Example: Figure out the trig function

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.330
We're asked to determine
a function of the form y

00:00:04.330 --> 00:00:11.150
equals a sine of bx or y
is equal to a cosine bx

00:00:11.150 --> 00:00:14.160
represented by the graph below.

00:00:14.160 --> 00:00:16.400
So we need to figure out
maybe what the a's are,

00:00:16.400 --> 00:00:17.816
what the b's are,
and whether this

00:00:17.816 --> 00:00:19.720
is a sine or a cosine function.

00:00:19.720 --> 00:00:22.420
So let's see what
clues there are.

00:00:22.420 --> 00:00:25.240
So the first thing
that I notice is

00:00:25.240 --> 00:00:29.260
that whatever this function
is, when x is equal to 0,

00:00:29.260 --> 00:00:31.080
it does not equal 0.

00:00:31.080 --> 00:00:34.440
It is equal to negative 2.

00:00:34.440 --> 00:00:37.410
So based on that,
based on what we

00:00:37.410 --> 00:00:39.240
know about sine and
cosine functions,

00:00:39.240 --> 00:00:41.590
do you think that this is
going to be of the form y

00:00:41.590 --> 00:00:45.030
is equal to a sine
of bx or of the form

00:00:45.030 --> 00:00:47.314
y is equal to a cosine of bx?

00:00:47.314 --> 00:00:49.230
Without even knowing
what the a's and b's are,

00:00:49.230 --> 00:00:52.460
do you think this is going to
be a sine function or cosine

00:00:52.460 --> 00:00:54.720
function?

00:00:54.720 --> 00:00:57.470
Well, let's think about
what sine of 0 is.

00:00:57.470 --> 00:01:01.700
If you take sine
of 0, we already

00:01:01.700 --> 00:01:04.489
know that that is equal to 0.

00:01:04.489 --> 00:01:06.490
What is cosine of 0?

00:01:06.490 --> 00:01:10.020
Cosine of 0 is equal to 1.

00:01:10.020 --> 00:01:13.260
So it would be very hard--
and especially in this form,

00:01:13.260 --> 00:01:15.400
it would be impossible--
if sine of 0

00:01:15.400 --> 00:01:20.510
is 0, to multiply 0 by
something to get to negative 2.

00:01:20.510 --> 00:01:25.580
So it can't be a sine
function of this form.

00:01:25.580 --> 00:01:28.920
You might say, well,
the cosine of 0 is 1.

00:01:28.920 --> 00:01:30.310
But here, it's negative 2.

00:01:30.310 --> 00:01:32.030
But at least if
you have a 1, you

00:01:32.030 --> 00:01:34.140
can then multiply
it by something

00:01:34.140 --> 00:01:37.000
to get to a negative 2.

00:01:37.000 --> 00:01:40.280
So what we now know is that
we are at least of this form.

00:01:40.280 --> 00:01:42.459
But now we have to figure
out what the a's and b's

00:01:42.459 --> 00:01:43.500
are going to be equal to.

00:01:43.500 --> 00:01:53.490
We know that this function is
y is equal to a cosine of bx.

00:01:53.490 --> 00:02:05.100
So the next question I ask
you is what is a going to be.

00:02:05.100 --> 00:02:06.870
Well, let's think about it.

00:02:06.870 --> 00:02:11.600
We already saw if we
just had cosine of bx,

00:02:11.600 --> 00:02:15.620
when x is equal to 0,
cosine of b times 0

00:02:15.620 --> 00:02:17.280
would just be cosine of 0.

00:02:17.280 --> 00:02:18.910
And it would get us to 1.

00:02:18.910 --> 00:02:19.750
But we're not 1.

00:02:19.750 --> 00:02:20.850
We're at negative 2.

00:02:20.850 --> 00:02:22.940
It looks like [? it ?]
took a cosine function,

00:02:22.940 --> 00:02:24.830
and at least, when
x is equal to 0,

00:02:24.830 --> 00:02:27.570
we multiplied it by negative 2.

00:02:27.570 --> 00:02:31.290
So this should be negative 2.

00:02:31.290 --> 00:02:33.590
So now we have a little
bit more filled in

00:02:33.590 --> 00:02:34.740
of what we actually have.

00:02:34.740 --> 00:02:45.440
We know that it's y is equal
to negative 2 cosine of bx.

00:02:45.440 --> 00:02:48.500
And this gels with what we see
right over here, the amplitude

00:02:48.500 --> 00:02:49.440
here.

