00:04:12.500
We could use opposite
and hypotenuse.

00:04:12.500 --> 00:04:14.660
We could use adjacent
and hypotenuse.

00:04:14.660 --> 00:04:16.427
Since we know that
XY is exactly 5

00:04:16.427 --> 00:04:18.510
and we don't have to deal
with this approximation,

00:04:18.510 --> 00:04:19.750
let's use that side.

00:04:19.750 --> 00:04:21.250
So what trigonometric
ratios deal

00:04:21.250 --> 00:04:23.850
with adjacent and hypotenuse?

00:04:23.850 --> 00:04:26.670
Well we see from
SohCahToa cosine deals

00:04:26.670 --> 00:04:30.080
with adjacent over hypotenuse.

00:04:30.080 --> 00:04:36.940
So we could say that
the cosine of 65 degrees

00:04:36.940 --> 00:04:40.090
is equal to the length of
the adjacent side, which

00:04:40.090 --> 00:04:44.690
is 5 over the length of
the hypotenuse, which

00:04:44.690 --> 00:04:46.910
has a length of b.

00:04:46.910 --> 00:04:49.010
And then we can
try to solve for b.

00:04:49.010 --> 00:04:51.120
You multiply both
sides times b, you're

00:04:51.120 --> 00:04:56.910
left with b times cosine of
65 degrees is equal to 5.

00:04:56.910 --> 00:04:59.290
And then to solve for b,
you could divide both sides

00:04:59.290 --> 00:05:01.910
by cosine of 65 degrees.

00:05:01.910 --> 00:05:03.100
This is just a number here.

00:05:03.100 --> 00:05:04.820
So we're just dividing--
we have to figure it

00:05:04.820 --> 00:05:06.570
out what our calculator,
but this is just

00:05:06.570 --> 00:05:08.120
going to evaluate
to some number.

00:05:08.120 --> 00:05:12.950
So we can divide both sides by
that, by cosine of 65 degrees.

00:05:16.680 --> 00:05:20.290
And we're left with
b is equal to 5

00:05:20.290 --> 00:05:24.470
over the cosine of 65 degrees.

00:05:24.470 --> 00:05:27.740
So let us now use our calculator
to figure out the length of b.

00:05:27.740 --> 00:05:35.210
Length of b is 5 divided
by cosine of 65 degrees.

00:05:35.210 --> 00:05:39.719
And I get, if I round to
the nearest tenth, 11.8.

00:05:39.720 --> 00:05:43.290
So b is approximately equal
to, rounded to the nearest

00:05:43.290 --> 00:05:46.210
tenth, 11.8.

00:05:46.210 --> 00:05:50.099
So b is equal to 11.8.

00:05:50.100 --> 00:05:52.730
And then we're almost done
solving this right triangle.

00:05:52.730 --> 00:05:55.105
And you could have figured
this out using the Pythagorean

00:05:55.105 --> 00:05:57.480
theorem as well, saying that
5 squared plus 10.7 squared

00:05:57.480 --> 00:05:58.750
should be equal to b squared.

00:05:58.750 --> 00:06:00.950
And hopefully you would
get the exact same answer.

00:06:00.950 --> 00:06:02.616
And the last thing
we have to figure out

00:06:02.616 --> 00:06:07.919
is the measure of angle
W right over here.

00:06:07.920 --> 00:06:09.560
So I'll give you a
few seconds to think

00:06:09.560 --> 00:06:14.170
about what the
measure of angle W is.

00:06:14.170 --> 00:06:15.910
Well here we just
have to remember

00:06:15.910 --> 00:06:20.500
that the sum of the angles of a
triangle add up to 180 degrees.

00:06:20.500 --> 00:06:26.940
So angle w plus 65 degrees,
that's this angle right up

00:06:26.940 --> 00:06:30.420
here, plus the right angle,
this is a right triangle,

00:06:30.420 --> 00:06:34.280
they're going to add
up to 180 degrees.

00:06:34.280 --> 00:06:37.371
So all we need to do
is-- well we can simplify

00:06:37.372 --> 00:06:38.830
the left-hand side
right over here.

00:06:38.830 --> 00:06:41.539
65 plus 90 is 155.

00:06:41.540 --> 00:06:48.380
So angle W plus 155 degrees
is equal to 180 degrees.

00:06:48.380 --> 00:06:52.920
And then we get angle W-- if we
subtract 155 from both sides--

00:06:52.920 --> 00:06:57.590
angle W is equal to 25 degrees.

00:06:57.590 --> 00:07:03.123
And we are done solving the
right triangle shown below.


Example: Using soh cah toa

WEBVTT
Kind: captions
Language: en

00:00:00.959 --> 00:00:03.974
我们来练习一下直角三角函数比，

00:00:03.974 --> 00:00:06.192
还有直角三角函数。

00:00:06.192 --> 00:00:11.775
我们来求一下角theta（∠θ）的正弦和余弦。

00:00:11.775 --> 00:00:15.617
先求余弦。

00:00:15.649 --> 00:00:19.866
现在我们知道这是角theta （∠θ）

00:00:19.938 --> 00:00:22.606
And it is indeed a right triangle.

00:00:22.606 --> 00:00:25.960
So I'll give you a few seconds to think about that.

00:00:26.621 --> 00:00:30.839
Well to answer this you just have to remember the definitions of the trig functions.

00:00:30.839 --> 00:00:35.674
To help us there, we'll use the mnemonic soh cah toa.

00:00:35.674 --> 00:00:37.389
Soh

00:00:37.389 --> 00:00:38.806
Cah

00:00:38.806 --> 00:00:42.657
Toa

00:00:42.657 --> 00:00:44.088
soh cah toa

00:00:44.088 --> 00:00:47.655
And the part of soh cah toa that relates to cosign is the cah part.

00:00:47.655 --> 00:00:55.472
This defines sine, that's why we have the 's'; this defines cosine, that's why it starts with a 'c'; this defines tangent, that's why it starts with a 't'.

00:00:55.472 --> 00:00:59.304
So if you look at cah, it says that cosine (let me use that same color)...

00:00:59.304 --> 00:01:09.689
it says that cosine of an angle is equal to the adjacent side over the hypotenuse.

00:01:09.689 --> 00:01:13.256
So in our example here, what is the adjacent side?

00:01:13.256 --> 00:01:18.389
Well, if we look at it, it's the side that's next to it and not the hypotenuse.

00:01:18.389 --> 00:01:20.938
This side is next to it and it's not the hypotenuse.

00:01:20.938 --> 00:01:28.805
This side up here is next to our angle, but it is the hypotenuse; it's the one that's opposite the right angle.

00:01:28.805 --> 00:01:33.789
So this is the hypotenuse up here.

00:01:33.789 --> 00:01:37.456
This is... if we're looking at angle theta, this is the adjacent side.

00:01:37.456 --> 00:01:46.905
And, while we're at it, if you want to think of the oppisite side (we don't have to deal with it for cosine), but it never hurts to label it right now. That is the opposite.

00:01:46.905 --> 00:01:49.489
And this is relative to angle theta.

00:01:49.489 --> 00:01:53.472
So, with that out of the way, we say that cosine of theta is equal to adjacent over the hypotenuse.

00:01:53.472 --> 00:01:59.573
Adjacent has length 4. What is the hypotenuse?

00:01:59.573 --> 00:02:04.157
Well we know what side is the hypotenuse, but they haven't given us the length yet.

00:02:04.157 --> 00:02:07.738
But we can figure it out using the Pythagorean Theorem.

00:02:07.738 --> 00:02:12.406
We have 2 sides of a right triangle, we can always figure out the third side.

00:02:12.406 --> 00:02:21.121
We know that the sum of the squares of the 2 shorter sides will be equal to the square of the hypotenuse.

00:02:21.121 --> 00:02:35.756
So, we have 4 squared, plus 7 squared is going to be equal to, I'll just call it h squared or hypotenuse squared.

00:02:35.756 --> 00:02:48.805
If 4 squred is 16, and 7 squared is 49 is going to be equal to h squared.

00:02:48.805 --> 00:02:54.339
And, let's see, 16+50 would be 66, so 16+49 is 65.

00:02:54.339 --> 00:03:01.556
So this side over here is 65. H squred is equal to 65.

00:03:01.556 --> 00:03:06.755
Or we can say that h is equal to the square root of 65.

00:03:06.755 --> 00:03:15.689
And it doesn't look like there's any perfect squares in here - 65 is 13 times 5, neither of those is perfect squares, so this is about as simplified as we can get this radical.

00:03:15.689 --> 00:03:19.306
So the hypotenous is equal to the square root of 65.

00:03:19.306 --> 00:03:28.671
So in this case cosine of theta is equal to the adjacent side, which has length 4, over the hypotenous which has the length square root of 65.

00:03:28.671 --> 00:03:34.407
Now, let's do the same thing with the sine. What is the sine of theta going to be?

00:03:34.407 --> 00:03:36.388
I'm going to give you a few seconds to think about it.

00:03:36.388 --> 00:03:45.721
Well, soh tells us that sine is equal to opposite over hypotenuse.

00:03:45.721 --> 00:03:52.408
In this case, relative to angle theta, the opposite side has length 7.

00:03:52.408 --> 00:03:58.306
And what is the hypotenuse, or what is the length of the hypotenuse? Well, we just figured it out.

00:03:58.306 --> 00:04:02.306
It is the square root of 65.


Converting an explicit formula of a geometric sequence to a recursive formula

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:07.040
So I have the function g
of x is equal to 9 times 8

00:00:07.040 --> 00:00:09.400
to the x minus 1 power.

00:00:09.400 --> 00:00:16.070
And it's defined for
x being a positive--

00:00:16.070 --> 00:00:19.950
or if x is a positive-- integer.

