Using Trig Functions

WEBVTT
Kind: captions
Language: en

00:00:01.180 --> 00:00:03.150
We're now going to do a few
examples to actually show

00:00:03.150 --> 00:00:06.320
you why the trig functions
are actually useful.

00:00:06.320 --> 00:00:08.760
So let's get started
with a problem.

00:00:08.760 --> 00:00:11.065
Let's say I have this
right triangle.

00:00:15.860 --> 00:00:17.040
That's my right triangle.

00:00:20.990 --> 00:00:22.470
There's the right angle.

00:00:22.470 --> 00:00:25.950
And let's say I know that
the measure of this angle

00:00:25.950 --> 00:00:31.710
is pi over 4 radians.

00:00:31.710 --> 00:00:33.980
And I'll just write
rad for short.

00:00:33.980 --> 00:00:37.230
If the measure of this angle is
pi over 4 radians, and I also

00:00:37.230 --> 00:00:40.900
know that this side of the
triangle-- this side right

00:00:40.900 --> 00:00:48.080
here-- is 10 square roots of 2.

00:00:48.080 --> 00:00:51.970
So I know this side
of the triangle.

00:00:51.970 --> 00:00:54.690
I know this angle, which
is pi over 4 radians.

00:00:54.690 --> 00:01:00.420
And now, the question is, what
is this side of the triangle?

00:01:00.420 --> 00:01:01.410
I'm going to highlight that.

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And let me make it in orange.

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So let's figure out what
we know and what we

00:01:07.900 --> 00:01:09.390
need to figure out.

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We know the angle,
pi over 4 radians.

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And actually, turns out if
you were to convert that

00:01:13.970 --> 00:01:16.290
to degrees, it would
be 45 degrees.

00:01:16.290 --> 00:01:18.120
And we know-- what
side is this?

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This is the hypotenuse
of the triangle, right?

00:01:21.440 --> 00:01:23.130
And what are we trying
to figure out?

00:01:23.130 --> 00:01:26.720
Are we trying to figure out
the hypotenuse, the adjacent

00:01:26.720 --> 00:01:30.390
side to the angle, or the
opposite side to the angle?

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Well, this is the hypotenuse,
we already know that.

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This is the opposite side.

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This is the opposite side.

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And this yellow side is
the adjacent side, right?

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It's just adjacent
to this angle.

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So we know the angle, we know
the hypotenuse, and we want to

00:01:50.150 --> 00:01:52.060
figure out the adjacent side.

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So let me ask you a question.

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What trig function deals
with the adjacent side

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and the hypotenuse?

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Because we have the adjacent
side is what we want

00:02:01.620 --> 00:02:04.005
to figure out, and we
know the hypotenuse.

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Well, let's write down
our mnemonic, just in

00:02:05.770 --> 00:02:08.390
case you forgot it.

00:02:08.390 --> 00:02:08.836
SOHCAHTOA.

00:02:16.820 --> 00:02:19.650
So which one uses
adjacent and hypotenuse?

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Right?

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It's CAH.

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And CAH, the c is for what?

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The c is for cosine.

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Cosine of an angle-- let's just
call it any angle-- is equal to

00:02:30.980 --> 00:02:35.870
the adjacent over
the hypotenuse.

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So let's use this information
to try to solve for this orange

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side, or this yellow side.

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So we know that cosine of pi
over 4 radians-- so let's say

00:02:48.350 --> 00:02:55.580
cosine of pi over 4-- must
equal this adjacent

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side right here.

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Let's just call that
a. a for adjacent.

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The adjacent side divided
by the hypotenuse.

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The hypotenuse is this side.

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And in the problem, we
were given that it's

00:03:07.590 --> 00:03:08.730
10 square roots of 2.

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So we can solve for a by
multiplying both sides of

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this equation by 10
square roots of 2.

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And we will get--
because, right?

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If we just multiply times
10 square root of 2,

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these cancel out.

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And then you get a 10
square root of 2 here.

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So you get a is equal to 10
square roots of 2 times

00:03:37.840 --> 00:03:42.710
the cosine of pi over 4.

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Now you're probably saying,
Sal, this does not look too

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simple, and I don't know how
big the cosine of pi over 4 is.

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What do I do?

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Well, no one has the trig
functions, or the values of

00:03:54.650 --> 00:03:56.450
the trig functions memorized.

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There's a couple
of ways to do it.

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Either I could give you what
the cosine of pi over 4 is.

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That's sometimes
given in a problem.

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Or you can make sure that your
calculator is set to radians

00:04:05.400 --> 00:04:09.400
and you can just type in pi
divided by 4-- which is roughly

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0.79-- and then press
the cosine button.

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You finally know
what it's good for.

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And you'll get a value.

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Or-- and this is kind of the
old school way of doing it--

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there are trig tables where
you could look up what cosine

00:04:20.554 --> 00:04:22.540
of pi over 4 is in a table.

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Since I don't have any of that
at my disposal right now,

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I'll just tell you what the
cosine of pi over 4 is.

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The cosine of pi over 4 is
square root of 2 over 2.

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So a, which is the adjacent
side-- a for adjacent-- is

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equal to 10 square roots of 2
times square root of 2 over 2.

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Remember, to get the square
root of 2 over 2, you might

00:04:48.750 --> 00:04:49.330
be a little confused.

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You're like, how
did Sal get that?

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All I said is, the cosine
of pi over 4 is square

00:04:54.420 --> 00:04:55.090
root of 2 over 2.

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And that's not something that--
well, actually, this one you

00:04:57.010 --> 00:04:59.420
might know offhand, because
of the 45 degree angle.

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But this isn't something
that people memorize.

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This is something you would
look up, or it's given in

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the problem, or you'd
use a calculator for.

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And a calculator, of course,
wouldn't give you square

00:05:07.610 --> 00:05:08.020
root of 2 over 2.

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It'd give you a decimal
number that's not obviously

00:05:11.910 --> 00:05:13.180
square root of 2 over 2.

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But anyway, I told you that the
cosine of pi over 4 is the

00:05:15.800 --> 00:05:17.310
square root of 2 over 2.

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And so if we multiply, what's
the square root of 2 over 2?

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What's the square root of 2
times the square root of 2?

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It's 2.

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So that's 2, and then that
cancels with that 2.

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And so everything cancels
except for the 10.

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So the adjacent side
is equal to 10.

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Let's do another one.

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Let me delete this.

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Give me 1 second.

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I'm actually-- this is one of
the few modules that I'm not

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generating the problems on the
fly, because I need to make

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sure that I actually have
the trig function values

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before I do the problem.

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So let's say I have
another right triangle.

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I probably shouldn't have
deleted that last one.

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So let's see, this is
my right triangle.

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How much time do I have--
about 4 minutes left.

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Should be enough.

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So this is my right triangle.

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And I know the angle--
let's call this--.

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I know this angle right
here is 0.54 radians.

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And I also know that this side
right here is 3 units long.

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And I want to figure
out this side.

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So what do I know?

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Well, this side is what side
relative to the angle?

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It's the opposite side, right?

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Because the angle is here, and
we go opposite the angle.

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So this is the opposite side.

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And what's this side?

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Is this the adjacent side,
or is it the hypotenuse?

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Well, this is the
hypotenuse, right?

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The long side, and it's
opposite the right angle.

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So this is the adjacent side.

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So what trig function uses
opposite and adjacent?

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Let's write down
SOHCAHTOA again.

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SOHCAHTOA.

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TOA uses opposite and adjacent.

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OA.

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So T for tangent, right?

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TOA.

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So tangent is equal to
opposite over adjacent.

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So let's use that.

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So let's take the tangent
of 0.54 radians.

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So the tangent of 0.54 will
equal the side opposite to it.

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So that's 3, right?

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The opposite side is 3.

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Over the adjacent side.

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Well, once again, the adjacent
side is what we don't know.

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So we have to solve for a.

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So if we multiply both sides
by a, we get a tan of 0.54--

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we could do that because we
know it's not 0-- equals 3.

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Or a is equal to 3 divided
by the tangent of 0.54.

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So once again, I don't have
memorized what the tangent of

00:08:22.500 --> 00:08:29.770
0.54 is, but I will tell you
what it is because you also

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don't have it memorized.

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Or you could use a calculator
to figure it out if you

00:08:31.990 --> 00:08:34.060
had a radian function.

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The tangent of 0.54 is equal
to-- let me make sure

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I have this right.

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Oh, right.

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The tangent of 0.54 is 3/5.

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So then a is equal
to 3 over 3/5.

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Right, the adjacent side--
now, once again, how

00:08:56.850 --> 00:08:57.810
did I get this 3/5?

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Well, I just told you.

00:08:58.820 --> 00:09:00.200
Or you can use a calculator
to know that the

00:09:00.200 --> 00:09:03.250
tangent of 0.54 is 3/5.

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And of course, I'm using
numbers that work out

00:09:05.130 --> 00:09:07.400
well, just so that the
fractions all cancel.

00:09:07.400 --> 00:09:09.580
So we know that the adjacent
side is equal to-- when you

00:09:09.580 --> 00:09:12.550
divide by fractions, it's like
multiplying by the numerator.

00:09:12.550 --> 00:09:14.420
Multiplying by the inverse.

00:09:14.420 --> 00:09:17.430
So times 5/3.

00:09:17.430 --> 00:09:20.370
So the adjacent side
is equal to 5.

00:09:20.370 --> 00:09:21.120
There.

00:09:21.120 --> 00:09:21.930
There you go.

00:09:21.930 --> 00:09:23.870
So let's just think
about what I always do.

00:09:23.870 --> 00:09:27.190
I think about what I have,
what sides I have, and what

00:09:27.190 --> 00:09:28.210
side I want to solve for.

00:09:28.210 --> 00:09:30.130
And in this case, it was the
opposite side I had, and I

00:09:30.130 --> 00:09:31.740
wanted to solve for
the adjacent side.

00:09:31.740 --> 00:09:35.170
And I said, what trig function
involves those 2 sides?

00:09:35.170 --> 00:09:36.940
The opposite and the adjacent.

00:09:36.940 --> 00:09:37.880
I wrote down SOHCAHTOA.

00:09:37.880 --> 00:09:39.410
I said, oh, TOA.

00:09:39.410 --> 00:09:40.370
Opposite and adjacent.

00:09:40.370 --> 00:09:41.320
That's tan.

00:09:41.320 --> 00:09:43.500
So I took the tan of the angle.

00:09:43.500 --> 00:09:45.960
And then I said, the tan of the
angle is equal to the opposite

00:09:45.960 --> 00:09:47.820
side divided by the
adjacent side.

00:09:47.820 --> 00:09:48.780
That's right here.

00:09:48.780 --> 00:09:50.430
And then I just solved
for the adjacent side.

00:09:50.430 --> 00:09:52.740
And of course, I used a
calculator, or I told you

00:09:52.740 --> 00:09:55.880
what the tangent of 0.54 is.

00:09:55.880 --> 00:09:58.110
I think I'll do a couple more
of these problems in the next

00:09:58.110 --> 00:09:59.960
module, but I'm out
of time for now.

00:09:59.960 --> 00:10:01.480
Have fun.


Radian and degree

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.770
Welcome to the presentation
on radians and degrees.

00:00:03.770 --> 00:00:07.010
So you all are probably already
reasonably familiar with

00:00:07.010 --> 00:00:07.950
the concept of degrees.

00:00:07.950 --> 00:00:10.310
I think in our angles models
we actually drill you

00:00:10.310 --> 00:00:12.820
through a bunch of problems.