00:02:49.440 --> 00:02:51.760
You see that the difference
between the maximum value

00:02:51.760 --> 00:02:53.968
and the minimum value, or
the minimum and the maximum

00:02:53.968 --> 00:02:56.684
is 4, 1/2 of that is 2.

00:02:56.684 --> 00:02:58.100
Or another way you
think about it,

00:02:58.100 --> 00:03:01.140
we're varying 2 from
this center point.

00:03:01.140 --> 00:03:03.202
And over here, if you
think about the amplitude--

00:03:03.202 --> 00:03:04.660
the amplitude is
the absolute value

00:03:04.660 --> 00:03:06.430
of this number right over here.

00:03:06.430 --> 00:03:10.000
The amplitude is equal
to the absolute value

00:03:10.000 --> 00:03:13.090
of this negative 2,
which is indeed equal 2.

00:03:13.090 --> 00:03:15.660
So it's consistent so far.

00:03:15.660 --> 00:03:18.510
Now let's think
about what b here is.

00:03:18.510 --> 00:03:20.340
And maybe we can
use our knowledge

00:03:20.340 --> 00:03:24.290
of what the period of
a periodic function

00:03:24.290 --> 00:03:28.496
is to think about
what b might be.

00:03:28.496 --> 00:03:29.620
Well, let's look over here.

00:03:29.620 --> 00:03:35.230
What is the period of
this periodic function?

00:03:35.230 --> 00:03:37.120
Well, let's draw one period.

00:03:37.120 --> 00:03:42.459
So if we use this as our
starting point-- or one cycle,

00:03:42.459 --> 00:03:43.000
I should say.

00:03:43.000 --> 00:03:43.980
Let's draw one cycle.

00:03:43.980 --> 00:03:47.310
If you view that as our
starting point, at 2 pi over 3,

00:03:47.310 --> 00:03:49.430
we have completed that cycle.

00:03:49.430 --> 00:03:51.170
And then we could
start the next cycle.

00:03:51.170 --> 00:03:53.870
We repeat the
pattern over again.

00:03:53.870 --> 00:03:55.840
Then you start the next cycle.

00:03:55.840 --> 00:03:59.112
So based on that,
what is the period?

00:03:59.112 --> 00:04:00.570
Well, it's the
length that you need

00:04:00.570 --> 00:04:03.120
to go in x to
complete one cycle.

00:04:03.120 --> 00:04:08.100
So that length right over
there, you see, is 2 pi over 3.

00:04:08.100 --> 00:04:13.850
So the period is 2 pi over 3.

00:04:13.850 --> 00:04:17.260
And given that the period
here is 2 pi over 3,

00:04:17.260 --> 00:04:22.420
can you figure out
what b is going to be?

00:04:22.420 --> 00:04:25.236
Well, the period
of this is going

00:04:25.236 --> 00:04:34.560
to be equal to 2 pi over
the absolute value of b.

00:04:34.560 --> 00:04:36.700
And you can solve
this multiple ways.

00:04:36.700 --> 00:04:41.270
You can multiply both sides by
3 and the absolute value of b.

00:04:41.270 --> 00:04:43.650
And you would be left
with the absolute value

00:04:43.650 --> 00:04:50.400
of b is equal to 3,
which means that b could

00:04:50.400 --> 00:04:54.200
be equal to positive
or negative 3.

00:04:54.200 --> 00:04:57.780
And so you might say,
well, Sal, what do I use?

00:04:57.780 --> 00:05:01.176
Does b equal positive
3 or negative 3?

00:05:01.176 --> 00:05:02.550
And so the next
question I'll ask

00:05:02.550 --> 00:05:06.880
you is, for a
cosine function, do

00:05:06.880 --> 00:05:11.900
you get different
values if you were

00:05:11.900 --> 00:05:24.980
to make this a cosine of 3x
or a cosine of negative 3x?

00:05:24.980 --> 00:05:28.144
Do you get different values?

00:05:28.144 --> 00:05:30.310
Well, if you play with the
unit circle a little bit.

00:05:34.880 --> 00:05:36.600
So I'm going to draw
a little rough unit

00:05:36.600 --> 00:05:38.200
circle right over here.

00:05:38.200 --> 00:05:39.870
Remember, cosine
is the x-coordinate

00:05:39.870 --> 00:05:42.600
where we intersect
the unit circle.