00:00:19.950 --> 00:00:27.000
If x is a positive integer.

00:00:27.000 --> 00:00:29.000
So we could say the
domain of this function,

00:00:29.000 --> 00:00:32.080
or all the valid inputs
here are positive integers.

00:00:32.080 --> 00:00:34.970
So 1, 2, 3, 4, 5,
on and on and on.

00:00:34.970 --> 00:00:37.360
So this is an explicitly
defined function.

00:00:37.360 --> 00:00:41.370
What I now want to do is to
write a recursive definition

00:00:41.370 --> 00:00:42.990
of this exact same function.

00:00:42.990 --> 00:00:46.110
That given an x, it'll give
the exact same outputs.

00:00:46.110 --> 00:00:48.410
So let's first just try
to understand the inputs

00:00:48.410 --> 00:00:49.650
and outputs here.

00:00:49.650 --> 00:00:51.340
So let's make a little table.

00:00:51.340 --> 00:00:53.560
Let's make a table here.

00:00:53.560 --> 00:00:56.620
And let's think
about what happens

00:00:56.620 --> 00:01:01.880
when we put in various x's
into this function definition.

00:01:01.880 --> 00:01:03.780
So the domain is
positive integers.

00:01:03.780 --> 00:01:05.030
So let's try a couple of them.

00:01:05.030 --> 00:01:08.240
1, 2, 3, 4.

00:01:08.240 --> 00:01:11.410
And then see what the
corresponding g of x is.

00:01:11.410 --> 00:01:14.100
g of x.

00:01:14.100 --> 00:01:17.080
So when x is equal
to 1, g of x is

00:01:17.080 --> 00:01:19.780
9 times 8 to the
1 minus 1 power,

00:01:19.780 --> 00:01:23.310
9 times 8 to the 0
power, or 9 times 1.

00:01:23.310 --> 00:01:27.050
So g of x is going to be just 9.

00:01:27.050 --> 00:01:32.610
When x is 2, what's
going to happen?

00:01:32.610 --> 00:01:35.830
It'll be 9 times 8
to the 2 minus 1.

00:01:35.830 --> 00:01:39.010
So that's the same thing as
9 times 8 to the 1st power.

00:01:39.010 --> 00:01:40.600
And that's just going
to be 9 times 8.

00:01:40.600 --> 00:01:43.135
So that is 72.

00:01:43.135 --> 00:01:44.760
Actually let me just
write it that way.

00:01:44.760 --> 00:01:47.260
Let me write it
as just 9 times 8.

00:01:47.260 --> 00:01:49.270
9 times 8.

00:01:49.270 --> 00:01:52.980
Then when x is equal to
3, what's going on here?

00:01:52.980 --> 00:01:55.120
Well this is going
to be 3 minus 1 is 2.

00:01:55.120 --> 00:01:56.690
So it's going to be 8 squared.

00:01:56.690 --> 00:01:59.380
So it's going to be
9 times 8 squared.

00:01:59.380 --> 00:02:04.290
So we could write that
as 9 times 8 times 8.

00:02:04.290 --> 00:02:06.650
I think you see a little
bit of a pattern forming.

00:02:06.650 --> 00:02:10.525
When x is 4, this is going to be
8 to the 4 minus 1 power, or 9

00:02:10.525 --> 00:02:11.950
to the 3rd power.

00:02:11.950 --> 00:02:19.090
So that's 9 times
8 times 8 times 8.

00:02:19.090 --> 00:02:20.726
So this gives a
good clue about how

00:02:20.726 --> 00:02:22.100
we would define
this recursively.

00:02:22.100 --> 00:02:25.670
Notice, if our first term,
when x equals 1 is 9,

00:02:25.670 --> 00:02:31.310
every term after that is 8
times the preceding term.

00:02:31.310 --> 00:02:33.850
Is 8 times the preceding term.

00:02:33.850 --> 00:02:35.700
8 times the preceding term.

00:02:35.700 --> 00:02:37.700
8 times the preceding term.

00:02:37.700 --> 00:02:39.730
So let's define that as
a recursive function.

00:02:39.730 --> 00:02:41.960
So first define our base case.

00:02:41.960 --> 00:02:45.560
So we could say
g of x-- and I'll

00:02:45.560 --> 00:02:48.670
do this is a new color
because I'm overusing the red.

00:02:48.670 --> 00:02:49.980
I like the blue.

00:02:49.980 --> 00:02:51.620
g of x.

00:02:51.620 --> 00:02:53.550
Well we can define
our base case.

00:02:53.550 --> 00:02:57.570
It's going to be equal
to 9 if x is equal to 1.

00:02:57.570 --> 00:03:04.580
g of x equals 9 if x equals 1.

00:03:04.580 --> 00:03:07.170
So that took care of
that right over there.

00:03:07.170 --> 00:03:11.140
And then if it
equals anything else

00:03:11.140 --> 00:03:13.570
it equals the previous g of x.

00:03:13.570 --> 00:03:17.760
So if we're looking at--
let's go all the way down

00:03:17.760 --> 00:03:21.410
to x minus 1, and then an x.

00:03:21.410 --> 00:03:33.120
So if this entry right over
here is g of x minus 1,

00:03:33.120 --> 00:03:35.209
however many times
you multiply the 8s

00:03:35.209 --> 00:03:38.230
and we have a 9 in front,
so this is g of x minus 1.

00:03:38.230 --> 00:03:42.360
We know that g of x-- we know
that this one right over here

00:03:42.360 --> 00:03:47.460
is going to be the previous
entry, g of x minus 1.

00:03:47.460 --> 00:03:55.794
The previous entry times 8.

00:03:55.794 --> 00:03:57.210
So we could write
that right here.

00:03:57.210 --> 00:03:59.390
Times 8.

00:03:59.390 --> 00:04:02.290
So for any other
x other than 1, g

00:04:02.290 --> 00:04:07.060
of x is equal to the previous
entry-- so it's g of-- I'll

00:04:07.060 --> 00:04:16.190
do that in a blue color--
g of x minus 1 times 8.

00:04:16.190 --> 00:04:29.190
If x is greater than 1, or
x is integer greater than 1.

00:04:29.190 --> 00:04:32.570
Now let's verify that
this actually works.

00:04:32.570 --> 00:04:35.070
So let's draw
another table here.

00:04:38.210 --> 00:04:41.230
So once again, we're
going to have x

00:04:41.230 --> 00:04:42.864
and we're going to have g of x.

00:04:42.864 --> 00:04:45.280
But this time we're going to
use this recursive definition

00:04:45.280 --> 00:04:45.880
for g of x.

00:04:45.880 --> 00:04:47.255
And the reason
why it's recursive

00:04:47.255 --> 00:04:49.390
is it's referring to itself.

00:04:49.390 --> 00:04:52.580
In its own definition, it's
saying hey, g of x, well if x

00:04:52.580 --> 00:04:54.580
doesn't equal 1 it's going
to be g of x minus 1.

00:04:54.580 --> 00:04:56.080
It's using the function itself.

00:04:56.080 --> 00:04:58.666
But we'll see that it
actually does work out.

00:04:58.666 --> 00:04:59.290
So let's see...

00:04:59.290 --> 00:05:04.790
When x is equal to 1, so g
of 1-- well if x equals 1,

00:05:04.790 --> 00:05:06.380
it's equal to 9.

00:05:06.380 --> 00:05:07.140
It's equal to 9.

00:05:07.140 --> 00:05:09.150
So that was pretty
straightforward.

00:05:09.150 --> 00:05:11.380
What happens when x equals 2?

00:05:11.380 --> 00:05:14.400
Well when x equals 2, this
case doesn't apply anymore.

00:05:14.400 --> 00:05:15.850
We go down to this case.

00:05:15.850 --> 00:05:17.840
So when x is equal
to 2 it's going

00:05:17.840 --> 00:05:20.750
to be equivalent
to g of 2 minus 1.

00:05:20.750 --> 00:05:22.250
Let me write this down.

00:05:22.250 --> 00:05:28.770
It's going to be equivalent
to g of 2 minus 1 times 8,

00:05:28.770 --> 00:05:33.590
which is the same thing
as g of 1 times 8.

00:05:33.590 --> 00:05:34.670
And what's g of 1?

00:05:34.670 --> 00:05:36.890
Well g of 1 is right over here.

00:05:36.890 --> 00:05:38.010
g of 1 is 9.

00:05:38.010 --> 00:05:40.820
So this is going to
be equal to 9 times 8.

00:05:40.820 --> 00:05:42.730
Exactly what we got over here.

00:05:42.730 --> 00:05:45.840
And of course this was
equivalent to g of 2.

00:05:45.840 --> 00:05:47.060
So let me write this.

00:05:47.060 --> 00:05:49.370
This is g of 2.

00:05:49.370 --> 00:05:50.870
Let me scroll over
a little bit so I

00:05:50.870 --> 00:05:52.710
don't get all scrunched up.

00:05:52.710 --> 00:05:54.780
So now let's go to 3.

00:05:54.780 --> 00:05:56.110
Let's go to 3.

00:05:56.110 --> 00:05:57.890
And right now I'll
write g of 3 first.

00:05:57.890 --> 00:06:01.650
So g of 3 is equal
to-- we're going

00:06:01.650 --> 00:06:09.130
to this case-- it's equal
to g of 3 minus 1 times 8.

00:06:09.130 --> 00:06:13.770
So that's equal
to g of 2 times 8.

00:06:13.770 --> 00:06:15.060
Well what's g of 2?

00:06:15.060 --> 00:06:18.060
Well g of 2, we already
figured out is 9 times 8.