00:00:12.820 --> 00:00:23.460
You're probably familiar that
a right angle is 90 degrees.

00:00:23.460 --> 00:00:28.650
Or half a right angle
-- 45 degrees.

00:00:28.650 --> 00:00:32.630
And you're also probably
familiar with the concept that

00:00:32.630 --> 00:00:36.610
in a circle -- and that's my
best adept at a circle -- in a

00:00:36.610 --> 00:00:41.010
circle, there are 360 degrees.

00:00:41.010 --> 00:00:44.750
So today I'm going to introduce
you to another measure or unit

00:00:44.750 --> 00:00:47.270
for angles and this
is called a radian.

00:00:52.160 --> 00:00:53.450
So what is a radian?

00:00:53.450 --> 00:00:55.650
So I'll start with the
definition and I think this

00:00:55.650 --> 00:00:57.105
might give you a little
intuition for why it's

00:00:57.105 --> 00:00:59.910
even called radian.

00:00:59.910 --> 00:01:02.850
Let me use this circle tool and
actually draw a nice circle.

00:01:10.060 --> 00:01:14.270
I'm still using the radian
tool, the circle tool.

00:01:14.270 --> 00:01:14.530
OK.

00:01:19.430 --> 00:01:21.630
This is a radius of length r.

00:01:21.630 --> 00:01:25.500
A radian is the angle
that subtends an arc.

00:01:25.500 --> 00:01:30.210
And all subtend means is if
this is angle, and this is

00:01:30.210 --> 00:01:34.520
the arc, this angle subtends
this arc and this arc

00:01:34.520 --> 00:01:36.020
subtends this angle.

00:01:36.020 --> 00:01:41.050
So a radian -- one radian -- is
the angle that subtends an arc

00:01:41.050 --> 00:01:44.130
that's the length
of the radius.

00:01:44.130 --> 00:01:46.780
So the length of
this is also r.

00:01:46.780 --> 00:01:50.440
And this angle is one radian.

00:01:50.440 --> 00:01:51.140
i think that's messy.

00:01:51.140 --> 00:01:52.490
Let me draw a bigger circle.

00:01:55.010 --> 00:01:56.640
Here you go.

00:01:56.640 --> 00:01:57.860
And I'm going to do this
because I was wondering

00:01:57.860 --> 00:01:58.780
why they do radians.

00:01:58.780 --> 00:02:00.300
We all know degrees.

00:02:00.300 --> 00:02:02.090
But actually when you think
about it it actually makes a

00:02:02.090 --> 00:02:03.100
reasonable amount of sense.

00:02:03.100 --> 00:02:05.873
So let me use the
line tool now.

00:02:12.980 --> 00:02:18.990
And let's say that this radius
is a length r and that this arc

00:02:18.990 --> 00:02:21.460
right here is also length r.

00:02:21.460 --> 00:02:28.210
Then this angle, what's called
theta, is equal to one radian.

00:02:28.210 --> 00:02:30.220
And now it makes sense that
they call it a radian.

00:02:30.220 --> 00:02:32.440
It's kind of like a radius.

00:02:32.440 --> 00:02:35.100
So let me ask a question:
how many radians are

00:02:35.100 --> 00:02:37.420
there in a circle?

00:02:37.420 --> 00:02:41.300
Well, if this is r, what is
the whole circumference

00:02:41.300 --> 00:02:42.050
of a circle?

00:02:44.630 --> 00:02:46.540
It's 2 pi r, right?

00:02:46.540 --> 00:02:50.050
You know that from the
basic geometry module.

00:02:50.050 --> 00:02:55.850
So if the radian is the angle
that sub tends an arc of r,

00:02:55.850 --> 00:03:03.650
then the angle that subtends an
arc of 2 pi r is 2 pi radians.

00:03:03.650 --> 00:03:06.970
So this angle is 2 pi radians.

00:03:12.510 --> 00:03:14.820
If you're still confused,
think of it this way.

00:03:14.820 --> 00:03:20.390
An angle of 2 pi radians going
all the way around subtends

00:03:20.390 --> 00:03:22.650
an arc of 2 pi radiuses.

00:03:22.650 --> 00:03:23.500
Or radii.

00:03:23.500 --> 00:03:26.460
I don't know how to say
the plural of radius.

00:03:26.460 --> 00:03:27.110
Maybe it's radians.

00:03:27.110 --> 00:03:30.130
And I don't know.

00:03:30.130 --> 00:03:32.630
So why am I going through all
of this mess and confusing you?

00:03:32.630 --> 00:03:35.580
I just want to one, give you an
intuition for why it's called

00:03:35.580 --> 00:03:38.130
a radian and kind of how
it relates to a circle.

00:03:38.130 --> 00:03:41.890
And then given that there 2 pi
radians in a circle, we can now

00:03:41.890 --> 00:03:46.980
figure out a relationship
between radians and degrees.

00:03:46.980 --> 00:03:49.920
Let me delete this.

00:03:49.920 --> 00:03:54.190
So we said in a circle,
there are 2 pi radians.

00:03:57.340 --> 00:03:58.970
And how many degrees
are there in a circle?

00:03:58.970 --> 00:04:00.800
If we went around a whole
circle how many degrees?

00:04:00.800 --> 00:04:04.360
Well that's equal
to 360 degrees.

00:04:07.080 --> 00:04:07.520
So there.

00:04:07.520 --> 00:04:09.620
We have an equation that sets
up a conversion between

00:04:09.620 --> 00:04:10.950
radians and degrees.

00:04:10.950 --> 00:04:19.390
So one radian is equal to
360 over 2 pi degrees.

00:04:19.390 --> 00:04:22.570
I just divided both
sides by 2 pi.

00:04:22.570 --> 00:04:27.040
Which equals 180
over pi degrees.

00:04:29.710 --> 00:04:31.080
Similarly, we could have
done the other way.

00:04:31.080 --> 00:04:34.000
We could have divided both
sides by 360 and we could have

00:04:34.000 --> 00:04:38.530
said 1 degree -- I'm just going
to divide both sides but

00:04:38.530 --> 00:04:39.970
360 and I'm flipping it.

00:04:39.970 --> 00:04:45.410
1 degree is equal to 2
pi over 360 radians.

00:04:48.570 --> 00:04:53.260
Which equals pi
over 180 radians.

00:04:53.260 --> 00:05:00.440
So then we have a conversion:
1 radian equals 180 over pi

00:05:00.440 --> 00:05:05.220
degrees and 1 degree equals
pi over 180 radians.

00:05:05.220 --> 00:05:06.980
Amd if you ever forget
these, it doesn't hurt

00:05:06.980 --> 00:05:08.740
to to memorize this.

00:05:08.740 --> 00:05:12.520
But if you ever forget it,
I always go back to this.

00:05:12.520 --> 00:05:15.810
That 2 pi radians is
equal to 360 degrees.

00:05:15.810 --> 00:05:21.450
Or another way that actually
makes the algebra a little

00:05:21.450 --> 00:05:26.545
simpler is if you just
think of a half circle.

00:05:26.545 --> 00:05:31.550
A half circle -- this angle
-- is a 180 degrees, right?

00:05:35.210 --> 00:05:36.120
That's a degree sign.

00:05:36.120 --> 00:05:37.810
I could also write degrees out.

00:05:37.810 --> 00:05:39.680
And that's also equal
to pi radians.

00:05:42.680 --> 00:05:46.250
So pi radians equal 180 degrees
and we can get to see the math.

00:05:46.250 --> 00:05:57.250
1 radian equals 180 over pi
degrees or 1 degree is equal

00:05:57.250 --> 00:06:00.940
to pi over 180 radians.

00:06:00.940 --> 00:06:02.495
So let's do a couple of
problems were you'll get

00:06:02.495 --> 00:06:03.540
the intuition for this.

00:06:03.540 --> 00:06:09.010
If I asked you 45 degrees --
to convert that into radians.

00:06:12.440 --> 00:06:18.410
Well, we know that 1 degree
os pi over 180 radians.

00:06:18.410 --> 00:06:32.910
So 45 degrees is equal to 45
times pi over 180 radians.

00:06:32.910 --> 00:06:36.850
And let's see, 45
divided by 180.

00:06:36.850 --> 00:06:42.360
45 goes into 180 four times so
this equals pi over 4 radians.

00:06:45.650 --> 00:06:49.600
45 degrees is equal to
pi over 4 radians.

00:06:49.600 --> 00:06:52.610
And just keep in mind, these
are just two different units

00:06:52.610 --> 00:06:55.070
or two different ways
of measuring angles.

00:06:55.070 --> 00:06:56.590
And the reason why I do this is
this is actually the

00:06:56.590 --> 00:06:59.700
mathematical standard for
measuring angles, although most

00:06:59.700 --> 00:07:01.690
of us are more familiar with
degrees just from

00:07:01.690 --> 00:07:03.030
everyday life.

00:07:03.030 --> 00:07:04.920
Let's do a couple
of other examples.

00:07:04.920 --> 00:07:06.690
Just always remember:
this 1 radian equals

00:07:06.690 --> 00:07:08.400
180 over pi degrees.

00:07:08.400 --> 00:07:10.200
1 degree equals pi
over 180 radians.

00:07:10.200 --> 00:07:12.630
If you ever get confused,
just write this out.

00:07:12.630 --> 00:07:15.400
this is what I do because I
always forget whether it's

00:07:15.400 --> 00:07:17.570
pi over 180 or 180 over pi.

00:07:17.570 --> 00:07:21.550
I just remember pi radians
is equal to 180 degrees.

00:07:21.550 --> 00:07:23.840
Let's do another one.

00:07:23.840 --> 00:07:33.060
So if I were to say pi
over 2 radians equals

00:07:33.060 --> 00:07:33.765
how many degrees?

00:07:37.480 --> 00:07:40.660
Well I already forgot what I
had just written so I just

00:07:40.660 --> 00:07:45.565
remind myself that pi radians
is equal to 180 degrees.

00:07:55.720 --> 00:07:57.930
Oh, my wife just got home, so
I'm just going to have to leave

00:07:57.930 --> 00:08:02.670
the presentation like that
and I will continue it later.

00:08:02.670 --> 00:08:05.120
Actually, let me just finish
this problem and then I'll

00:08:05.120 --> 00:08:07.270
go attend to my wife.

00:08:07.270 --> 00:08:12.140
But we know that pi radians is
equal to 180 degrees, right?

00:08:12.140 --> 00:08:18.840
So one radian is equal to 180
over -- that's one radian -- is

00:08:18.840 --> 00:08:21.660
equal to 180 over pi degrees.

00:08:21.660 --> 00:08:23.470
I just figure out the
formula again because

00:08:23.470 --> 00:08:24.490
I always forget it.

00:08:24.490 --> 00:08:25.500
So let's go back here.

00:08:25.500 --> 00:08:33.160
So pi over 2 radians is
equal to pi over 2 times

00:08:33.160 --> 00:08:38.510
180 over pi degrees.

00:08:38.510 --> 00:08:41.585
And that equals 90 degrees.

00:08:47.240 --> 00:08:48.830
I'll do one more example.

00:08:54.480 --> 00:08:55.915
Let's say 30 degrees.

00:09:00.950 --> 00:09:03.200
Once again, I forgot the
formula so I just remember

00:09:03.200 --> 00:09:10.960
that pi radians is
equal to 180 degrees.

00:09:10.960 --> 00:09:19.150
So 1 degree is equal to
pi over 180 radians.