00:05:42.600 --> 00:05:44.990
And if we go in the
positive angle direction,

00:05:44.990 --> 00:05:47.787
if we go in that direction, our
x-coordinate-- it starts at 1,

00:05:47.787 --> 00:05:49.370
and then it gets a
little bit shorter.

00:05:49.370 --> 00:05:51.494
If we go in the negative
direction, it starts at 1,

00:05:51.494 --> 00:05:53.380
and then it gets a
little bit shorter.

00:05:53.380 --> 00:05:55.480
And so you can experiment
this with a good bit.

00:05:55.480 --> 00:05:58.080
But you'll see
that cosine of 3x--

00:05:58.080 --> 00:05:59.650
and this is only
the case for cosine,

00:05:59.650 --> 00:06:05.710
not for sine-- cosine of 3x is
equal to cosine of negative 3x.

00:06:05.710 --> 00:06:09.700
So you can actually pick either
positive 3 or negative 3.

00:06:09.700 --> 00:06:12.930
But for simplicity in
this case right over here,

00:06:12.930 --> 00:06:15.010
I'll just go with
the positive 3.

00:06:15.010 --> 00:06:17.920
So this could be the
graph of-- and now we

00:06:17.920 --> 00:06:22.690
get our drum roll-- y is
equal to negative 2 times

00:06:22.690 --> 00:06:26.020
the cosine of-- I said I
wouldn't do the negative--

00:06:26.020 --> 00:06:30.190
the cosine of positive 3x.

00:06:30.190 --> 00:06:33.077
And we are done.


Example: Amplitude and period cosine transformations

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.445
We are asked to
graph the function y

00:00:03.445 --> 00:00:07.940
is equal to negative
2.5 cosine of 1/3

00:00:07.940 --> 00:00:12.150
x on the interval, 0 to 6
pi, including the endpoints.

00:00:12.150 --> 00:00:15.610
So let me do my best
attempt at graphing that.

00:00:15.610 --> 00:00:18.890
And to start off, I'm going
to graph with the simplest

00:00:18.890 --> 00:00:21.390
function, or the
simplest version of this,

00:00:21.390 --> 00:00:24.690
or the root of this,
which is just cosine of x.

00:00:24.690 --> 00:00:27.290
So let me just graph,
and eventually you

00:00:27.290 --> 00:00:31.760
can kind of-- let me
just graph cosine of x.

00:00:31.760 --> 00:00:33.880
So that's my y-axis.

00:00:33.880 --> 00:00:36.130
And I want to have some space
here so I can eventually

00:00:36.130 --> 00:00:39.380
graph this entire thing.

00:00:39.380 --> 00:00:43.710
So let's say that this is
negative 1, this is negative 2.

00:00:43.710 --> 00:00:47.970
This is positive 1,
this is positive 2.

00:00:47.970 --> 00:00:54.740
And let's say that this
right over here is 2 pi.

00:00:54.740 --> 00:00:57.587
And then of course that
could be pi right over there.

00:00:57.587 --> 00:00:59.170
Now, the first thing
I'm going to do--

00:00:59.170 --> 00:01:02.340
let me copy this because
I could use it later

00:01:02.340 --> 00:01:03.720
to graph the whole thing.

00:01:06.510 --> 00:01:07.920
So let's start off.

00:01:07.920 --> 00:01:12.400
So I'm just going to graph
y is equal to cosine of x.

00:01:12.400 --> 00:01:14.500
So when x is equal
to 0-- and I'm just

00:01:14.500 --> 00:01:16.590
going to do it between
the interval 0 and 2 pi.

00:01:16.590 --> 00:01:17.964
Obvious it's a
periodic function,

00:01:17.964 --> 00:01:20.620
it'll keep going in the negative
and the positive directions.

00:01:20.620 --> 00:01:22.630
So what happens when
x is equal to 0?

00:01:22.630 --> 00:01:24.320
What is cosine of x?

00:01:24.320 --> 00:01:27.760
Well cosine of 0 is 1.

00:01:27.760 --> 00:01:35.270
What about when
x is equal to pi?

00:01:35.270 --> 00:01:37.632
What is cosine of pi?

00:01:37.632 --> 00:01:39.790
Well cosine of pi is negative 1.

00:01:42.360 --> 00:01:44.850
And then what's cosine of 2 pi?

00:01:44.850 --> 00:01:46.050
Well that's 1 again.