00:06:18.060 --> 00:06:22.400
So it's equal to 9 times 8--
that's g of 2-- times 8 again.

00:06:22.400 --> 00:06:24.970
And so you see we get
the exact same results.

00:06:24.970 --> 00:06:29.690
So this is the recursive
definition of this function.


Die rolling probability

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.120
Find the probability
of rolling doubles

00:00:03.120 --> 00:00:07.134
on two six-sided dice
numbered from 1 to 6.

00:00:07.134 --> 00:00:09.050
So when they're talking
about rolling doubles,

00:00:09.050 --> 00:00:10.883
they're just saying,
if I roll the two dice,

00:00:10.883 --> 00:00:12.840
I get the same number
on the top of both.

00:00:12.840 --> 00:00:15.240
So, for example, a 1
and a 1, that's doubles.

00:00:15.240 --> 00:00:19.540
A 2 and a 2, that is doubles.

00:00:19.540 --> 00:00:22.400
A 3 and a 3, a 4 and a 4,
a 5 and a 5, a 6 and a 6,

00:00:22.400 --> 00:00:25.280
all of those are
instances of doubles.

00:00:25.280 --> 00:00:29.030
So the event in question
is rolling doubles

00:00:29.030 --> 00:00:33.250
on two six-sided dice
numbered from 1 to 6.

00:00:33.250 --> 00:00:35.585
So let's think about all
of the possible outcomes.

00:00:35.585 --> 00:00:37.210
Or another way to
think about it, let's

00:00:37.210 --> 00:00:39.950
think about the
sample space here.

00:00:39.950 --> 00:00:42.350
So what can we roll
on the first die.

00:00:42.350 --> 00:00:46.740
So let me write this
as die number 1.

00:00:46.740 --> 00:00:48.132
What are the possible rolls?

00:00:48.132 --> 00:00:49.590
Well, they're
numbered from 1 to 6.

00:00:49.590 --> 00:00:56.650
It's a six-sided die, so I can
get a 1, a 2, a 3, a 4, a 5,

00:00:56.650 --> 00:00:58.010
or a 6.

00:00:58.010 --> 00:01:02.780
Now let's think about the
second die, so die number 2.

00:01:02.780 --> 00:01:04.129
Well, exact same thing.

00:01:04.129 --> 00:01:11.810
I could get a 1, a 2,
a 3, a 4, a 5, or a 6.

00:01:11.810 --> 00:01:14.880
Now, given these possible
outcomes for each of the die,

00:01:14.880 --> 00:01:17.660
we can now think of the
outcomes for both die.

00:01:17.660 --> 00:01:21.400
So, for example, in this--
let me draw a grid here just

00:01:21.400 --> 00:01:22.740
to make it a little bit neater.

00:01:22.740 --> 00:01:28.410
So let me draw a line there and
then a line right over there.

00:01:28.410 --> 00:01:30.130
Let me draw actually
several of these,

00:01:30.130 --> 00:01:33.840
just so that we could really
do this a little bit clearer.

00:01:33.840 --> 00:01:38.070
So let me draw a full grid.

00:01:38.070 --> 00:01:40.040
All right.

00:01:40.040 --> 00:01:44.460
And then let me draw the
vertical lines, only a few

00:01:44.460 --> 00:01:47.170
more left.

00:01:47.170 --> 00:01:49.020
There we go.

00:01:49.020 --> 00:01:52.350
Now, all of this top row,
these are the outcomes

00:01:52.350 --> 00:01:54.590
where I roll a 1
on the first die.

00:01:54.590 --> 00:01:57.080
So I roll a 1 on the first die.

00:01:57.080 --> 00:02:00.084
These are all of those outcomes.

00:02:00.084 --> 00:02:02.000
And this would be I run
a 1 on the second die,

00:02:02.000 --> 00:02:02.950
but I'll fill that in later.

00:02:02.950 --> 00:02:04.866
These are all of the
outcomes where I roll a 2

00:02:04.866 --> 00:02:07.710
on the first die.

00:02:07.710 --> 00:02:10.020
This is where I roll
a 3 on the first die.

00:02:13.396 --> 00:02:17.680
4-- I think you get the
idea-- on the first die.

00:02:17.680 --> 00:02:22.220
And then a 5 on
the first to die.

00:02:22.220 --> 00:02:24.900
And then finally, this last
row is all the outcomes

00:02:24.900 --> 00:02:29.100
where I roll a 6
on the first die.

00:02:29.100 --> 00:02:31.060
Now, we can go
through the columns,

00:02:31.060 --> 00:02:34.590
and this first column is where
we roll a 1 on the second die.

00:02:39.130 --> 00:02:42.440
This is where we roll
a 2 on the second die.

00:02:42.440 --> 00:02:45.900
So let's draw that out, write
it out, and fill in the chart.

00:02:45.900 --> 00:02:47.900
Here's where we roll
a 3 on the second die.

00:02:52.219 --> 00:02:54.510
This is a comma that I'm
doing between the two numbers.

00:02:54.510 --> 00:02:55.593
Here is where we have a 4.

00:03:01.320 --> 00:03:04.530
And then here is where
we roll a 5 on the second

00:03:04.530 --> 00:03:06.390
die, just filling this in.

00:03:12.870 --> 00:03:15.360
This last column is where we
roll a 6 on the second die.

00:03:22.550 --> 00:03:26.594
Now, every one of these
represents a possible outcome.

00:03:26.594 --> 00:03:29.010
This outcome is where we roll
a 1 on the first die and a 1

00:03:29.010 --> 00:03:29.800
on the second die.

00:03:29.800 --> 00:03:31.970
This outcome is where we
roll a 3 on the first die,

00:03:31.970 --> 00:03:33.510
a 2 on the second die.

00:03:33.510 --> 00:03:35.750
This outcome is where we
roll a 4 on the first die

00:03:35.750 --> 00:03:37.480
and a 5 on the second die.

00:03:37.480 --> 00:03:41.430
And you can see here, there are
36 possible outcomes, 6 times

00:03:41.430 --> 00:03:43.500
6 possible outcomes.

00:03:43.500 --> 00:03:47.880
Now, with this out of the way,
how many of these outcomes

00:03:47.880 --> 00:03:55.470
satisfy our criteria of rolling
doubles on two six-sided dice?

00:03:55.470 --> 00:03:58.650
How many of these outcomes
are essentially described

00:03:58.650 --> 00:04:01.830
by our event?

00:04:01.830 --> 00:04:03.250
Well, we see them right here.

00:04:03.250 --> 00:04:07.230
Doubles, well, that's rolling
a 1 and 1, that's a 2 and a 2,

00:04:07.230 --> 00:04:12.960
a 3 and a 3, a 4 and a 4, a
5 and a 5, and a 6 and a 6.

00:04:12.960 --> 00:04:19.610
So we have 1, 2, 3, 4, 5, 6
events satisfy this event,

00:04:19.610 --> 00:04:23.170
or are the outcomes that are
consistent with this event.

00:04:23.170 --> 00:04:25.570
Now given that, let's
answer our question.

00:04:25.570 --> 00:04:28.900
What is the probability
of rolling doubles

00:04:28.900 --> 00:04:33.690
on two six-sided die
numbered from 1 to 6?

00:04:33.690 --> 00:04:35.290
Well, the probability
is going to be

00:04:35.290 --> 00:04:40.690
equal to the number of outcomes
that satisfy our criteria,

00:04:40.690 --> 00:04:42.900
or the number of outcomes
for this event, which

00:04:42.900 --> 00:04:46.680
are 6-- we just figured
that out-- over the total--

00:04:46.680 --> 00:04:51.300
I want to do that pink
color-- number of outcomes,

00:04:51.300 --> 00:04:54.350
over the size of
our sample space.

00:04:54.350 --> 00:04:57.840
So this right over here,
we have 36 total outcomes.

00:04:57.840 --> 00:05:01.660
So we have 36 outcomes,
and if you simplify this,

00:05:01.660 --> 00:05:06.440
6/36 is the same thing as 1/6.

00:05:06.440 --> 00:05:08.900
So the probability
of rolling doubles

00:05:08.900 --> 00:05:13.734
on two six-sided dice
numbered from 1 to 6 is 1/6.


Inverting 3x3 part 2: Determinant and adjugate of a matrix

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.120
We're nearing the home
stretch of our quest

00:00:03.120 --> 00:00:07.020
to find the inverse of this
three-by-three matrix here.

00:00:07.020 --> 00:00:08.610
And the next thing
that we can do

00:00:08.610 --> 00:00:10.680
is find the determinant
of it, which we already

00:00:10.680 --> 00:00:12.810
have a good bit
of practice doing.

00:00:12.810 --> 00:00:18.860
So the determinant
of C, of our matrix--

00:00:18.860 --> 00:00:21.290
I'll do that same
color-- C, there

00:00:21.290 --> 00:00:23.310
are several ways
that you could do it.

00:00:30.450 --> 00:00:32.759
You could take this
top row of the matrix

00:00:32.759 --> 00:00:35.290
and take the value of
each of those terms

00:00:35.290 --> 00:00:39.060
times the cofactor-- times
the corresponding cofactor--

00:00:39.060 --> 00:00:40.190
and take the sum there.

00:00:40.190 --> 00:00:41.300
That's one technique.

00:00:41.300 --> 00:00:42.760
Or you could do
the technique where

00:00:42.760 --> 00:00:45.620
you rewrite these
first two columns,

00:00:45.620 --> 00:00:51.061
and then you take the product
of the top to left diagonals,

00:00:51.061 --> 00:00:53.995
sum those up, and
then subtract out

00:00:53.995 --> 00:00:55.462
the top right to
the bottom left.