00:09:19.150 --> 00:09:27.220
So 30 degrees is equal to 30
times pi over 180 radians

00:09:27.220 --> 00:09:31.320
which equals -- 30 goes
into 180 six times.

00:09:31.320 --> 00:09:36.160
That equals pi over 6 radians.

00:09:36.160 --> 00:09:39.630
Hopefully you have a sense of
how to convert between degrees

00:09:39.630 --> 00:09:42.070
and radians now and even why
it's called a radian because

00:09:42.070 --> 00:09:45.880
it's very closely related to
a radius and you'll feel

00:09:45.880 --> 00:09:50.210
comfortable when someone asks
you to, I don't know, deal with

00:09:50.210 --> 00:09:52.410
radians as opposed to degrees.

00:09:52.410 --> 00:09:54.671
I'll see you in the
next presentation.


Basic Trigonometry II

WEBVTT
Kind: captions
Language: en

00:00:01.060 --> 00:00:03.280
Welcome to the second part
of the presentation on

00:00:03.280 --> 00:00:03.770
basic trigonometry.

00:00:03.770 --> 00:00:06.580
In the last 10 minutes, I had
trouble getting in a lot of

00:00:06.580 --> 00:00:09.280
examples, so I want to do a
couple more with you guys.

00:00:09.280 --> 00:00:09.780
OK.

00:00:09.780 --> 00:00:13.000
So, let me start over just
because this got messy.

00:00:15.850 --> 00:00:17.740
And we're just going to do what
we did in the last time around.

00:00:17.740 --> 00:00:20.310
So let me just draw
another right triangle.

00:00:20.310 --> 00:00:21.845
And make sure it's not
going to be too big.

00:00:25.410 --> 00:00:27.860
Here's my right triangle.

00:00:27.860 --> 00:00:30.270
And let me just throw
out some random sides.

00:00:30.270 --> 00:00:31.660
Let me say that this is 6.

00:00:35.930 --> 00:00:39.960
Let's make this side 5.

00:00:39.960 --> 00:00:42.730
And then, if this is a right
triangle, Pythagorean Theorem

00:00:42.730 --> 00:00:45.890
tells us that this would be
the square root of what?

00:00:45.890 --> 00:00:55.420
36 plus 25 is equal to
the square root of 61.

00:00:55.420 --> 00:00:56.600
I think that's right.

00:00:56.600 --> 00:00:59.270
I've gotten feedback on some of
my other videos that I tend to

00:00:59.270 --> 00:01:01.840
get this type of
addition wrong.

00:01:01.840 --> 00:01:03.390
I malfunction sometimes.

00:01:03.390 --> 00:01:03.950
But anyway.

00:01:03.950 --> 00:01:06.230
So this side is the square
root of 61, and that's

00:01:06.230 --> 00:01:07.830
the hypotenuse.

00:01:07.830 --> 00:01:10.450
So let's get started
with some problems.

00:01:10.450 --> 00:01:12.720
If I were to give you--
if I were to ask you.

00:01:12.720 --> 00:01:13.300
Well, let's see.

00:01:13.300 --> 00:01:18.560
Let's call this angle theta.

00:01:18.560 --> 00:01:25.070
And I want to know what
is the tangent of theta?

00:01:25.070 --> 00:01:29.390
And we'll shorten that as
tangent of-- tan of theta.

00:01:29.390 --> 00:01:33.080
What is the tangent of
this angle right here?

00:01:33.080 --> 00:01:35.580
Well, you probably already
forgot what the definition

00:01:35.580 --> 00:01:36.160
of tangent is.

00:01:36.160 --> 00:01:37.560
So I will repeat it.

00:01:37.560 --> 00:01:39.580
In fact, I will write
up in this corner.

00:01:39.580 --> 00:01:40.700
Soh cah toa.

00:01:44.980 --> 00:01:47.610
So I think now your brain might
be refreshed and you'll

00:01:47.610 --> 00:01:51.760
remember that toa is the
mnemonic for tangent.

00:01:51.760 --> 00:01:54.000
And it says that tangent
is equal to the opposite

00:01:54.000 --> 00:01:55.600
over the adjacent.

00:01:55.600 --> 00:01:58.660
So the tangent of theta is
equal to the opposite side--

00:01:58.660 --> 00:02:01.900
well, that's this side, that's
the side of length 5--

00:02:01.900 --> 00:02:04.160
over the adjacent side.

00:02:04.160 --> 00:02:05.350
That's this side.

00:02:05.350 --> 00:02:08.010
The side of length 6.

00:02:08.010 --> 00:02:09.110
That's pretty easy, huh?

00:02:09.110 --> 00:02:12.910
The tangent of theta is 5/6.

00:02:12.910 --> 00:02:14.410
And we'll just do
a couple more.

00:02:14.410 --> 00:02:14.860
All right?

00:02:14.860 --> 00:02:16.550
We'll just go through all of
the trig functions, or at least

00:02:16.550 --> 00:02:18.300
the basic trig functions.

00:02:18.300 --> 00:02:23.250
What is the sine of theta?

00:02:23.250 --> 00:02:24.780
Well, let's go back
to our mnemonic.

00:02:24.780 --> 00:02:25.480
Soh cah toa.

00:02:25.480 --> 00:02:27.180
This is one of the few things
in mathematics that you

00:02:27.180 --> 00:02:28.630
should probably memorize.

00:02:28.630 --> 00:02:30.680
It's kind of a
funny word anyway.

00:02:30.680 --> 00:02:33.710
And soh-- to find sine.

00:02:33.710 --> 00:02:36.090
It tells us that sine is
opposite over hypotenuse.

00:02:40.670 --> 00:02:44.840
Well, the opposite side,
once again, is 5.

00:02:44.840 --> 00:02:46.420
And what's the hypotenuse?

00:02:46.420 --> 00:02:47.970
Well, the hypotenuse, we
just figured out, was

00:02:47.970 --> 00:02:49.145
the square root of 61.

00:02:53.900 --> 00:02:57.170
And a lot of people don't like
irrational denominators.

00:02:57.170 --> 00:02:59.150
So we can rationalize
the denominator.

00:02:59.150 --> 00:03:01.390
And we do that by multiplying
the numerator and the

00:03:01.390 --> 00:03:03.280
denominator by the
square root of 61.

00:03:03.280 --> 00:03:09.570
So if we say that this is equal
to 5 over the square root of 61

00:03:09.570 --> 00:03:13.080
times the square root of
61, over the square

00:03:13.080 --> 00:03:13.720
root of 61, right?

00:03:13.720 --> 00:03:14.600
We're just multiplying it by 1.

00:03:14.600 --> 00:03:17.940
Because this is the same
thing top and bottom.

00:03:17.940 --> 00:03:24.440
This equals 5 square roots of
61 over-- what's the square

00:03:24.440 --> 00:03:26.350
root of 61 times the
square root of 61?

00:03:26.350 --> 00:03:29.140
Oh yeah, it's 61.

00:03:29.140 --> 00:03:35.110
So the sine of theta is 5
square roots of 61 over 61.

00:03:35.110 --> 00:03:38.610
And then finally, let me
make some space here.

00:03:38.610 --> 00:03:39.915
Let me erase some stuff.

00:03:43.550 --> 00:03:46.420
Let me erase this
one right here.

00:03:46.420 --> 00:03:48.930
And you're probably still
wondering, OK, I kind

00:03:48.930 --> 00:03:52.090
of get this whole sine,
tangent, cosine thing.

00:03:52.090 --> 00:03:53.430
What is it useful for?

00:03:53.430 --> 00:03:57.140
And all I can tell you right
now is, get to know how to use

00:03:57.140 --> 00:03:59.500
these, soh cah toa, and in the
next presentation and onwards,

00:03:59.500 --> 00:04:01.410
we're going to show you that
trigonometry is actually

00:04:01.410 --> 00:04:03.220
probably one of the most
obviously useful

00:04:03.220 --> 00:04:04.660
things in math.

00:04:04.660 --> 00:04:05.900
You can figure out
all sorts of things.

00:04:05.900 --> 00:04:08.930
How far planets are, how
tall buildings are.

00:04:08.930 --> 00:04:10.130
I mean, there's tons of
things you could figure

00:04:10.130 --> 00:04:12.010
out with trigonometry.

00:04:12.010 --> 00:04:14.260
And then later, we'll study
sine waves and cosine

00:04:14.260 --> 00:04:15.460
waves, and all that.

00:04:15.460 --> 00:04:19.650
You'll learn that it actually
describes almost everything.

00:04:19.650 --> 00:04:20.990
But anyway.

00:04:20.990 --> 00:04:22.440
Going back to the problem.

00:04:22.440 --> 00:04:25.850
All we have left now is cosine.

00:04:25.850 --> 00:04:28.200
Oh, look how big that is.

00:04:28.200 --> 00:04:33.770
Cosine of theta equals-- we'll
go back to our mnemonic.

00:04:33.770 --> 00:04:34.840
Soh cah toa.

00:04:34.840 --> 00:04:37.820
Well, cosine is adjacent
over hypotenuse.

00:04:37.820 --> 00:04:40.300
So once again, what's
the adjacent side?

00:04:40.300 --> 00:04:42.530
Well, this is the angle we're
finding the cosine of, so the

00:04:42.530 --> 00:04:44.480
adjacent side is right here.

00:04:44.480 --> 00:04:46.100
So length 6.

00:04:46.100 --> 00:04:49.200
So it equals the adjacent
side, which is 6, right?

00:04:49.200 --> 00:04:51.300
And we figured out what
the hypotenuse was.

00:04:51.300 --> 00:04:52.200
That's this side.

00:04:52.200 --> 00:04:54.940
And its length,
square root of 61.

00:04:54.940 --> 00:04:58.130
And if we rationalize this
denominator, we get 6 square

00:04:58.130 --> 00:05:01.320
roots of 61 over 61.

00:05:01.320 --> 00:05:03.110
It's kind of messy numbers.

00:05:03.110 --> 00:05:07.730
But I think now you get the
hang of figuring out-- if you

00:05:07.730 --> 00:05:12.390
know the sides of a triangle--
figuring out what the sine, the

00:05:12.390 --> 00:05:16.380
cosine, or the tangent of any
given angle in that

00:05:16.380 --> 00:05:17.290
right triangle is.

00:05:17.290 --> 00:05:21.890
And obviously, you can't figure
it out for this angle, because

00:05:21.890 --> 00:05:25.010
for this angle the opposite
and the hypotenuse are

00:05:25.010 --> 00:05:26.400
actually the same number.

00:05:26.400 --> 00:05:27.420
So actually-- never mind.

00:05:27.420 --> 00:05:28.950
You actually can figure it out.

00:05:28.950 --> 00:05:30.350
But it actually gives
something-- an

00:05:30.350 --> 00:05:32.180
interesting number.

00:05:32.180 --> 00:05:36.830
So with that said, I will
finish this presentation.

00:05:36.830 --> 00:05:40.560
And in the next presentation, I
will show you how-- if we know

00:05:40.560 --> 00:05:44.620
what the sine, or the cosine,
or the tangent of an angle is,

00:05:44.620 --> 00:05:48.860
and we know one of the sides--
how we can figure out

00:05:48.860 --> 00:05:51.500
the other sides.

00:05:51.500 --> 00:05:53.020
See you in the next
presentation.

00:05:53.020 --> 00:05:54.320
Bye.


Basic Trigonometry

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.630
Welcome to the presentation
on basic trigonometry.

00:00:04.630 --> 00:00:07.250
Sorry it's taken so long to get
a new video out, but I had

00:00:07.250 --> 00:00:08.440
a lot of family in town.