00:00:55.462 --> 00:00:57.160
I'll do the second
one just so that you

00:00:57.160 --> 00:00:59.179
can see that you
get the same result.

00:00:59.179 --> 00:00:59.720
So let's see.

00:00:59.720 --> 00:01:02.910
The determinant is going
to be equal to-- I'll

00:01:02.910 --> 00:01:05.610
rewrite all of these
things-- so negative

00:01:05.610 --> 00:01:12.870
1, negative 2, 2,
2, 1, 1, 3, 4, 5.

00:01:12.870 --> 00:01:16.150
And let me now, just to make
it a little bit simpler,

00:01:16.150 --> 00:01:18.260
rewrite these first two columns.

00:01:18.260 --> 00:01:22.802
So negative 1,
negative 2, 2, 1, 3, 4.

00:01:22.802 --> 00:01:25.290
So the determinant
is going to be

00:01:25.290 --> 00:01:28.030
equal to-- so let
me write this down.

00:01:28.030 --> 00:01:32.100
So you have negative
1 times 1 times 5.

00:01:32.100 --> 00:01:34.200
Well that's just going
to be negative 5,

00:01:34.200 --> 00:01:36.230
taking that product.

00:01:36.230 --> 00:01:42.590
Then you have negative
2 times 1 times 3.

00:01:42.590 --> 00:01:44.690
Well that's negative 6.

00:01:44.690 --> 00:01:46.760
So we'll have negative 6.

00:01:46.760 --> 00:01:48.830
Or you could say plus
negative 6 there.

00:01:48.830 --> 00:01:52.430
And then you have
2 times 2 times 4.

00:01:52.430 --> 00:01:56.150
Well that's just 4 times
4, which is just 16.

00:01:56.150 --> 00:01:58.650
So we have plus 16.

00:01:58.650 --> 00:02:01.980
And then we do the top
right to the bottom left.

00:02:01.980 --> 00:02:05.460
So you have negative
2 times 2 times 5.

00:02:05.460 --> 00:02:07.650
Well that's negative 4 times 5.

00:02:07.650 --> 00:02:11.070
So that is negative 20.

00:02:11.070 --> 00:02:13.150
But we're going to
subtract negative 20.

00:02:13.150 --> 00:02:15.430
So that's negative 4
times 5, negative 20,

00:02:15.430 --> 00:02:18.682
but we're going to
subtract negative 20.

00:02:18.682 --> 00:02:22.530
Obviously that's going to
turn into adding positive 20.

00:02:22.530 --> 00:02:26.860
Then you have negative 1 times
1 times 4, which is negative 4.

00:02:26.860 --> 00:02:29.670
But we're going to
subtract these products.

00:02:29.670 --> 00:02:32.080
We're going to
subtract negative 4.

00:02:32.080 --> 00:02:36.450
And then you have 2 times
1 times 3, which is 6.

00:02:36.450 --> 00:02:37.640
But we have to subtract it.

00:02:37.640 --> 00:02:40.300
So we have subtracting 6.

00:02:40.300 --> 00:02:44.800
And so this simplifies
to negative 5 minus 6

00:02:44.800 --> 00:02:52.570
is negative 11, plus 16
gets us to positive 5.

00:02:52.570 --> 00:02:56.050
So all of this
simplifies to positive 5.

00:02:56.050 --> 00:03:02.180
And then we have plus 20 plus 4.

00:03:02.180 --> 00:03:04.016
Actually, let me do
that green color,

00:03:04.016 --> 00:03:06.480
so we don't get confused.

00:03:06.480 --> 00:03:13.169
So we have plus
20 plus 4 minus 6.

00:03:13.169 --> 00:03:14.210
So what does this get us?

00:03:14.210 --> 00:03:24.720
5 plus 20 is 25, plus 4 is
29, minus 6 gets us to 23.

00:03:24.720 --> 00:03:31.077
So our determinant right
over here is equal to 23.

00:03:34.490 --> 00:03:37.620
So now we are really
in the home stretch.

00:03:37.620 --> 00:03:40.360
The inverse of this
matrix is going

00:03:40.360 --> 00:03:45.030
to be 1 over our determinant
times the transpose

00:03:45.030 --> 00:03:46.970
of this cofactor matrix.

00:03:46.970 --> 00:03:49.220
And the transpose of
the cofactor matrix

00:03:49.220 --> 00:03:50.800
is called the adjugate.

00:03:50.800 --> 00:03:51.780
So let's do that.

00:03:51.780 --> 00:03:54.704
So let's write
the adjugate here.

00:03:54.704 --> 00:03:55.620
This is the drum roll.

00:03:55.620 --> 00:03:57.070
We're really in
the home stretch.

00:03:57.070 --> 00:04:01.130
C inverse is equal to
1 over the determinant,

00:04:01.130 --> 00:04:09.830
so it's equal to 1/23,
times the adjugate of C.

00:04:09.830 --> 00:04:18.829
And so this is going to be equal
to 1/23 times the transpose

00:04:18.829 --> 00:04:21.810
of our cofactor matrix.

00:04:21.810 --> 00:04:24.470
So we have our cofactor
matrix right over here.

00:04:24.470 --> 00:04:27.480
So each row now
becomes a column.

00:04:27.480 --> 00:04:33.970
So this row now
becomes a column.

00:04:33.970 --> 00:04:39.730
So it becomes 1, negative 7,
5 becomes the first column.

00:04:39.730 --> 00:04:45.640
The second row becomes
the second column-- 18,

00:04:45.640 --> 00:04:49.320
negative 11, negative 2.

00:04:49.320 --> 00:04:53.890
And then finally, the third
row becomes the third column.

00:04:53.890 --> 00:04:58.716
You have negative 4, 5, and 3.

00:04:58.716 --> 00:05:00.090
And now we just
have to multiply,

00:05:00.090 --> 00:05:04.410
or you could say divide, each of
these by 23, and we are there.

00:05:04.410 --> 00:05:09.085
So this is the inverse of
our original matrix C, home

00:05:09.085 --> 00:05:09.990
stretch.

00:05:09.990 --> 00:05:16.870
1 divided by 23 is just 1/23.

00:05:16.870 --> 00:05:18.194
Then you have 18/23.

00:05:18.194 --> 00:05:20.610
Actually, let me give myself
a little bit more real estate

00:05:20.610 --> 00:05:21.506
to do this in.

00:05:25.210 --> 00:05:27.010
So there we go.

00:05:27.010 --> 00:05:38.590
So 1 divided by 23-- 1/23,
18/23, negative 4/23,

00:05:38.590 --> 00:05:52.830
negative 7/23, negative 11/23,
5/23, 5/23, negative 2/23.

00:05:52.830 --> 00:05:54.840
And then finally,
assuming we haven't made

00:05:54.840 --> 00:05:56.780
any careless mistakes,
which would shock me

00:05:56.780 --> 00:06:01.250
if we haven't, we get to 3/23.

00:06:01.250 --> 00:06:02.320
And we are done.

00:06:02.320 --> 00:06:06.610
We have successfully inverted
a three-by-three matrix.

00:06:06.610 --> 00:06:10.650
Once again, something I
strongly believe better done

00:06:10.650 --> 00:06:13.030
by a computer and
probably should not

00:06:13.030 --> 00:06:17.690
be part of a typical Algebra
2 curriculum, because it tends

00:06:17.690 --> 00:06:21.467
to be displayed in a,
non-contextual way.


Inverting 3x3 part 1: Calculating matrix of minors and cofactor matrix

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.060
I'm now going to do one of
my least favorite things

00:00:03.060 --> 00:00:07.310
to do by hand, and that is
to invert a 3 by 3 matrix.

00:00:07.310 --> 00:00:10.575
And it can be useful because
you can solve systems that way.

00:00:10.575 --> 00:00:12.700
But you'll see it's very
computationally-intensive.

00:00:12.700 --> 00:00:14.480
And it is better to
be done by a computer.

00:00:14.480 --> 00:00:16.604
And the only thing that's
more painful is doing a 4

00:00:16.604 --> 00:00:20.620
by 4 or a 4 by 5 matrix, which
would-- or a 4 by 4 or a 5

00:00:20.620 --> 00:00:23.132
by 5 matrix, which
could take all day.

00:00:23.132 --> 00:00:25.340
And I'd probably, definitely
make a careless mistake.

00:00:25.340 --> 00:00:28.160
But let's just take
it step-by-step.

00:00:28.160 --> 00:00:31.150
So the first thing I'm going to
do, this is my 3 by 3 matrix,

00:00:31.150 --> 00:00:34.820
is I'm going to construct
a matrix of minors.

00:00:34.820 --> 00:00:44.750
So let me construct
here a matrix of minors.

00:00:44.750 --> 00:00:48.370
And I'm going to draw this
really big, right over here,

00:00:48.370 --> 00:00:50.520
to give ourselves
some real estate.

00:00:50.520 --> 00:00:54.560
And the matrix of minors, what
you do is, for each element

00:00:54.560 --> 00:00:57.930
in this matrix, you cross
out the corresponding row,

00:00:57.930 --> 00:00:59.510
the corresponding column.

00:00:59.510 --> 00:01:02.070
And you replace it
with the determinant

00:01:02.070 --> 00:01:04.170
of the elements that are left.

00:01:04.170 --> 00:01:05.790
So what are left
when you get rid

00:01:05.790 --> 00:01:09.860
of this row and this column,
the minor is 1, 1, 4, 5.

00:01:09.860 --> 00:01:14.190
So the determinant
of 1, 1, 4, 5.

00:01:14.190 --> 00:01:15.730
Let's keep doing that.

00:01:15.730 --> 00:01:17.970
This will be
replaced-- well I'll

00:01:17.970 --> 00:01:19.380
let you think about that first.

00:01:19.380 --> 00:01:22.120
What is this going
to be replaced with?

00:01:22.120 --> 00:01:24.920
Well it's going to be replaced--
this row, this column.

00:01:24.920 --> 00:01:27.615
The determinant of 2, 1, 3, 5.

00:01:34.010 --> 00:01:35.720
Let's keep going.

00:01:35.720 --> 00:01:38.480
Let's do this element.

00:01:38.480 --> 00:01:42.610
It's going to be replaced with
the determinant of, we get rid

00:01:42.610 --> 00:01:45.120
of this row, this
column, 2, 1, 3, 4.

00:01:49.050 --> 00:01:52.040
We're a third of the way done,
at least with this first stage

00:01:52.040 --> 00:01:53.180
of it.

00:01:53.180 --> 00:01:54.680
So for this element
right here, it's

00:01:54.680 --> 00:01:56.980
going to be replaced
with its minor.

00:01:56.980 --> 00:01:59.840
So we get rid of this
row, this column.

00:01:59.840 --> 00:02:02.050
The determinant of
negative 2, 2, 4, 5.

00:02:08.810 --> 00:02:11.660
Then we have-- I'm
trying to switch up

00:02:11.660 --> 00:02:13.840
the colors reasonably--
this element.

00:02:13.840 --> 00:02:15.910
Get rid of the middle
row, middle column.

00:02:15.910 --> 00:02:22.425
You're left with the
determinant negative 1, 2, 3, 5.

00:02:25.820 --> 00:02:28.080
Now we move on
to-- and I'm really

00:02:28.080 --> 00:02:30.800
running out of colors-- this
element right over here,

00:02:30.800 --> 00:02:35.580
where its minor is-- we'll get
rid of this row, this column--

00:02:35.580 --> 00:02:41.130
negative 1, negative 2, 3, 4.

00:02:44.330 --> 00:02:46.810
So let me see that.

00:02:46.810 --> 00:02:49.160
Sorry, someone's car alarm
started ringing outside.

00:02:49.160 --> 00:02:51.320
So let me make sure I
didn't lose my focus here.

00:02:51.320 --> 00:02:53.170
I don't want to make
any careless mistakes.

00:02:53.170 --> 00:02:56.700
This row, this column,
negative 1, negative 2, 3, 4.

00:02:56.700 --> 00:02:57.870
All right.

00:02:57.870 --> 00:03:00.100
Now let's move over here.

00:03:00.100 --> 00:03:01.930
Get rid of the first
column, last row.

00:03:01.930 --> 00:03:04.290
You have negative 2, 2, 1, 1.

00:03:04.290 --> 00:03:09.040
So we have negative 2, 2, 1, 1.

00:03:09.040 --> 00:03:13.460
Now let's move to this one.

00:03:13.460 --> 00:03:17.800
The middle column, bottom row,
you have negative 1, 2, 2, 1.

00:03:17.800 --> 00:03:23.840
So we have negative 1, 2, 2, 1.

00:03:23.840 --> 00:03:25.540
And then we are in
the home stretch

00:03:25.540 --> 00:03:27.570
for at least the
matrix of minors.

00:03:27.570 --> 00:03:30.990
And so we are looking at
this element right over here.

00:03:30.990 --> 00:03:32.610
Get rid of the last
column, last row.

00:03:32.610 --> 00:03:36.940
You're left with negative
1, negative 2, 2, 1.

00:03:36.940 --> 00:03:43.870
So the determinant of
negative 1, negative 2, 2, 1.

00:03:43.870 --> 00:03:47.470
And from here we just have
to evaluate each of these

00:03:47.470 --> 00:03:49.695
to get the actual
matrix of minors.

00:03:49.695 --> 00:03:51.400
This is just a
representation of it.

00:03:51.400 --> 00:03:54.150
So let's do that.

00:03:54.150 --> 00:03:55.860
So once again, we're
still at the stage

00:03:55.860 --> 00:03:58.914
of getting our matrix of minors.

00:03:58.914 --> 00:04:01.080
And actually I don't have
to write it as big anymore

00:04:01.080 --> 00:04:02.630
because now they're going
to have numeric values.

00:04:02.630 --> 00:04:04.671
They're not going to be
these little determinants

00:04:04.671 --> 00:04:05.910
of two-by-twos.

00:04:05.910 --> 00:04:11.540
So what is the determinant
over here on the top left?

00:04:11.540 --> 00:04:15.440
Well it's going to be 1
times 5 minus 1 times 4.

00:04:15.440 --> 00:04:17.360
1 times 5 minus 4 times 1.

00:04:17.360 --> 00:04:22.440
So it's going to be 5
minus 4, which is 1.

00:04:22.440 --> 00:04:27.220
What is the determinant over
here, this blue determinant?

00:04:27.220 --> 00:04:31.410
Well it's going to be 2 times
5, which is 10, minus 3 times 1.

00:04:31.410 --> 00:04:35.390
So 10 minus 3 is 7.

00:04:35.390 --> 00:04:38.100
What is this determinant,
on the top right,

00:04:38.100 --> 00:04:39.850
going to evaluate to?

00:04:39.850 --> 00:04:43.640
Well you have 2 times 4
is 8, minus 3 times 1.

00:04:43.640 --> 00:04:46.140
So it's 8 minus 3, which is 5.

00:04:48.820 --> 00:04:51.130
Then we go over here.

00:04:51.130 --> 00:04:54.080
What is this determinant
going to evaluate to?

00:04:54.080 --> 00:04:59.350
We have negative 2 times 5 is
negative 10, minus 4 times 2.

00:04:59.350 --> 00:05:02.950
So it's negative 10 minus
8, which is negative 18.

00:05:06.020 --> 00:05:10.300
Then we have negative 1
times 5, which is negative 5,

00:05:10.300 --> 00:05:12.110
minus 3 times 2.

00:05:12.110 --> 00:05:16.420
So it's negative 5 minus
6, which is negative 11.

00:05:16.420 --> 00:05:22.120
I want to do that in
white, negative 11.

00:05:22.120 --> 00:05:25.170
What's this determinant
going to evaluate to?

00:05:25.170 --> 00:05:30.200
We have negative 1 times 4,
negative 4, minus negative 6.

00:05:30.200 --> 00:05:35.810
So that's negative 4 plus
6, which is positive 2.

00:05:35.810 --> 00:05:39.870
I want to do that in that--
so that is positive 2.

00:05:39.870 --> 00:05:42.440
We have three left.

00:05:42.440 --> 00:05:44.970
What does this evaluate to?

00:05:44.970 --> 00:05:49.960
Negative 2 times 1 is
negative 2, minus 1 times 2.

00:05:49.960 --> 00:05:54.880
So it's negative 2 minus
2 gets us to negative 4.

00:05:54.880 --> 00:05:56.330
Home stretch now.

00:05:56.330 --> 00:06:02.770
Negative 1 times negative 1 is
negative 1, minus 2 times 2.

00:06:02.770 --> 00:06:08.500
So it's negative 1 minus
4, which is negative 5.

00:06:08.500 --> 00:06:12.910
And then finally we
have negative 1 times 1

00:06:12.910 --> 00:06:16.480
is negative 1, minus
2 times negative 2,

00:06:16.480 --> 00:06:18.850
so minus negative 4.

00:06:18.850 --> 00:06:22.930
So it's negative 1
minus negative 4.

00:06:22.930 --> 00:06:25.260
That's the same
thing as adding a 4.

00:06:25.260 --> 00:06:26.940
So it's negative 1 plus 4.

00:06:26.940 --> 00:06:29.600
So it's going to
be a positive 3.

00:06:29.600 --> 00:06:33.630
So this right over here is
our true matrix of minors.

00:06:33.630 --> 00:06:37.140
And from there we
can get our cofactor.

00:06:37.140 --> 00:06:40.270
We can get our
cofactor matrix just

00:06:40.270 --> 00:06:42.950
by remembering a
checkerboard pattern.

00:06:42.950 --> 00:06:44.620
So a checkerboard
pattern tells us

00:06:44.620 --> 00:06:48.910
positive, negative, positive,
negative, positive, negative,

00:06:48.910 --> 00:06:50.340
positive, negative, positive.

00:06:50.340 --> 00:06:51.970
It's a little
self-explanatory why

00:06:51.970 --> 00:06:53.650
that's called a checkerboard.

00:06:53.650 --> 00:06:58.010
So if we sign this matrix
of minors in this pattern,

00:06:58.010 --> 00:07:00.390
then we get our cofactor matrix.

00:07:00.390 --> 00:07:05.580
So let's set up our cofactor
matrix right over here.

00:07:05.580 --> 00:07:06.898
So this is our cofactor.

00:07:10.030 --> 00:07:11.980
A lot of terminology,
but hopefully it's

00:07:11.980 --> 00:07:13.450
making a little bit of sense.

00:07:13.450 --> 00:07:14.950
Our cofactor matrix.

00:07:14.950 --> 00:07:17.730
So we just have to apply
these signs to these values,

00:07:17.730 --> 00:07:19.610
to the matrix of minors.

00:07:19.610 --> 00:07:23.590
So 1 is now going to have
applied a positive sign to it.

00:07:23.590 --> 00:07:26.030
So it's still just going
to be a positive 1.

00:07:26.030 --> 00:07:27.850
You're going to have
7, but it's going

00:07:27.850 --> 00:07:29.560
to have a negative
sign applied to it.

00:07:29.560 --> 00:07:32.190
So it's going to
be a negative 7.

00:07:32.190 --> 00:07:35.800
You have a 5, positive 5.

00:07:35.800 --> 00:07:37.140
The 5 is already positive.