00:00:08.440 --> 00:00:11.120
So let's get started
with trigonometry.

00:00:11.120 --> 00:00:13.590
Let me get the pen
tools all set up.

00:00:13.590 --> 00:00:14.830
I'm a little rusty.

00:00:14.830 --> 00:00:17.200
I'll use green.

00:00:17.200 --> 00:00:19.820
Trigonometry.

00:00:19.820 --> 00:00:27.790
I think it means-- I think it's
from Ancient Greek, and it

00:00:27.790 --> 00:00:30.230
means triangle measure.

00:00:30.230 --> 00:00:32.290
I think that I read it on
Wikipedia a couple days ago,

00:00:32.290 --> 00:00:33.610
so I believe that's the case.

00:00:33.610 --> 00:00:38.040
But all trigonometry is is
really the study of right

00:00:38.040 --> 00:00:40.570
triangles and the relationship
between the sides and the

00:00:40.570 --> 00:00:42.520
angles of a right triangle.

00:00:42.520 --> 00:00:43.890
That might sound a
little confusing, but

00:00:43.890 --> 00:00:45.300
I'll get started.

00:00:45.300 --> 00:00:47.850
And actually, you've probably
seen a lot of these things that

00:00:47.850 --> 00:00:49.940
we're going to go over now, and
you'll finally know what

00:00:49.940 --> 00:00:52.840
those buttons on the
calculator actually do.

00:00:52.840 --> 00:00:56.730
So let's start with
a right triangle.

00:00:56.730 --> 00:00:59.016
Let's see, so it's a triangle.

00:01:01.620 --> 00:01:05.420
And it's a right triangle.

00:01:05.420 --> 00:01:08.470
Just for simplicity, let's say
that this side is 3, this side

00:01:08.470 --> 00:01:12.730
is 4, and the hypotenuse is 5.

00:01:12.730 --> 00:01:17.540
So the trig functions tell you
that for any angle, it tells

00:01:17.540 --> 00:01:20.170
you what the ratios of the
sides of the triangle are

00:01:20.170 --> 00:01:21.130
relative to that angle.

00:01:21.130 --> 00:01:22.550
Let me give you an example.

00:01:22.550 --> 00:01:25.490
Let's call this angle theta.

00:01:25.490 --> 00:01:29.510
Theta is the Greek alphabet
people tend to use for the

00:01:29.510 --> 00:01:33.000
angle that you're going to
find the trig function of.

00:01:33.000 --> 00:01:41.530
Let's say you wanted to
find the sine and s-i-n

00:01:41.530 --> 00:01:44.020
is short for sine.

00:01:44.020 --> 00:01:48.310
Let's say you wanted to
find the sine of theta.

00:01:48.310 --> 00:01:50.500
So before we solve the sine
of theta, I'm just going to

00:01:50.500 --> 00:01:53.310
throw out a mnemonic that I
remembered when I was learning

00:01:53.310 --> 00:01:55.700
this in school, and I carried
it every time, and every time I

00:01:55.700 --> 00:01:59.320
do a trig problem, I actually
write it down on the page, or I

00:01:59.320 --> 00:02:00.940
at least repeat it to myself.

00:02:00.940 --> 00:02:09.500
And this is SOH CAH TOA.

00:02:09.500 --> 00:02:11.960
I have vague memories of my
math teacher in high school

00:02:11.960 --> 00:02:14.990
telling a story about some
Indian princess, who was named

00:02:14.990 --> 00:02:16.290
Sohcahtoa, but I forget.

00:02:16.290 --> 00:02:19.560
But all you have to
remember is SOHCAHTOA.

00:02:19.560 --> 00:02:21.680
Now you might say, well,
what it's SOHCAHTOA?

00:02:21.680 --> 00:02:26.150
Well, SOHCAHTOA says that sine
is opposite over hypotenuse,

00:02:26.150 --> 00:02:31.160
cosine is adjacent over
hypotenuse, and tangent is

00:02:31.160 --> 00:02:33.110
opposite over adjacent.

00:02:33.110 --> 00:02:35.000
Now, that's going to be
confusing right now, but

00:02:35.000 --> 00:02:36.420
we're going to do a lot
of examples, and I think

00:02:36.420 --> 00:02:37.510
it's going to make sense.

00:02:37.510 --> 00:02:38.760
So let's go back
to this problem.

00:02:38.760 --> 00:02:40.280
We want to know what's
the sine of theta.

00:02:40.280 --> 00:02:43.380
Theta is this angle
in the triangle.

00:02:43.380 --> 00:02:45.300
So let's go to our
mnemonic SOHCAHTOA.

00:02:45.300 --> 00:02:46.700
So which one is sine?

00:02:46.700 --> 00:02:49.930
Well, S for sine,
so we use SOH.

00:02:49.930 --> 00:02:56.040
And we know that sine from
this mnemonic, sine of, let's

00:02:56.040 --> 00:03:01.150
say, theta, is equal to
opposite over hypotenuse.

00:03:01.150 --> 00:03:04.980
Opposite over hypotenuse.

00:03:04.980 --> 00:03:06.860
OK, so let's just figure
out what the opposite

00:03:06.860 --> 00:03:08.590
and the hypotenuse are.

00:03:08.590 --> 00:03:11.790
Well, what is the opposite
side of this angle?

00:03:14.450 --> 00:03:17.500
Well, if we just go opposite
the angle, let's go here,

00:03:17.500 --> 00:03:20.890
the opposite side is 4,
is this length of 4.

00:03:20.890 --> 00:03:23.100
I'll make that in a color.

00:03:23.100 --> 00:03:25.110
Oh, I thought I was
changing colors.

00:03:25.110 --> 00:03:30.150
Yeah, so this side is the
opposite, and I'll circle it.

00:03:30.150 --> 00:03:31.380
Now, which side is
the hypotenuse?

00:03:31.380 --> 00:03:32.620
And you know this one.

00:03:32.620 --> 00:03:35.610
We've been doing this in the
Pythagorean theorem modules.

00:03:35.610 --> 00:03:38.320
The long side, or the side
opposite the right angle,

00:03:38.320 --> 00:03:40.640
is the hypotenuse.

00:03:40.640 --> 00:03:43.160
So that is the hypotenuse.

00:03:43.160 --> 00:03:45.370
So now I think we're
ready to figure out what

00:03:45.370 --> 00:03:47.750
the sine of theta is.

00:03:47.750 --> 00:03:50.390
The sine-- whoops,
I stayed in pink.

00:03:50.390 --> 00:03:55.920
Sine of theta is equal to the
opposite side, 4, over the

00:03:55.920 --> 00:03:59.670
hypotenuse, which is 5.

00:03:59.670 --> 00:04:01.160
We're done.

00:04:01.160 --> 00:04:04.960
Let's figure out what-- let me
erase part of this, and we'll

00:04:04.960 --> 00:04:08.652
figure out some more things
about this triangle.

00:04:08.652 --> 00:04:09.840
Let me erase this.

00:04:14.800 --> 00:04:17.210
I think if you practice this,
you'll realize that this is

00:04:17.210 --> 00:04:18.660
probably one of the easier
things you learn in

00:04:18.660 --> 00:04:22.090
mathematics, and it's actually
shocking that they take-- that

00:04:22.090 --> 00:04:27.790
they wait until Precalculus to
teach this, because a smart

00:04:27.790 --> 00:04:30.210
middle-schooler could, I think,
easily handle this.

00:04:30.210 --> 00:04:32.860
Not to make you feel bad if
you're not getting it, just to

00:04:32.860 --> 00:04:35.130
give you confidence that you
will get it, and you'll realize

00:04:35.130 --> 00:04:38.450
that it is very simple.

00:04:38.450 --> 00:04:43.520
OK, so let's figure out
what the cosine-- and

00:04:43.520 --> 00:04:46.010
cos is short for cosine.

00:04:46.010 --> 00:04:50.530
I'll write it out, but I'm
sure you've seen it before.

00:04:50.530 --> 00:04:54.500
So what is the cosine of theta?

00:04:54.500 --> 00:04:58.390
Well, we go back to our
mnemonic: SOHCAHTOA.

00:04:58.390 --> 00:05:01.770
Well, cosine is the CAH, right?

00:05:01.770 --> 00:05:05.570
And that tells us that
cosine of theta is equal to

00:05:05.570 --> 00:05:06.680
adjacent over hypotenuse.

00:05:11.730 --> 00:05:14.780
Well, once again, let's figure
out what the adjacent side is.

00:05:14.780 --> 00:05:20.000
Well, the adjacent side-- this
side was the opposite side,

00:05:20.000 --> 00:05:22.390
right, because it's
opposite the angle.

00:05:22.390 --> 00:05:24.480
This side is the hypotenuse,
because it's the longest side,

00:05:24.480 --> 00:05:27.250
and then, you could, just by
deductive reasoning, but also

00:05:27.250 --> 00:05:29.740
just by looking at it, you see
that this side right here, the

00:05:29.740 --> 00:05:32.080
side of length 3, is adjacent
to the angle, right?

00:05:32.080 --> 00:05:34.600
Adjacent means right beside it.

00:05:34.600 --> 00:05:36.950
So that's the adjacent side.

00:05:36.950 --> 00:05:40.110
We already figured out that
the hypotenuse is that

00:05:40.110 --> 00:05:42.400
side that I wrote in pink.

00:05:42.400 --> 00:05:46.770
So we're ready to figure out
what cosine of theta equals.

00:05:46.770 --> 00:05:54.610
Cosine of theta is equal to the
adjacent side, that's 3, over

00:05:54.610 --> 00:05:59.060
the hypotenuse, which
is this pink side, 5.

00:05:59.060 --> 00:06:00.910
Pretty straightforward,
isn't it?

00:06:00.910 --> 00:06:03.320
Let's do another one.

00:06:03.320 --> 00:06:05.050
OK, I don't want to
erase the whole thing.

00:06:05.050 --> 00:06:07.670
I just want to erase part of
the page because I want to

00:06:07.670 --> 00:06:08.773
keep using this triangle.

00:06:18.310 --> 00:06:19.920
OK, one left.

00:06:19.920 --> 00:06:21.860
The TOA.

00:06:21.860 --> 00:06:26.150
So if you remember what I said
a little while ago-- well,

00:06:26.150 --> 00:06:27.000
we'll figure it out.

00:06:27.000 --> 00:06:31.870
But what is the-- oh,
look how big that is.

00:06:31.870 --> 00:06:37.140
What is the tan of theta,
or the tangent of theta?

00:06:37.140 --> 00:06:39.740
Well, let's go back
to our mnemonic.

00:06:39.740 --> 00:06:41.320
TOA, right?

00:06:41.320 --> 00:06:43.510
TOA is for tangent,
or t for tangent.

00:06:43.510 --> 00:06:47.630
So it tells us that tangent is
the opposite over the adjacent.

00:06:47.630 --> 00:06:54.220
So tan of theta is equal to
opposite over adjacent.

00:06:54.220 --> 00:06:56.960
Well, that equals-- what
was the opposite side?

00:06:56.960 --> 00:07:00.360
Right, the opposite side was 4.

00:07:00.360 --> 00:07:01.130
And what was the adjacent side?

00:07:01.130 --> 00:07:01.780
Well, we just saw that.

00:07:01.780 --> 00:07:03.520
It was 3.

00:07:03.520 --> 00:07:08.890
So the tangent of
this angle is 4/3.

00:07:08.890 --> 00:07:10.910
Now let's do another
angle on this.