00:07:37.140 --> 00:07:38.530
You multiply it
times a positive,

00:07:38.530 --> 00:07:40.550
it's going to be a positive.

00:07:40.550 --> 00:07:43.140
You have negative
18, but then we

00:07:43.140 --> 00:07:45.040
have to multiply that
times a negative.

00:07:45.040 --> 00:07:48.360
So you get positive 18.

00:07:48.360 --> 00:07:52.430
You have negative 11, multiply
that times a positive 1.

00:07:52.430 --> 00:07:55.560
And so you still
have negative 11.

00:07:55.560 --> 00:08:00.680
You have positive 2,
multiply by a negative 1.

00:08:00.680 --> 00:08:03.270
You're going to have negative 2.

00:08:03.270 --> 00:08:07.840
Then you have negative 4,
multiply by a positive 1.

00:08:07.840 --> 00:08:10.160
You still have negative 4.

00:08:10.160 --> 00:08:11.980
And then you have negative 5.

00:08:11.980 --> 00:08:15.210
What is this going to be
in the cofactor matrix?

00:08:15.210 --> 00:08:17.800
Well you take a negative 5,
multiply by a negative 1.

00:08:17.800 --> 00:08:19.810
You have a positive 5.

00:08:19.810 --> 00:08:22.160
And then finally, you
have a positive 3,

00:08:22.160 --> 00:08:24.080
multiply by a positive 1.

00:08:24.080 --> 00:08:26.510
You're still going
to have a positive 3.

00:08:26.510 --> 00:08:28.650
So we've gone pretty
far in our journey,

00:08:28.650 --> 00:08:30.187
this very
computationally-intensive

00:08:30.187 --> 00:08:31.770
journey-- one that
I don't necessarily

00:08:31.770 --> 00:08:35.370
enjoy doing-- of finding
our inverse by getting

00:08:35.370 --> 00:08:37.081
to our cofactor matrix.

00:08:37.081 --> 00:08:38.830
Now we just have to
take this determinant,

00:08:38.830 --> 00:08:41.288
multiply this times 1 over the
determinant and we're there.

00:08:41.288 --> 00:08:45.663
We've figured out the
inverse of matrix C.


Finding the determinant of a 3x3 matrix method 2

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.220
As a hint, I will take the
determinant of another 3

00:00:04.220 --> 00:00:05.240
by 3 matrix.

00:00:05.240 --> 00:00:07.890
But it's the exact same
process for the 3 by 3 matrix

00:00:07.890 --> 00:00:09.880
that you're trying to
find the determinant of.

00:00:09.880 --> 00:00:13.100
So here is matrix A.
Here, it's these digits.

00:00:13.100 --> 00:00:15.020
This is a 3 by 3 matrix.

00:00:15.020 --> 00:00:17.672
And now let's evaluate
its determinant.

00:00:17.672 --> 00:00:19.880
So what we have to remember
is a checkerboard pattern

00:00:19.880 --> 00:00:21.770
when we think of
3 by 3 matrices:

00:00:21.770 --> 00:00:24.470
positive, negative, positive.

00:00:24.470 --> 00:00:29.490
So first we're going to
take positive 1 times 4.

00:00:29.490 --> 00:00:33.245
So we could just
write plus 4 times 4,

00:00:33.245 --> 00:00:35.045
the determinant of 4 submatrix.

00:00:35.045 --> 00:00:36.670
And when you say,
what's the submatrix?

00:00:36.670 --> 00:00:42.320
Well, get rid of the column
for that digit, and the row,

00:00:42.320 --> 00:00:44.700
and then the submatrix
is what's left over.

00:00:44.700 --> 00:00:46.670
So we'll take the
determinant of its submatrix.

00:00:46.670 --> 00:00:51.620
So it's 5, 3, 0, 0.

00:00:51.620 --> 00:00:56.550
Then we move on to the
second item in this row,

00:00:56.550 --> 00:00:57.806
in this top row.

00:00:57.806 --> 00:00:59.180
But the checkerboard
pattern says

00:00:59.180 --> 00:01:00.850
we're going to take
the negative of it.

00:01:00.850 --> 00:01:04.060
So it's going to be
negative of negative 1--

00:01:04.060 --> 00:01:09.080
let me do that in a slightly
different color-- of negative 1

00:01:09.080 --> 00:01:11.190
times the determinant
of its submatrix.

00:01:14.650 --> 00:01:16.970
You get rid of this
row, and this column.

00:01:16.970 --> 00:01:19.520
You're left with 4,
3, negative 2, 0.

00:01:25.200 --> 00:01:27.750
And then finally, you
have positive again.

00:01:27.750 --> 00:01:32.340
Positive times 1.

00:01:32.340 --> 00:01:33.330
This 1 right over here.

00:01:33.330 --> 00:01:35.890
Let me put the positive
in that same blue color.

00:01:35.890 --> 00:01:42.330
So positive 1, or plus
1 or positive 1 times 1.

00:01:42.330 --> 00:01:44.580
Really the negative is where
it got a little confusing

00:01:44.580 --> 00:01:45.550
on this middle term.

00:01:45.550 --> 00:01:50.200
But positive 1 times 1 times the
determinant of its submatrix.

00:01:50.200 --> 00:01:52.170
So it's submatrix is
this right over here.

00:01:52.170 --> 00:01:54.045
You get rid of the row,
get rid of the column

00:01:54.045 --> 00:01:55.365
4, 5, negative 2, 0.

00:02:01.930 --> 00:02:04.724
So now we just have to evaluate
these 2 by 2 determinants.

00:02:04.724 --> 00:02:06.140
So the determinant
right over here

00:02:06.140 --> 00:02:12.170
is going to be 5 times
0 minus 3 times 0.

00:02:12.170 --> 00:02:15.590
And all of that is going
to be multiplied times 4.

00:02:15.590 --> 00:02:17.110
Well this is going
to be 0 minus 0.

00:02:17.110 --> 00:02:18.510
So this is all just a 0.

00:02:18.510 --> 00:02:20.740
So 4 times 0 is just a 0.

00:02:20.740 --> 00:02:23.960
So this all simplifies to 0.

00:02:23.960 --> 00:02:25.860
Now let's do this term.

00:02:25.860 --> 00:02:28.080
We get negative negative 1.

00:02:28.080 --> 00:02:29.550
So that's positive 1.

00:02:29.550 --> 00:02:32.000
So let me just make
these positive.

00:02:32.000 --> 00:02:34.587
Positive 1, or we
could just write plus.

00:02:34.587 --> 00:02:35.670
Let me just write it here.

00:02:35.670 --> 00:02:41.000
So positive 1 times
4 times 0 is 0.

00:02:41.000 --> 00:02:45.510
So 4 times 0 minus
3 times negative 2.

00:02:45.510 --> 00:02:48.310
3 times negative
2 is negative 6.

00:02:48.310 --> 00:02:52.570
So you have 4-- oh, sorry,
you have 0 minus negative 6,

00:02:52.570 --> 00:02:54.100
which is positive 6.

00:02:54.100 --> 00:02:56.940
Positive 6 times 1 is just 6.

00:02:56.940 --> 00:02:58.330
So you have plus 6.

00:02:58.330 --> 00:03:01.310
And then finally you have
this last determinant.

00:03:01.310 --> 00:03:05.860
You have-- so it's going
to be plus 1 times 4 times

00:03:05.860 --> 00:03:10.580
0 minus 5 times negative 2.

00:03:13.760 --> 00:03:15.789
So this is going to be
equal to-- it's just

00:03:15.789 --> 00:03:17.830
going to be equal with--
1 times anything is just

00:03:17.830 --> 00:03:18.680
the same thing.

00:03:18.680 --> 00:03:20.320
4 times 0 is 0.

00:03:20.320 --> 00:03:23.390
And then 5 times negative
2 is negative 10.

00:03:23.390 --> 00:03:25.140
But we're going to
subtract a negative 10.

00:03:25.140 --> 00:03:27.100
So you get positive 10.

00:03:27.100 --> 00:03:30.670
So this just simplifies
to 10, positive 10.

00:03:30.670 --> 00:03:32.710
So you're left with,
let me be clear.

00:03:32.710 --> 00:03:36.480
This is 0, all of this
simplifies to plus 6,

00:03:36.480 --> 00:03:40.640
and all of this
simplifies to plus 10.

00:03:40.640 --> 00:03:44.540
And so you are left with, if
you add these up, 6 plus 10

00:03:44.540 --> 00:03:48.050
is equal to 16.

00:03:48.050 --> 00:03:49.970
So the trick here
is to just make

00:03:49.970 --> 00:03:51.880
sure you remember the
checkerboard pattern,

00:03:51.880 --> 00:03:54.510
and you don't mess up with all
the negative numbers and all

00:03:54.510 --> 00:03:55.738
of the multiplying.