00:07:10.910 --> 00:07:17.180
Let's call this angle
here-- I don't know.

00:07:17.180 --> 00:07:18.990
Let's call it x.

00:07:18.990 --> 00:07:21.700
I don't know any
other Greek letters.

00:07:21.700 --> 00:07:23.840
Let's call that angle x.

00:07:23.840 --> 00:07:27.570
So if we want to figure out the
tan of x, let's see if it's

00:07:27.570 --> 00:07:29.770
the same as the tan of theta.

00:07:29.770 --> 00:07:31.620
The tan of x.

00:07:31.620 --> 00:07:33.780
Well, now what's
the opposite side?

00:07:33.780 --> 00:07:36.340
Well, now the opposite side
is the white side, right?

00:07:36.340 --> 00:07:39.980
Because opposite this
angle is the 3 side.

00:07:39.980 --> 00:07:43.890
So we see here tan is opposite
over adjacent, so opposite is

00:07:43.890 --> 00:07:48.130
3, and then adjacent is 4.

00:07:48.130 --> 00:07:49.910
This is interesting.

00:07:49.910 --> 00:07:54.230
The tangent of this angle
is the inverse of the

00:07:54.230 --> 00:07:54.830
tangent to that angle.

00:07:54.830 --> 00:07:56.520
I don't want to confuse you too
much, but I just want to show

00:07:56.520 --> 00:07:59.930
you that when you take the trig
functions, it matters which

00:07:59.930 --> 00:08:03.050
angle of the right angle you're
taking the functions of.

00:08:03.050 --> 00:08:05.310
And you might be saying, well,
this is all good and well,

00:08:05.310 --> 00:08:06.810
Sal, but what use is this?

00:08:06.810 --> 00:08:10.990
Well, we'll later show you that
if you have some of the

00:08:10.990 --> 00:08:13.390
information, so you know an
angle, and you know a side, or

00:08:13.390 --> 00:08:15.940
you know a couple of sides, you
can figure out-- and if you

00:08:15.940 --> 00:08:20.920
have either a slide ruler or a
trig table or a good

00:08:20.920 --> 00:08:24.320
calculator, you can figure
out-- given the sides of a

00:08:24.320 --> 00:08:26.510
triangle, you can figure out
the angles, or given an angle

00:08:26.510 --> 00:08:28.220
and a side, you could
figure out other sides.

00:08:28.220 --> 00:08:31.010
And we're actually going to
do that in the next module.

00:08:31.010 --> 00:08:33.020
So, hopefully, this gives you a
little bit of an introduction.

00:08:33.020 --> 00:08:36.900
I'm almost out of time on the
YouTube 10-minute limit, so I'm

00:08:36.900 --> 00:08:39.130
going to wait to do a couple
more examples in

00:08:39.130 --> 00:08:40.840
the next video.

00:08:40.840 --> 00:08:42.900
See you in the next
presentation.

00:08:42.900 --> 00:08:44.200
Bye!


Quadratic inequalities (visual explanation)

WEBVTT
Kind: captions
Language: en

00:00:01.850 --> 00:00:05.900
Welcome to the presentation
on quadratic inequalities.

00:00:05.900 --> 00:00:09.660
Before we get to quadratic
inequalities, let's just start

00:00:09.660 --> 00:00:11.945
graphing some functions and
interpret them and then we'll

00:00:11.945 --> 00:00:13.880
slowly move to the
inequalities.

00:00:13.880 --> 00:00:26.360
Let's say I had f of x is equal
to x squared plus x minus 6.

00:00:26.360 --> 00:00:29.210
Well, if we wanted to figure
out where this function

00:00:29.210 --> 00:00:33.070
intersects the x-axis or the
roots of it, we learned in our

00:00:33.070 --> 00:00:35.210
factoring quadratics that we
could just set f of x

00:00:35.210 --> 00:00:36.735
is equal to 0, right?

00:00:36.735 --> 00:00:39.460
Because f of x equals 0 when
you're intersecting the x-axis.

00:00:39.460 --> 00:00:46.990
So you would say x squared
plus x minus 6 is equal to 0.

00:00:46.990 --> 00:00:48.800
And you just factor
this quadratic.

00:00:48.800 --> 00:00:57.470
x plus 3 times x
minus 2 equals 0.

00:00:57.470 --> 00:01:03.270
And you would learn that the
roots of this quadratic

00:01:03.270 --> 00:01:11.600
function are x is equal to
minus 3, and x is equal to 2.

00:01:11.600 --> 00:01:13.970
How would we visualize this?

00:01:13.970 --> 00:01:16.970
Well let's draw this
quadratic function.

00:01:21.380 --> 00:01:24.510
Those are my very uneven lines.

00:01:24.510 --> 00:01:27.200
So the roots are x is
equal to negative 3.

00:01:27.200 --> 00:01:33.180
So this is, right here, x is at
minus 3y0 -- by definition one

00:01:33.180 --> 00:01:36.790
of the roots is where
f of x is equal to 0.

00:01:36.790 --> 00:01:41.370
So the y, or the f of
x axis here is 0.

00:01:41.370 --> 00:01:42.600
The coordinate is 0.

00:01:42.600 --> 00:01:47.130
And this point here
is 2 comma 0.

00:01:47.130 --> 00:01:53.360
Once again, this is the x-axis,
and this is the f of x-axis.

00:01:53.360 --> 00:01:56.270
We also know that the y
intercept is minus 6.

00:01:56.270 --> 00:01:57.800
This isn't the vertex,
this is the y intercept.

00:01:57.800 --> 00:02:03.710
And that the graph is going to
look something like this -- not

00:02:03.710 --> 00:02:05.640
as bumpy as what I'm drawing,
which I think you get the

00:02:05.640 --> 00:02:10.180
general idea if you've ever
seen a clean parabola.

00:02:10.180 --> 00:02:16.200
It looks like that with x minus
3 here, and x is 2 here.

00:02:16.200 --> 00:02:17.080
Pretty straightforward.

00:02:17.080 --> 00:02:19.480
We figured out the roots, we
figured out what it looks like.

00:02:19.480 --> 00:02:22.170
Now what if we, instead of
wanting to know where f of x is

00:02:22.170 --> 00:02:24.960
equal to 0, which is these two
points, what if we wanted

00:02:24.960 --> 00:02:29.290
to know where f of x
is greater than 0?

00:02:29.290 --> 00:02:33.200
What x values make f
of x greater than 0?

00:02:33.200 --> 00:02:36.050
Or another way of saying
it, what values make

00:02:36.050 --> 00:02:37.410
the statement true?

00:02:37.410 --> 00:02:42.740
x squared plus x minus 6
is greater than 0, Right,

00:02:42.740 --> 00:02:44.730
this is just f of x.

00:02:44.730 --> 00:02:49.470
Well if we look at the graph,
when is f of x greater than 0?

00:02:49.470 --> 00:02:52.010
Well this is the f of x
axis, and when are we

00:02:52.010 --> 00:02:52.910
in positive territory?

00:02:52.910 --> 00:02:55.360
Well f of x is greater than
0 here -- let me draw that

00:02:55.360 --> 00:03:00.890
another color -- is greater
than 0 here, right?

00:03:00.890 --> 00:03:04.980
Because it's above the x-axis.

00:03:04.980 --> 00:03:06.830
And f of x is greater
than 0 here.

00:03:11.560 --> 00:03:17.070
So just visually looking at it,
what x values make this true?

00:03:17.070 --> 00:03:23.580
Well, this is true whenever x
is less than minus 3, right, or

00:03:23.580 --> 00:03:26.440
whenever x is greater than 2.

00:03:26.440 --> 00:03:31.560
Because when x is greater than
2, f of x is greater than 0,

00:03:31.560 --> 00:03:35.910
and when x is less than
negative 3, f of x

00:03:35.910 --> 00:03:37.430
is greater than 0.

00:03:37.430 --> 00:03:41.460
So we would say the solution to
this quadratic inequality, and

00:03:41.460 --> 00:03:46.910
we pretty much solved this
visually, is x is less than

00:03:46.910 --> 00:03:52.970
minus 3, or x is
greater than 2.

00:03:52.970 --> 00:03:53.890
And you could test it out.

00:03:53.890 --> 00:03:56.990
You could try out the number
minus 4, and you should get f

00:03:56.990 --> 00:03:58.800
of x being greater than 0.

00:03:58.800 --> 00:04:00.890
You could try it out here.

00:04:00.890 --> 00:04:04.110
Or you could try the number 3
and make sure that this works.

00:04:04.110 --> 00:04:06.820
And you can just make sure
that, you could, for example,

00:04:06.820 --> 00:04:10.410
try out the number 0 and make
sure that 0 doesn't work,

00:04:10.410 --> 00:04:12.790
right, because 0 is
between the two roots.

00:04:12.790 --> 00:04:15.030
It actually turns out that
when x is equal to 0, f

00:04:15.030 --> 00:04:19.040
of x is minus 6, which is
definitely less than 0.

00:04:19.040 --> 00:04:22.400
So I think this will give you a
visual intuition of what this

00:04:22.400 --> 00:04:24.000
quadratic inequality means.

00:04:24.000 --> 00:04:26.510
Now with that visual intuition
in the back of your mind, let's

00:04:26.510 --> 00:04:29.090
do some more problems and maybe
we won't have to go through the

00:04:29.090 --> 00:04:32.600
exercise of drawing it, but
maybe I will draw it just to

00:04:32.600 --> 00:04:35.190
make sure that the
point hits home.

00:04:35.190 --> 00:04:37.140
Let me give you a slightly
trickier problem.

00:04:37.140 --> 00:04:49.120
Let's say I had minus x squared
minus 3x plus 28, let me

00:04:49.120 --> 00:04:52.190
say, is greater than 0.

00:04:52.190 --> 00:04:53.600
Well I want to get rid of
this negative sign in

00:04:53.600 --> 00:04:54.350
front of the x squared.

00:04:54.350 --> 00:04:56.420
I just don't like it there
because it makes it look

00:04:56.420 --> 00:04:58.080
more confusing to factor.

00:04:58.080 --> 00:05:00.140
I'm going to multiply
everything by negative 1.

00:05:00.140 --> 00:05:00.780
Both sides.

00:05:00.780 --> 00:05:08.210
I get x squared plus 3x minus
28, and when you multiply or

00:05:08.210 --> 00:05:10.160
divide by a negative, with any
inequality you have

00:05:10.160 --> 00:05:11.440
to swap the sign.

00:05:11.440 --> 00:05:16.860
So this is now going
to be less than 0.

00:05:16.860 --> 00:05:25.130
And if we were to factor this,
we get x plus 7 times x

00:05:25.130 --> 00:05:29.880
minus 4 is less than 0.

00:05:29.880 --> 00:05:32.460
So if this was equal to 0, we
would know that the two roots

00:05:32.460 --> 00:05:37.400
of this function -- let's
define the function f of x --

00:05:37.400 --> 00:05:40.540
let's define the function as f
of x is equal to -- well we can

00:05:40.540 --> 00:05:42.670
define it as this or this
because they're the same thing.

00:05:42.670 --> 00:05:47.080
But for simplicity let's define
it as x plus 7 times x minus 4.

00:05:47.080 --> 00:05:49.390
That's f of x, right?

00:05:49.390 --> 00:05:53.260
Well, after factoring it, we
know that the roots of this,

00:05:53.260 --> 00:06:05.600
the roots are x is equal to
minus 7, and x is equal to 4.