Finding the determinant of a 3x3 matrix method 1

WEBVTT
Kind: captions
Language: en

00:00:01.081 --> 00:00:04.699
As another hint, I will take the same matrix, matrix A

00:00:04.699 --> 00:00:10.215
and take its determinant again but I will do it using a different technique, either technique is valid

00:00:10.215 --> 00:00:12.132
so here we saying what is the determinant of

00:00:12.132 --> 00:00:17.431
the 3X3 Matrix A and we can is we can rewrite first two column

00:00:17.431 --> 00:00:25.264
so first column right over here we could rewrite it as 4 4 -2 and then the second column right over here

00:00:25.264 --> 00:00:27.282
we could rewrite it

00:00:27.282 --> 00:00:31.682
-1 5 0

00:00:31.682 --> 00:00:35.049
and we could do is we could take the sum of the products

00:00:35.049 --> 00:00:38.099
of the first three top left bottom left diagonal,

00:00:38.099 --> 00:00:39.299
let me show you

00:00:39.299 --> 00:00:40.881
so the product of that

00:00:40.881 --> 00:00:42.299
that

00:00:42.299 --> 00:00:43.514
plus

00:00:43.514 --> 00:00:45.196
that

00:00:45.196 --> 00:00:46.732
plus

00:00:46.732 --> 00:00:48.090
that

00:00:48.090 --> 00:00:50.223
trying my best to draw this neatly

00:00:50.223 --> 00:00:51.881
and then from that subtract

00:00:51.881 --> 00:00:54.632
top right to bottom left diagonal

00:00:54.632 --> 00:00:56.131
so from that subtract

00:00:56.131 --> 00:00:57.690
let get a color

00:00:57.690 --> 00:00:59.291
I havent used

00:00:59.291 --> 00:01:01.075
subtract that

00:01:01.075 --> 00:01:03.998
and that

00:01:03.998 --> 00:01:06.964
and that

00:01:06.964 --> 00:01:08.490
sounds really confusing with all the diagonals

00:01:08.490 --> 00:01:09.343
i have drawn

00:01:09.343 --> 00:01:10.876
lets look at the blue ones first

00:01:10.876 --> 00:01:13.149
4 times 5 times 0

00:01:13.149 --> 00:01:15.548
4 times 5 times 0

00:01:15.548 --> 00:01:17.581
plus -1 times 3 times -2

00:01:17.581 --> 00:01:22.132
plus -1 times 3 times -2

00:01:22.132 --> 00:01:24.866
let me put these in parenthesis

00:01:24.866 --> 00:01:29.708
plus 1 times 4 times 0

00:01:29.708 --> 00:01:32.151
and then we gonna subtract all of these orange

00:01:32.151 --> 00:01:34.950
diagonal, we go from the top right to the bottom left

00:01:34.950 --> 00:01:36.361
so we gonna subtract

00:01:36.361 --> 00:01:38.682
we could do 1 times 5 times -2

00:01:38.682 --> 00:01:43.567
1 times 5 times -2 and then we can subtract

00:01:43.567 --> 00:01:47.949
subtract 4 times 3 times 0

00:01:47.949 --> 00:01:50.564
4 times 3 times 0

00:01:50.564 --> 00:01:52.665
and then we can subtract

00:01:52.665 --> 00:01:54.649
-1 times 4 times 0

00:01:54.649 --> 00:01:56.798
-1 times 4 times 0

00:01:56.798 --> 00:01:59.382
and now we just evaluate this over here

00:01:59.382 --> 00:02:01.348
4 times 5 times 0

00:02:01.348 --> 00:02:03.182
is just 0

00:02:03.182 --> 00:02:05.533
-1 times 3 times -2

00:02:05.533 --> 00:02:10.565
is +6 so this is +6

00:02:10.565 --> 00:02:13.082
1 times 4 times 0 is 0 once again

00:02:13.082 --> 00:02:16.182
and then we have

00:02:16.182 --> 00:02:18.348
1 times 5 times -2

00:02:18.348 --> 00:02:23.631
is -10 but we have this negative out here so it becomes a +10

00:02:23.631 --> 00:02:27.181
then 4 times 3 times , well thats just going to be 0.

00:02:27.181 --> 00:02:31.283
and then we have -1 times 4 times 0, which is just 0

00:02:31.283 --> 00:02:34.555
so we are left with +6 + 10

00:02:34.555 --> 00:02:38.555
which is equal to +16


Finding the determinant of a 2x2 matrix

WEBVTT
Kind: captions
Language: en

00:00:01.459 --> 00:00:06.399
As a hint, I'll take the determinant of a very similar two by two matrix.

00:00:06.399 --> 00:00:10.514
So we wanna figure out the determinant of this matrix and we can denote it

00:00:10.514 --> 00:00:13.371
by the determinant of the 2x2 matrix five,

00:00:13.371 --> 00:00:17.642
three,negative one and four

00:00:17.642 --> 00:00:21.892
and the definition of determinant of the 2x2 matrix like this, as we can first take

00:00:21.892 --> 00:00:24.888
this diagonal, five times four

00:00:24.888 --> 00:00:29.532
so write it down, just going to be equal to

00:00:29.532 --> 00:00:35.777
5 * 4 minus of product of that diagonal

00:00:45.792 --> 00:00:46.721
...so what is this

00:00:46.721 --> 00:00:50.347
give us? So, five times four is twenty

00:00:50.347 --> 00:00:55.362
and then three times negative one will be negative three

00:00:55.362 --> 00:00:57.521
this is negative three...

00:00:57.521 --> 00:01:01.771
but making negative the negative three will make the positive three

00:01:02.969 --> 00:01:08.397
so the determinant of this matrix is twenty three.


Hydrolysis

WEBVTT
Kind: captions
Language: en

00:00:00.350 --> 00:00:01.950
- In the video on dehydration synthesis,

00:00:01.950 --> 00:00:04.590
we saw how we could start with a glucose molecule,

00:00:04.590 --> 00:00:06.820
and through dehydration synthesis

00:00:06.820 --> 00:00:09.990
form a bond with another glucose molecule.

00:00:09.990 --> 00:00:14.020
And just by doing that, you'd form the disaccharide maltose

00:00:14.020 --> 00:00:16.290
if these were both glucose molecules.

00:00:16.450 --> 00:00:17.350
But then you could keep going,

00:00:17.350 --> 00:00:21.700
and you could form longer chains of glucose molecules.

00:00:22.140 --> 00:00:24.990
And to these things, where you would take a monosaccharide,

00:00:24.990 --> 00:00:27.210
glucose is the most common example of that,

00:00:27.210 --> 00:00:28.590
and you create chains of these,

00:00:28.590 --> 00:00:30.440
we call these polysaccharides.

00:00:30.850 --> 00:00:35.290
Polysaccharides, this is a polysaccharide.

00:00:35.290 --> 00:00:36.990
And there's all sorts of interesting examples

00:00:36.990 --> 00:00:39.190
of polysaccharides all around you,

00:00:39.190 --> 00:00:41.920
especially polysaccharides of glucose,

00:00:41.920 --> 00:00:44.220
or things that are derived from glucose.

00:00:44.220 --> 00:00:46.650
This right here this is a bowl of mashed potatoes,

00:00:46.650 --> 00:00:48.550
which is mostly starch.

00:00:48.550 --> 00:00:51.400
Which is mainly just chains of glucose.

00:00:51.790 --> 00:00:57.470
So this right over here, that is starch.

00:00:59.920 --> 00:01:04.850
The shell of a lot of insects and things like lobsters,

00:01:04.850 --> 00:01:07.650
and the wings of these insects right over here,

00:01:07.650 --> 00:01:09.540
that's made of something called chitin.

00:01:09.920 --> 00:01:14.500
And chitin is also a polysaccharide.

00:01:14.920 --> 00:01:19.480
It's made of chains, a modification of glucose

00:01:19.480 --> 00:01:22.440
chains of that, that's chitin right over there.

00:01:22.810 --> 00:01:26.450
Very similar to starch, in our muscles we have glycogen,

00:01:26.450 --> 00:01:28.920
which is our store of energy in our muscles.

00:01:28.920 --> 00:01:31.700
You have cellulose,

00:01:32.290 --> 00:01:35.420
which is probably all around you right now.

00:01:35.420 --> 00:01:37.720
Cellulose are things like--

00:01:38.750 --> 00:01:40.450
Because this is something that's all around you,

00:01:40.450 --> 00:01:41.480
and you don't even realize it.

00:01:41.480 --> 00:01:44.090
Cellulose, this is what constitutes things

00:01:44.090 --> 00:01:46.720
like paper and wood.

00:01:46.720 --> 00:01:50.350
It's involved in the cell walls of plants.

00:01:50.350 --> 00:01:53.990
This right over here is a picture of cotton,

00:01:53.990 --> 00:01:55.685
cotton in its natural form.

00:01:55.685 --> 00:01:58.785
And cotton is actually one of the purest forms of cellulose,

00:01:58.785 --> 00:02:00.675
it's roughly 90 percent cellulose.

00:02:00.955 --> 00:02:04.585
And if you take a zoom in on a cotton fiber,

00:02:04.585 --> 00:02:07.085
actually a fiber of cellulose,

00:02:07.085 --> 00:02:11.285
you'll see chains of glucose molecules.

00:02:11.285 --> 00:02:16.385
So you see this right over here, that is a glucose molecule.

00:02:16.385 --> 00:02:20.245
Then you see another glucose molecule.

00:02:20.245 --> 00:02:21.825
And this chain has been formed

00:02:21.825 --> 00:02:23.985
through dehydration synthesis.

00:02:23.985 --> 00:02:25.985
And difference between starch and cellulose,

00:02:25.985 --> 00:02:26.885
for the main difference,

00:02:26.885 --> 00:02:28.585
in terms of how this bonding has.

00:02:28.585 --> 00:02:30.825
With starch, the glucose molecules just keep forming

00:02:30.825 --> 00:02:33.655
the way that you saw in the video on dehydration synthesis.

00:02:33.655 --> 00:02:36.255
While in cellulose, they get flipped over.

00:02:36.255 --> 00:02:40.985
So you can see here, this oxygen is pointing that way,

00:02:40.985 --> 00:02:42.385
this oxygen is pointing that way,

00:02:42.385 --> 00:02:43.755
that oxygen is pointing that way.

00:02:43.755 --> 00:02:45.755
And you could look up more about cellulose.

00:02:45.755 --> 00:02:47.885
But it's really interesting, what gives it its structure

00:02:47.885 --> 00:02:50.185
are these hydrogen bonds that form

00:02:50.185 --> 00:02:53.455
between the partially negative,

00:02:53.455 --> 00:02:56.355
the very electronegative oxygens on one strand.

00:02:56.355 --> 00:02:59.985
And the partially positive hydrogens on another strand,

00:02:59.985 --> 00:03:01.985
and that's actually what give its structure.

00:03:01.985 --> 00:03:05.525
So really, really interesting things, these polysaccharides.

00:03:05.525 --> 00:03:06.985
The question is,

00:03:06.985 --> 00:03:09.055
how do you actually break these things down?

00:03:09.055 --> 00:03:10.855
If I were to eat these mashed potatoes,

00:03:10.855 --> 00:03:14.425
how do I eventually turn this thing into glucose,

00:03:14.425 --> 00:03:16.485
so I could use it for energy?

00:03:16.485 --> 00:03:23.075
And the way that happens is through hydrolysis.

00:03:23.925 --> 00:03:25.925
And you could break down this word.

00:03:25.925 --> 00:03:29.985
The "hydro", if you see hydro, the prefix hydro,

00:03:29.985 --> 00:03:33.165
that's a good clue that it has something to do with water.

00:03:33.525 --> 00:03:36.025
And then if you see "lysis", if you're lysing something,

00:03:36.025 --> 00:03:37.245
this means that you're gonna break it down.

00:03:37.245 --> 00:03:40.285
So this is breaking down something using water.

00:03:40.285 --> 00:03:42.955
And that's exactly what happens with hydrolysis.

00:03:42.955 --> 00:03:44.915
If you have this polysaccharide,

00:03:44.915 --> 00:03:48.785
and let's throw a water molecule in there,

00:03:50.545 --> 00:03:51.955
this water molecule is going to be able

00:03:51.955 --> 00:03:55.245
to break one of these bonds.

00:03:56.025 --> 00:03:59.075
So we might end up with something like--

00:04:00.155 --> 00:04:01.955
This chain could keep going in both directions,

00:04:01.955 --> 00:04:04.435
but we could end up with something that looks like this.

00:04:05.615 --> 00:04:10.335
That looks something like that.

00:04:10.855 --> 00:04:13.285
So half of this water molecule gets broken up,

00:04:13.285 --> 00:04:14.825
essentially to break this bond.

00:04:14.825 --> 00:04:17.485
It's the opposite of dehydration synthesis.

00:04:17.485 --> 00:04:18.855
So let's see if we can understand,

00:04:18.855 --> 00:04:22.325
get an overview of exactly how that happens.

00:04:22.325 --> 00:04:28.515
So this right over here, this is maltose right over here.

00:04:28.515 --> 00:04:30.485
It's disaccharide, it's just two glucose molecules

00:04:30.485 --> 00:04:31.525
attached to each other.

00:04:31.525 --> 00:04:34.005
If we kept doing this, if this kept going,

00:04:35.025 --> 00:04:37.855
if this guy had bond to another glucose molecule,

00:04:37.855 --> 00:04:39.625
and this guy had a bond to another glucose molecule,

00:04:39.625 --> 00:04:41.655
then we'd be dealing with starch.

00:04:41.655 --> 00:04:43.385
Or we could be dealing with glycogen.

00:04:43.385 --> 00:04:44.455
If this was flipped over,

00:04:44.455 --> 00:04:46.185
and they kept flipping over and over,

00:04:46.185 --> 00:04:48.755
then we could be talking about cellulose.

00:04:48.755 --> 00:04:51.055
But let's just thing about how this is the mechanism.

00:04:51.055 --> 00:04:55.125
The mechanism by which this bond can actually be broken.

00:04:55.125 --> 00:04:59.835
It's really just the reverse of dehydration synthesis.

00:05:00.855 --> 00:05:02.655
This is going to just be an overview of it.

00:05:03.225 --> 00:05:07.845
This oxygen right over here, it's got two lone pairs.

00:05:07.845 --> 00:05:10.355
There's always a chance that if it bumps into something

00:05:10.355 --> 00:05:14.525
in just the right way, it could nab a hydrogen proton

00:05:14.525 --> 00:05:19.225
that is just sitting out there in the fluid.

00:05:19.225 --> 00:05:20.355
We're assuming that this is happening

00:05:20.355 --> 00:05:23.585
in an aqueous solution, it's happening in water.

00:05:23.585 --> 00:05:25.115
So it can just grab a hydrogen proton

00:05:25.115 --> 00:05:28.485
from a passing hydronium molecule.

00:05:28.485 --> 00:05:34.985
And so if it does that, it would form a covalent bond,

00:05:34.985 --> 00:05:36.685
and have a positive charge.

00:05:36.685 --> 00:05:39.485
And now relative to actually both carbons,

00:05:39.485 --> 00:05:41.425
but let's focus on this carbon right over here.

00:05:41.425 --> 00:05:43.625
This guy would be, what we call in organic chemistry,

00:05:43.625 --> 00:05:46.005
a good leaving group.

00:05:46.625 --> 00:05:49.855
So these electrons,

00:05:49.855 --> 00:05:51.985
the oxygen might want to just take these back,

00:05:51.985 --> 00:05:53.825
because it's got a positive charge,

00:05:53.825 --> 00:05:55.425
oxygen is really electronegative,

00:05:55.425 --> 00:05:58.525
so things just bump in exactly the right way.

00:05:58.525 --> 00:06:01.025
If things interact in exactly the right way,

00:06:01.025 --> 00:06:02.485
you might have another water molecule.

00:06:02.485 --> 00:06:05.085
And this is where that extra water molecule is valuable

00:06:05.085 --> 00:06:06.825
in our hydrolysis.

00:06:07.025 --> 00:06:09.025
So let's say this is just another water molecule

00:06:09.025 --> 00:06:11.285
just passing by in exactly the right way.

00:06:11.955 --> 00:06:19.255
This could form a bond with that carbon

00:06:19.255 --> 00:06:20.605
right over there.

00:06:20.815 --> 00:06:22.655
And just as it forms a bond with that carbon,

00:06:22.655 --> 00:06:24.525
the carbon says, "Okay, I'm getting to share

00:06:24.525 --> 00:06:28.175
"some other electrons, let me let go of these electrons."

00:06:28.245 --> 00:06:30.245
So it lets go of these electrons.

00:06:30.845 --> 00:06:33.785
And what do you have left?

00:06:33.785 --> 00:06:36.025
Well, we can go over here,

00:06:36.025 --> 00:06:40.455
and so now this carbon has, let me color code it,

00:06:40.455 --> 00:06:42.455
so this bond that was just forming,

00:06:42.455 --> 00:06:46.005
that is this bond right over here.

00:06:46.625 --> 00:06:52.095
This oxygen is this oxygen right over there.

00:06:52.625 --> 00:06:54.425
It actually has another hydrogen attached to it,

00:06:54.425 --> 00:06:55.555
so let me do that.

00:06:55.555 --> 00:06:56.945
So right when it makes the bond,

00:06:56.945 --> 00:06:58.715
it will have a positive charge.

00:06:58.715 --> 00:07:04.155
And then, this bond right over goes back to this oxygen.

00:07:04.155 --> 00:07:05.955
This oxygen right over here,

00:07:05.955 --> 00:07:08.905
is that oxygen right over there.

00:07:09.415 --> 00:07:11.335
Now when it started off, this guy grabbed--

00:07:15.285 --> 00:07:17.525
The hydrogen proton that I grabbed, I showed in orange,

00:07:17.525 --> 00:07:18.705
that's this one.

00:07:19.185 --> 00:07:20.935
That's this one right over here.

00:07:24.085 --> 00:07:25.925
Now, this one grabbed a hydrogen proton,

00:07:25.925 --> 00:07:27.485
and now this one can actually give back

00:07:27.485 --> 00:07:28.855
a hydrogen proton through solution.

00:07:29.385 --> 00:07:31.955
If a water molecule passing by could just grab

00:07:31.955 --> 00:07:36.125
this hydrogen proton and then become a hydronium molecule.

00:07:36.125 --> 00:07:39.165
So it took a hydrogen proton, it's giving it back,

00:07:39.165 --> 00:07:40.585
and so what we are left with--

00:07:41.285 --> 00:07:45.555
It took up this water molecule right over here

00:07:45.555 --> 00:07:46.905
to break the bond.

00:07:47.355 --> 00:07:49.455
And so this is a positive charge.

00:07:49.455 --> 00:07:50.985
It could be a passing hydronium molecule,

00:07:51.945 --> 00:07:54.635
and it'll just hand it off to that.

00:07:55.255 --> 00:07:56.155
And there you have it,

00:07:56.155 --> 00:08:00.585
we have two standalone glucose molecules right over there.

00:08:00.585 --> 00:08:01.725
We have broken the bond.

00:08:01.725 --> 00:08:03.255
And these could be parts of chains,

00:08:03.255 --> 00:08:05.025
in which case, we've just broken the chain.

00:08:05.455 --> 00:08:07.085
Or, if we're just dealing with maltose,

00:08:07.085 --> 00:08:08.315
now we've broken it down

00:08:08.315 --> 00:08:13.215
into the individual glucose molecules.

00:08:13.215 --> 00:08:14.285
And the example here is with glucose,

00:08:14.285 --> 00:08:16.655
but it could've been the case with maltose,

00:08:16.655 --> 00:08:17.925
and it could've been the case with sucrose

00:08:17.925 --> 00:08:20.655
where we break sucrose down using hydrolysis

00:08:20.655 --> 00:08:23.038
into a glucose molecule and a fructose molecule.

00:08:23.038 --> 00:08:27.087
So it's a very important reaction in biology.