00:06:05.600 --> 00:06:07.900
Now what we want to know
is what x values make

00:06:07.900 --> 00:06:10.000
this inequality true?

00:06:10.000 --> 00:06:12.060
If this was any
equality we'd be done.

00:06:12.060 --> 00:06:14.650
But we want to know what
makes this inequality true.

00:06:14.650 --> 00:06:17.600
I'll give you a little bit of a
trick, it's always going to be

00:06:17.600 --> 00:06:21.160
the numbers in between the
two roots or outside

00:06:21.160 --> 00:06:23.120
of the two roots.

00:06:23.120 --> 00:06:25.810
So what I do whenever I'm doing
this on a test or something, I

00:06:25.810 --> 00:06:28.510
just test numbers that are
either between the roots or

00:06:28.510 --> 00:06:30.610
outside of the two roots.

00:06:30.610 --> 00:06:34.660
So let's pick a number that's
between x equals minus

00:06:34.660 --> 00:06:36.360
7 and x equals 4.

00:06:36.360 --> 00:06:41.560
Well let's try x equals 0.

00:06:41.560 --> 00:06:46.660
Well, f of 0 is equal to -- we
could do it right here -- f of

00:06:46.660 --> 00:06:57.360
0 is 0 plus 7 times 0 minus 4
is just 7 times minus

00:06:57.360 --> 00:07:00.280
4, which is minus 28.

00:07:00.280 --> 00:07:04.100
So f of 0 is minus 28.

00:07:04.100 --> 00:07:08.800
Now is this -- this is the
function we're working with

00:07:08.800 --> 00:07:11.790
-- is this less than 0?

00:07:11.790 --> 00:07:13.140
Well yeah, it is.

00:07:13.140 --> 00:07:16.090
So it actually turns that a
number, an x value between

00:07:16.090 --> 00:07:17.470
the two roots works.

00:07:17.470 --> 00:07:19.940
So actually I immediately
know that the answer here

00:07:19.940 --> 00:07:23.340
is all of the x's that are
between the two roots.

00:07:23.340 --> 00:07:29.170
So we could say that the
solution to this is

00:07:29.170 --> 00:07:34.720
minus 7 is less than x
which is less than 4.

00:07:34.720 --> 00:07:35.460
Because now the other way.

00:07:35.460 --> 00:07:38.240
You could have tried a number
that's outside of the roots,

00:07:38.240 --> 00:07:41.460
either less than minus 7 or
greater than 4 and

00:07:41.460 --> 00:07:42.650
have tried it out.

00:07:42.650 --> 00:07:45.820
Let's say if you
had tried out 5.

00:07:45.820 --> 00:07:48.140
Try x equals 5.

00:07:48.140 --> 00:07:55.780
Well then f of 5 would
be 12 times 1, right,

00:07:55.780 --> 00:07:58.690
which is equal to 12.

00:07:58.690 --> 00:07:59.570
f of 5 is 12.

00:07:59.570 --> 00:08:02.300
Is that less than 0?

00:08:02.300 --> 00:08:03.110
No.

00:08:03.110 --> 00:08:04.060
So that wouldn't have worked.

00:08:04.060 --> 00:08:06.000
So once again, that gives
us a confidence that we

00:08:06.000 --> 00:08:07.260
got the right interval.

00:08:07.260 --> 00:08:11.750
And if we wanted to think about
this visually, because we got

00:08:11.750 --> 00:08:14.920
this answer, when you do it
visually it actually makes, I

00:08:14.920 --> 00:08:18.640
think, a lot of sense,
but maybe I'm biased.

00:08:26.180 --> 00:08:28.770
If you look at it visually
it looks like this.

00:08:35.130 --> 00:08:40.890
If you drive visually and this
is the parabola, this is f of

00:08:40.890 --> 00:08:52.580
x, the roots here are minus 7,
0 and 4, 0, we're saying that

00:08:52.580 --> 00:08:56.230
for all x values between these
two numbers, f of

00:08:56.230 --> 00:08:57.390
x is less than 0.

00:08:57.390 --> 00:08:59.770
And that makes sense, because
when is f of x less than 0?

00:08:59.770 --> 00:09:02.420
Well this is the
graph of f of x.

00:09:06.110 --> 00:09:07.570
And when is f of x less than 0?

00:09:07.570 --> 00:09:08.480
Right here.

00:09:08.480 --> 00:09:10.880
So what x values give us that?

00:09:10.880 --> 00:09:14.140
Well the x values that give
us that are right here.

00:09:14.140 --> 00:09:15.410
I hope I'm not confusing
you too much with

00:09:15.410 --> 00:09:16.960
these visual graphs.

00:09:16.960 --> 00:09:19.140
And you're probably saying,
well how do I know

00:09:19.140 --> 00:09:20.180
I don't include 0?

00:09:20.180 --> 00:09:23.116
Well you could try it out, but
if you -- oh, well how come

00:09:23.116 --> 00:09:24.690
I don't include the roots?

00:09:24.690 --> 00:09:28.030
Well at the roots, f
of x is equal to 0.

00:09:28.030 --> 00:09:31.640
So if this was this, if this
was less than or equal to 0,

00:09:31.640 --> 00:09:36.290
then the answer would be
negative 7 is less than

00:09:36.290 --> 00:09:39.230
or equal to x is less
than or equal to 4.

00:09:39.230 --> 00:09:40.620
I hope that gives you a sense.

00:09:40.620 --> 00:09:42.460
You pretty much just have to
try number in between the

00:09:42.460 --> 00:09:45.250
roots, and try number outside
of the roots, and that tells

00:09:45.250 --> 00:09:49.300
you what interval will
make the inequality true.

00:09:49.300 --> 00:09:51.640
I'll see you in the
next presentation.


Introduction to Logarithms

WEBVTT
Kind: captions
Language: en

00:00:01.040 --> 00:00:03.750
Welcome to the logarithm
presentation.

00:00:03.750 --> 00:00:06.000
Let me write down the word
logarithm just because it is

00:00:06.000 --> 00:00:08.930
another strange and unusual
word like hypotenuse and it's

00:00:08.930 --> 00:00:10.700
good to at least, see it once.

00:00:10.700 --> 00:00:14.520
Let me get the pen
tool working.

00:00:14.520 --> 00:00:14.980
Logarithm.

00:00:19.680 --> 00:00:24.620
This is one of my most
misspelled words.

00:00:24.620 --> 00:00:27.300
I went to MIT and actually one
of the a cappella groups there,

00:00:27.300 --> 00:00:30.050
they were called
the Logarhythms.

00:00:30.050 --> 00:00:31.710
Like rhythm, like music.

00:00:31.710 --> 00:00:33.990
But anyway, I'm digressing.

00:00:33.990 --> 00:00:35.730
So what is a logarithm?

00:00:35.730 --> 00:00:37.790
Well, the easiest way to
explain what a logarithm is is

00:00:37.790 --> 00:00:41.370
to have first-- I guess it's
just to say it's the inverse of

00:00:41.370 --> 00:00:43.140
taking the exponent
of something.

00:00:43.140 --> 00:00:44.250
Let me explain.

00:00:44.250 --> 00:00:49.900
If I said that 2 to the third
power-- well, we know that

00:00:49.900 --> 00:00:52.120
from the exponent modules.

00:00:52.120 --> 00:00:54.770
2 the third power, well
that's equal to 8.

00:00:54.770 --> 00:00:56.830
And once again, this
is a 2, it's not a z.

00:00:56.830 --> 00:00:59.530
2 to the third power is 8, so
it actually turns out that

00:00:59.530 --> 00:01:04.580
log-- and log is short
for the word logarithm.

00:01:04.580 --> 00:01:13.200
Log base 2 of eight
is equal to 3.

00:01:13.200 --> 00:01:15.130
I think when you look at that
you're trying to say oh,

00:01:15.130 --> 00:01:17.210
that's trying to make
a little bit of sense.

00:01:17.210 --> 00:01:22.730
What this says, if I were to
ask you what log base 2 of

00:01:22.730 --> 00:01:27.770
8 is, this says 2 to the
what power is equal to 8?

00:01:27.770 --> 00:01:31.060
So the answer to a logarithm--
you can say the answer to this

00:01:31.060 --> 00:01:33.880
logarithm expression, or if you
evaluate this logarithm

00:01:33.880 --> 00:01:36.440
expression, you should get a
number that is really the

00:01:36.440 --> 00:01:42.320
exponent that you would have
to raised 2 to to get 8.

00:01:42.320 --> 00:01:43.970
And once again, that's 3.

00:01:43.970 --> 00:01:47.560
Let's do a couple more examples
and I think you might get it.

00:01:47.560 --> 00:01:54.690
If I were to say log--
what happened to my pen?

00:01:54.690 --> 00:02:03.710
log base 4 of 64 is equal to x.

00:02:03.710 --> 00:02:09.820
Another way of rewriting this
exact equation is to say 4 to

00:02:09.820 --> 00:02:14.230
the x power is equal to 64.

00:02:14.230 --> 00:02:16.790
Or another way to think
about it, 4 to what

00:02:16.790 --> 00:02:18.290
power is equal to 64?

00:02:18.290 --> 00:02:20.960
Well, we know that 4 to
the third power is 64.

00:02:20.960 --> 00:02:25.950
So we know that in this
case, this equals 3.

00:02:25.950 --> 00:02:36.120
So log base 4 of
64 is equal to 3.

00:02:36.120 --> 00:02:39.320
Let me do a bunch of more
examples and I think the more

00:02:39.320 --> 00:02:42.260
examples you see, it'll
start to make some sense.

00:02:42.260 --> 00:02:45.720
Logarithms are a simple idea,
but I think they can get

00:02:45.720 --> 00:02:48.980
confusing because they're the
inverse of exponentiation,

00:02:48.980 --> 00:02:52.390
which is sometimes itself,
a confusing concept.

00:02:52.390 --> 00:03:05.780
So what is log base 10 of
let's say, 1,000,000.

00:03:05.780 --> 00:03:08.540
Put some commas
here to make sure.

00:03:08.540 --> 00:03:12.490
So this equals question mark.

00:03:12.490 --> 00:03:15.960
Well, all we have to ask
ourselves is 10 to what power

00:03:15.960 --> 00:03:17.770
is equal to 1,000,000.

00:03:17.770 --> 00:03:22.060
And 10 to any power is actually
equal to 1 followed by the

00:03:22.060 --> 00:03:24.900
power of-- if you say 10 of
the fifth power, that's equal

00:03:24.900 --> 00:03:26.930
to 1 followed by five 0's.

00:03:26.930 --> 00:03:29.550
So if we have 1 followed by six
0's this is the same thing

00:03:29.550 --> 00:03:31.350
as 10 to the sixth power.

00:03:31.350 --> 00:03:34.590
So 10 to the sixth power
is equal to 1,000,000.

00:03:34.590 --> 00:03:47.170
So since 10 to the sixth power
is equal to 1,000,000 log base

00:03:47.170 --> 00:03:54.060
10 of 1,000,000 is equal to 6.

00:03:54.060 --> 00:03:57.740
Just remember, this 6 is an
exponent that we raise 10

00:03:57.740 --> 00:03:59.640
to to get the 1,000,000.

00:03:59.640 --> 00:04:01.460
I know I'm saying this in a
hundred different ways and

00:04:01.460 --> 00:04:04.200
hopefully, one or two of these
million different ways that I'm

00:04:04.200 --> 00:04:06.310
explaining it actually
will make sense.

00:04:06.310 --> 00:04:08.830
Let's do some more.

00:04:08.830 --> 00:04:12.570
Actually, I'll do even a
slightly confusing one.

00:04:12.570 --> 00:04:19.790
log base 1/2 of 1/8.

00:04:23.252 --> 00:04:25.820
Let's say that that equals x.

00:04:25.820 --> 00:04:27.760
So let's just remind
ourselves, that's just

00:04:27.760 --> 00:04:32.050
like saying 1/2-- whoops.

00:04:32.050 --> 00:04:32.670
1/2.

00:04:32.670 --> 00:04:34.280
That's supposed to
be parentheses.

00:04:34.280 --> 00:04:37.020
To the x power is equal to 1/8.

00:04:40.500 --> 00:04:44.490
Well, we know that 1/2 to the
third power is equal to 1/8.

00:04:44.490 --> 00:04:54.766
So log base 1/2 of
1/8 is equal to 3.

00:04:54.766 --> 00:04:56.275
Let me do a bunch
of more problems.

00:05:00.850 --> 00:05:02.290
Actually, let me mix
it up a little bit.

00:05:02.290 --> 00:05:13.680
Let's say that log base
x of 27 is equal to 3.

00:05:13.680 --> 00:05:16.480
What's x?

00:05:16.480 --> 00:05:20.520
Well, just like what we did
before, this says that x to the

00:05:20.520 --> 00:05:22.790
third power is equal to 27.

00:05:25.350 --> 00:05:34.060
Or x is equal to the
cubed root of 27.

00:05:34.060 --> 00:05:36.170
And all that means is that
there's some number times

00:05:36.170 --> 00:05:38.160
itself three times
that equals 27.

00:05:38.160 --> 00:05:39.740
And I think at this point
you know that that

00:05:39.740 --> 00:05:41.370
number would be 3.

00:05:41.370 --> 00:05:43.150
x equals 3.

00:05:43.150 --> 00:05:51.060
So we could write log base
3 of 27 is equal to 3.

00:05:54.100 --> 00:05:55.830
Let me think of
another example.

00:05:55.830 --> 00:05:57.750
I'm only doing relatively small
numbers because I don't have

00:05:57.750 --> 00:06:00.050
a calculator with me and I
have to do them in my head.

00:06:00.050 --> 00:06:07.710
So what is log-- let
me think about this.

00:06:07.710 --> 00:06:14.440
What is log base 100 of 1?

00:06:14.440 --> 00:06:16.690
This is a trick problem.

00:06:16.690 --> 00:06:18.380
So once again, let's just
say that this is equal

00:06:18.380 --> 00:06:22.440
to question mark.

00:06:22.440 --> 00:06:25.330
So remember this is log
base 100 hundred of 1.

00:06:25.330 --> 00:06:30.250
So this says 100 to the
question mark power

00:06:30.250 --> 00:06:32.720
is equal to 1.

00:06:32.720 --> 00:06:34.960
Well, what do we have to
raise-- if we have any number

00:06:34.960 --> 00:06:37.530
and we raise it to what
power, when do we get 1?

00:06:37.530 --> 00:06:39.790
Well, if you remember from the
exponent rules, or actually not

00:06:39.790 --> 00:06:42.470
the exponent rules, from the
exponent modules, anything to

00:06:42.470 --> 00:06:44.880
the 0-th power is equal to 1.

00:06:44.880 --> 00:06:51.330
So we could say 100 to
the 0 power equals 1.

00:06:51.330 --> 00:07:00.410
So we could say log base 100
hundred of 1 is equal to 0

00:07:00.410 --> 00:07:04.930
because 100 to the 0-th
power is equal to 1.

00:07:04.930 --> 00:07:07.860
Let me ask another question.

00:07:07.860 --> 00:07:16.120
What if I were to ask you
log, let's say base 2 of 0?

00:07:16.120 --> 00:07:18.060
So what is that equal to?

00:07:18.060 --> 00:07:20.330
Well, what I'm asking you,
I'm saying 2-- let's

00:07:20.330 --> 00:07:22.160
say that equals x.

00:07:22.160 --> 00:07:25.770
2 to some power x
is equal to 0.

00:07:25.770 --> 00:07:28.430
So what is x?

00:07:28.430 --> 00:07:30.580
Well, is there anything
that I can raise 2 to

00:07:30.580 --> 00:07:32.850
the power of to get 0?

00:07:32.850 --> 00:07:33.790
No.

00:07:33.790 --> 00:07:35.830
So this is undefined.

00:07:35.830 --> 00:07:38.710
Undefined or no solution.

00:07:38.710 --> 00:07:41.990
There's no number that
I can raise 2 to the

00:07:41.990 --> 00:07:44.440
power of and get 0.

00:07:44.440 --> 00:07:51.320
Similarly if I were to
ask you log base 3 of

00:07:51.320 --> 00:07:54.210
let's say, negative 1.

00:07:54.210 --> 00:07:56.810
And we're assuming we're
dealing with the real numbers,

00:07:56.810 --> 00:07:58.630
which are most of the numbers
that I think at this point

00:07:58.630 --> 00:08:00.440
you have dealt with.

00:08:00.440 --> 00:08:02.660
There's nothing I can raise
three 3 to the power of to

00:08:02.660 --> 00:08:04.240
get a negative number,
so this is undefined.

00:08:10.510 --> 00:08:14.620
So as long as you have a
positive base here, this

00:08:14.620 --> 00:08:21.380
number, in order to be defined,
has to be greater than-- well,

00:08:21.380 --> 00:08:23.680
it has to be greater
than or equal-- no.

00:08:23.680 --> 00:08:25.590
It has to be greater than 0.

00:08:25.590 --> 00:08:26.210
Not equal to.

00:08:26.210 --> 00:08:28.980
It cannot be 0 and it
cannot be negative.

00:08:28.980 --> 00:08:30.020
Let's do a couple
more problems.

00:08:30.020 --> 00:08:32.350
I think I have another
minute and a half.

00:08:32.350 --> 00:08:36.390
You're already prepared to do
the level 1 logarithms module,

00:08:36.390 --> 00:08:39.240
but let's do a couple of more.

00:08:39.240 --> 00:08:47.130
What is log base 8-- I'm
going to do a slightly

00:08:47.130 --> 00:08:52.510
tricky one-- of 1/64.

00:08:52.510 --> 00:08:53.940
Interesting.

00:08:53.940 --> 00:09:00.010
We know that log base 8 of
64 would equal 2, right?

00:09:00.010 --> 00:09:02.800
Because 8 squared
is equal to 64.

00:09:02.800 --> 00:09:06.240
But 8 to what power
equals 1/64?

00:09:06.240 --> 00:09:09.320
Well, we learned from the
negative exponent module that

00:09:09.320 --> 00:09:13.030
that is equal to negative 2.

00:09:13.030 --> 00:09:17.610
If you remember, 8 to the
negative 2 power is the same

00:09:17.610 --> 00:09:20.230
thing as 1/8 to the 2 power.

00:09:20.230 --> 00:09:24.960
8 squared, which
is equal to 1/64.

00:09:24.960 --> 00:09:26.960
Interesting.

00:09:26.960 --> 00:09:29.590
I'll leave this for
you to think about.

00:09:29.590 --> 00:09:31.590
When you take the inverse of
whatever you're taking the

00:09:31.590 --> 00:09:33.830
logarithm of, it turns
the answer negative.

00:09:33.830 --> 00:09:36.260
And we'll do a lot more
logarithm problems and explore

00:09:36.260 --> 00:09:38.880
a lot more of the properties of
logarithms in future modules.

00:09:38.880 --> 00:09:43.120
But I think you're ready at
this point to do the level 1

00:09:43.120 --> 00:09:45.770
logarithm set of exercises.

00:09:45.770 --> 00:09:47.600
See you in the next module.


30-60-90 Triangles II

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.250
Let's continue with the
30, 60, 90 triangles.

00:00:06.480 --> 00:00:09.640
So just review what we just
learned, or hopefully learned--

00:00:09.640 --> 00:00:15.910
at minimum what we just saw,
--is if we have a 30, 60, 90 --

00:00:15.910 --> 00:00:18.380
and once again, remember: this
is only applies to 30, 60, 90

00:00:18.380 --> 00:00:26.560
triangles --and if I were to
say the hypotenuse is of length

00:00:26.560 --> 00:00:31.320
h, we learned that the side
opposite the 30-degree angle,

00:00:31.320 --> 00:00:34.340
and this is the shortest side
of the triangle, is going to be

00:00:34.340 --> 00:00:37.270
h over 2, or 1/2 times
the hypotenuse.

00:00:37.270 --> 00:00:40.240
And we also learned that the
longer side, or the side

00:00:40.240 --> 00:00:42.810
opposite the 60-degree side,
is equal to the square

00:00:42.810 --> 00:00:46.840
root of 3 over 2 times h.

00:00:46.840 --> 00:00:50.640
So let's do a problem where
we use this information.

00:00:50.640 --> 00:00:56.370
Let's say I had this
triangle right here.

00:00:56.370 --> 00:00:58.010
It's a 90-degree triangle;
let's say that this

00:00:58.010 --> 00:01:00.690
is 30 degrees.

00:01:00.690 --> 00:01:02.750
And we could also figure out
obviously if that's 30, this

00:01:02.750 --> 00:01:07.040
is 90, that this is
also 60 degrees.

00:01:07.040 --> 00:01:10.510
And let's say that the
hypotenuse is 12.

00:01:10.510 --> 00:01:12.300
The length is 12 and we know
that this is the hypotenuse

00:01:12.300 --> 00:01:14.980
because it's opposite
the right angle.

00:01:14.980 --> 00:01:18.630
What is the side right here?

00:01:18.630 --> 00:01:21.840
Well, is the side opposite the
60-degree angle, or is it

00:01:21.840 --> 00:01:23.910
opposite the 30-degree angle?

00:01:23.910 --> 00:01:26.460
It's the 30-degree angle
that opens into it, right?

00:01:26.460 --> 00:01:28.650
I drew this triangle a little
bit different on purpose.

00:01:28.650 --> 00:01:32.050
The 30-degree angle opens up
into this side, and it's

00:01:32.050 --> 00:01:34.060
also the shortest side.

00:01:34.060 --> 00:01:37.360
We learned that the side
opposite the 30-degree angle is

00:01:37.360 --> 00:01:40.680
half the hypotenuse, so the
hypotenuse is 12;

00:01:40.680 --> 00:01:42.860
this would be 6.

00:01:42.860 --> 00:01:46.310
And this side, which is
opposite the 60-degree side, is

00:01:46.310 --> 00:01:49.730
equal to the square root of 3
over 2 times the hypotenuse.

00:01:49.730 --> 00:01:54.690
So it's the square root of 3
over 2 times 12, or it's just

00:01:54.690 --> 00:01:58.150
equal to 6 square roots of 3.

00:01:58.150 --> 00:02:01.150
Another interesting thing is,
of course, the longer

00:02:01.150 --> 00:02:04.600
non-hypotenuse side is square
root of 3 times longer

00:02:04.600 --> 00:02:06.270
than the short side.

00:02:06.270 --> 00:02:07.810
I don't confuse you too much.

00:02:07.810 --> 00:02:08.660
Let's do another one.

00:02:15.010 --> 00:02:20.800
Let's say this is 30 degrees--
it's our right triangle --and I

00:02:20.800 --> 00:02:28.390
were to tell you that this side
right here is 5, what is

00:02:28.390 --> 00:02:29.900
the length of this side?

00:02:33.970 --> 00:02:35.750
Well first of all let's
figure out what we have.

00:02:35.750 --> 00:02:37.390
5 is which side?

00:02:37.390 --> 00:02:39.540
So if this is 30 degrees,
we know that this is

00:02:39.540 --> 00:02:41.990
going to be 60 degrees.

00:02:41.990 --> 00:02:47.010
So 5 is opposite the 60-degree
side, and x is the hypotenuse.

00:02:47.010 --> 00:02:49.840
Since x is opposite the
90-degree side, it's also

00:02:49.840 --> 00:02:53.010
the longest side of
the right triangle.

00:02:53.010 --> 00:02:57.910
So we know from our formula
that 5 is equal to the square

00:02:57.910 --> 00:03:00.940
root of 3 over 2 times the
hypotenuse, which in

00:03:00.940 --> 00:03:02.850
this example is x.

00:03:02.850 --> 00:03:04.240
And now we just solve for x.

00:03:04.240 --> 00:03:06.770
We can multiply both
sides by the inverse

00:03:06.770 --> 00:03:07.865
of this coefficient.

00:03:07.865 --> 00:03:19.710
So if you just multiply 2 times
the square root of 3-- can

00:03:19.710 --> 00:03:25.030
ignore this --we get 10 over
the square root of three here.

00:03:25.030 --> 00:03:27.140
And, of course, this 2
cancels out with this 2.

00:03:27.140 --> 00:03:28.667
This square root of 3 cancels
out this square root

00:03:28.667 --> 00:03:30.970
of 3 is equal to x.

00:03:30.970 --> 00:03:33.510
And now if you watched the last
couple of presentations, you

00:03:33.510 --> 00:03:36.690
realize that this could be the
right answer, but we have a

00:03:36.690 --> 00:03:39.660
square root of 3 in the
denominator, which people don't

00:03:39.660 --> 00:03:42.980
like because it's an irrational
number in the denominator.

00:03:42.980 --> 00:03:44.690
And I guess we could
have a debate as to

00:03:44.690 --> 00:03:46.010
why that might be bad.

00:03:46.010 --> 00:03:49.870
So let's rationalize
this denominator.

00:03:49.870 --> 00:03:55.150
We say x is equal to 10 over
the square to 3; to rationalize

00:03:55.150 --> 00:03:57.750
this denominator we can
multiply the numerator and the

00:03:57.750 --> 00:03:59.910
denominator by the
square root of 3.

00:03:59.910 --> 00:04:02.670
Because as long as we multiply
the numerator and the

00:04:02.670 --> 00:04:05.280
denominator by the same thing,
it's like multiplying by 1.

00:04:05.280 --> 00:04:09.790
So this is equal to 10 square
roots of 3 over square root of

00:04:09.790 --> 00:04:12.996
3 times square of 3;
well that's just 3.

00:04:12.996 --> 00:04:16.212
So x equals 10 square
roots of 3 over 3.

00:04:16.212 --> 00:04:17.870
That's the hypotenuse.

00:04:17.870 --> 00:04:18.990
I know I confused you.

00:04:18.990 --> 00:04:22.920
And, of course, if this is 10
square root of 3 over 3--

00:04:22.920 --> 00:04:26.600
that's the hypotenuse --we know
that the 30-degree side-- this

00:04:26.600 --> 00:04:28.820
is 30 degrees --we know the
30-degree side is half of

00:04:28.820 --> 00:04:35.430
that, so it's 5 square
root of 3 over 3.

00:04:35.430 --> 00:04:38.100
Anyway, I think that might
give you a sense of the

00:04:38.100 --> 00:04:40.230
30, 60, 90 triangles.

00:04:40.230 --> 00:04:43.980
I think you might be ready now
to try some of the level two

00:04:43.980 --> 00:04:46.080
Pythagorean Theorem problems.

00:04:46.080 --> 00:04:47.600
Have fun.


Intro to 30-60-90 triangles

WEBVTT
Kind: captions
Language: en

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Sorry for starting the
presentation with a cough.

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I think I still have a little
bit of a bug going around.

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But now I want to continue
with the 45-45-90 triangles.

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So in the last presentation we
learned that either side of a

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45-45-90 triangle that isn't
the hypotenuse is equal to the

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square route of 2 over 2
times the hypotenuse.

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Let's do a couple
of more problems.

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So if I were to tell you that
the hypotenuse of this

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triangle-- once again,
this only works for

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45-45-90 triangles.

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And if I just draw one 45
you know the other angle's

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got to be 45 as well.

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If I told you that the
hypotenuse here is,

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let me say, 10.

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We know this is a hypotenuse
because it's opposite

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the right angle.

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And then I would ask you
what this side is, x.

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Well we know that x is equal
to the square root of 2 over

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2 times the hypotenuse.

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So that's square root
of 2 over 2 times 10.

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Or, x is equal to 5
square roots of 2.

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Right?

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10 divided by 2.

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So x is equal to 5
square roots of 2.

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And we know that this side
and this side are equal.

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Right?

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I guess we know this is an
isosceles triangle because

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these two angles are the same.

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So we also that this
side is 5 over 2.

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And if you're not
sure, try it out.

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Let's try the
Pythagorean theorem.

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We know from the Pythagorean
theorem that 5 root 2 squared,

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plus 5 root 2 squared is equal
to the hypotenuse squared,

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where the hypotenuse is 10.

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Is equal to 100.

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Or this is just 25 times 2.

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So that's 50.

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But this is 100 up here.

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Is equal to 100.

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And we know, of course,
that this is true.

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So it worked.

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We proved it using the
Pythagorean theorem, and

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that's actually how we
came up with this formula

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in the first place.

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Maybe you want to go back to
one of those presentations

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if you forget how we
came up with this.

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I'm actually now going
to introduce another

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type of triangle.

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And I'm going to do it the same
way, by just posing a problem

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to you and then using the
Pythagorean theorem

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to figure it out.

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This is another type
of triangle called a

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30-60-90 triangle.

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And if I don't have time
for this I will do

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another presentation.

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Let's say I have a
right triangle.

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That's not a pretty one,
but we use what we have.

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That's a right angle.

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And if I were to tell you that
this is a 30 degree angle.

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Well we know that the
angles in a triangle

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have to add up to 180.

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So if this is 30, this is 90,
and let's say that this is x.

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x plus 30 plus 90 is equal to
180, because the angles in

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a triangle add up to 180.

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We know that x is equal to 60.

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Right?

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So this angle is 60.

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And this is why it's called a
30-60-90 triangle-- because

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that's the names of the three
angles in the triangle.

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And if I were to tell you that
the hypotenuse is-- instead of

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calling it c, like we always
do, let's call it h-- and I

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want to figure out the other
sides, how do we do that?

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Well we can do that
using pretty much the

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Pythagorean theorem.

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And here I'm going to
do a little trick.

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Let's draw another copy of this
triangle, but flip it over

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draw it the other side.

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And this is the same triangle,
it's just facing the

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other direction.

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Right?

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If this is 90 degrees
we know that these two

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angles are supplementary.

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You might want to review the
angles module if you forget

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that two angles that share kind
of this common line would

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add up to 180 degrees.

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So this is 90, this
will also be 90.

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And you can eyeball it.

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It makes sense.

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And since we flip it, this
triangle is the exact

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same triangle as this.

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It's just flipped
over the other side.

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We also know that this
angle is 30 degrees.

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And we also know that this
angle is 60 degrees.

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Right?

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Well if this angle is 30
degrees and this angle is 30

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degrees, we also know that this
larger angle-- goes all the way

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from here to here--
is 60 degrees.

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Right?

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Well if this angle is 60
degrees, this top angle is 60

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degrees, and this angle on the
right is 60 degrees, then we

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know from the theorem that we
learned when we did 45-45-90

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triangles that if these two
angles are the same then the

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sides that they don't share
have to be the same as well.

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So what are the sides
they don't share?

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Well, it's this side
and this side.

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So if this side is h
then this side is h.

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Right?

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But this angle is
also 60 degrees.

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So if we look at this 60
degrees and this 60 degrees, we

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know that the sides that they
don't share are also equal.

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Well they share this side, so
the sides that they don't share

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are this side and this side.

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So this side is h, we also
know that this side is h.

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Right?

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So it turns out that if you
have 60 degrees, 60 degrees,

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and 60 degrees that all the
sides have the same lengths, or

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it's an equilateral triangle.

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And that's something
to keep in mind.

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And that makes sense too,
because an equilateral triangle

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is symmetric no matter
how you look at it.

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So it makes sense that all of
the angles would be the same

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and all of the sides would
have the same length.

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But, hm.

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When we originally did this
problem we only used half of

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this equilateral triangle.

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So we know this whole side
right here is of length h.

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But if that whole side is of
length h, well then this side

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right here, just the base of
our original triangle-- and I'm

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trying to be messy on purpose.

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We tried another color.

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This is going to be
half of that side.

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Right?

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Because that's h over 2,
and this is also h over 2.

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Right over here.

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So if we go back to our
original triangle, and we said

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that this is 30 degrees and
that this is the hypotenuse,

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because it's opposite the right
angle, we know that the side

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opposite the 30 degree side
is 1/2 of the hypotenuse.

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And just a reminder,
how did we do that?

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Well we doubled the triangle.

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Turned it into an
equilateral triangle.

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Figured out this whole
side has to be the same

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as the hypotenuse.

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And this is 1/2 of
that whole side.

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So it's 1/2 of the hypotenuse.

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So let's remember that.

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The side opposite the 30 degree
side is 1/2 of the hypotenuse.

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Let me redraw that on another
page, because I think

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this is getting messy.

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So going back to what
I had originally.

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This is a right angle.

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This is the hypotenuse--
this side right here.

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If this is 30 degrees, we just
derived that the side opposite

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the 30 degrees-- it's like what
the angle is opening into--

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that this is equal to
1/2 the hypotenuse.

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If this is equal to 1/2
the hypotenuse then what

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is this side equal to?

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Well, here we can use the
Pythagorean theorem again.

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We know that this side squared
plus this side squared-- let's

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call this side A-- is
equal to h squared.

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So we have 1/2 h squared plus A
squared is equal to h squared.

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This is equal to h squared
over 4 plus A squared,

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is equal to h squared.

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Well, we subtract h
squared from both sides.

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We get A squared is equal to h
squared minus h squared over 4.

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So this equals h squared
times 1 minus 1/4.

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This is equal to 3/4 h squared.

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And once going that's
equal to A squared.

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I'm running out of space,
so I'm going to go all

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the way over here.

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So take the square root of both
sides, and we get A is equal

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to-- the square root of 3/4
is the same thing as the

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square root of 3 over 2.

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And then the square root
of h squared is just h.

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And this A-- remember,
this is an area.

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This is what decides the
length of the side.

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I probably shouldn't
have used A.

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But this is equal to the square
root of 3 over 2, times h.

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So there.

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We've derived what all the
sides relative to the

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hypotenuse are of a
30-60-90 triangle.

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So if this is a 60 degree side.

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So if we know the hypotenuse
and we know this is a 30-60-90

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triangle, we know the side
opposite the 30 degree side

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is 1/2 the hypotenuse.

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And we know the side opposite
the 60 degree side is the

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square root of 3 over 2,
times the hypotenuse.

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In the next module I'll show
you how using this information,

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which you may or may not want
to memorize-- it's probably

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good to memorize and practice
with, because it'll make you

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very fast on standardized
tests-- how we can use this

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information to solve the sides
of a 30-60-90 triangle

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very quickly.

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See you in the next
presentation.