00:01:46.050 --> 00:01:48.510
We get back-- we've
completed a period,

00:01:48.510 --> 00:01:50.740
or we've completed
an entire cycle.

00:01:50.740 --> 00:01:54.580
And 2 pi is the
period of cosine of x.

00:01:54.580 --> 00:01:58.260
So this is one cycle
right over here.

00:01:58.260 --> 00:02:01.070
I could keep going if I wanted
to, but the whole point,

00:02:01.070 --> 00:02:06.140
I just wanted to graph this
one cycle between 0 and 2 pi.

00:02:06.140 --> 00:02:09.580
Now what I want to think about
is, what happens to this graph?

00:02:09.580 --> 00:02:12.620
Instead of graphing
y equals cosine

00:02:12.620 --> 00:02:15.260
of x-- let me draw
some graph paper again.

00:02:15.260 --> 00:02:18.640
Instead of drawing y is
equal to cosine of x,

00:02:18.640 --> 00:02:25.047
I'm going to draw y is
equal to cosine of 1/3 x.

00:02:25.047 --> 00:02:26.880
So the only difference
between that and that

00:02:26.880 --> 00:02:29.470
is now I'm multiplying
the x by 1/3.

00:02:29.470 --> 00:02:33.240
What's going to happen
to the graph over here?

00:02:33.240 --> 00:02:35.410
How is this going to change
instead of being an x,

00:02:35.410 --> 00:02:36.460
if it's a 1/3 x?

00:02:36.460 --> 00:02:38.845
What's going to
happen over here?

00:02:38.845 --> 00:02:40.970
And now I'm going to do it
over the entire interval

00:02:40.970 --> 00:02:41.840
between zero and six pi.

00:02:41.840 --> 00:02:43.715
So let me just make sure
I have enough space.

00:02:43.715 --> 00:02:49.370
So that's 3 pi, 4
pi, 5 pi, and 6 pi.

00:02:49.370 --> 00:02:52.660
What's going to
happen to this graph?

00:02:52.660 --> 00:02:54.740
Well, there's a couple of
ways to think about it.

00:02:54.740 --> 00:02:57.350
The easiest might
just be to say, well

00:02:57.350 --> 00:03:02.704
to complete an entire cycle,
we're going to go 1/3 as fast.

00:03:02.704 --> 00:03:04.370
Or we're going to go
three times slower.

00:03:04.370 --> 00:03:06.980
Or if you just want to
think about the period here,

00:03:06.980 --> 00:03:12.090
what's the period
of cosine of 1/3 x?

00:03:12.090 --> 00:03:16.860
Well the period is
going to be 2 pi divided

00:03:16.860 --> 00:03:19.540
by the absolute value of this
coefficient right over here.

00:03:19.540 --> 00:03:22.610
So it's the absolute value
of 1/3, which is just 1/3.

00:03:22.610 --> 00:03:25.280
So the period is 2
pi over 1/3, which

00:03:25.280 --> 00:03:29.140
is the same thing as 2 pi
times 3, which is 6 pi.

00:03:29.140 --> 00:03:31.780
Which gels with the intuition.

00:03:31.780 --> 00:03:34.160
That's going to take
three times as much

00:03:34.160 --> 00:03:37.370
time to get whatever we input
into the cosine function

00:03:37.370 --> 00:03:38.525
to get back to 2 pi.

00:03:38.525 --> 00:03:41.150
Because whatever we take
x, we're taking 1/3 of it.

00:03:41.150 --> 00:03:45.440
So to get to 2 pi, you can't
just have x equals 2 pi.

00:03:45.440 --> 00:03:48.740
x now has to equal 6
pi to get 2 pi inputted

00:03:48.740 --> 00:03:50.290
into the cosine function.

00:03:50.290 --> 00:03:53.590
So the period is now 6 pi.

00:03:53.590 --> 00:04:02.410
At x is equal to 0, 1/3 times 0
is 0, and the cosine of 0 is 1.

00:04:02.410 --> 00:04:06.990
When x is equal to 6 pi, you
have 6 pi divided by 3 is 2 pi.

00:04:06.990 --> 00:04:11.540
Cosine of 2 pi is equal to 1.

00:04:11.540 --> 00:04:14.120
And if you want to go
in between, over here

00:04:14.120 --> 00:04:15.520
to go in between, we tried pi.

00:04:15.520 --> 00:04:17.410
But over here, we
could try 3 pi.

00:04:17.410 --> 00:04:21.420
When x is 3 pi, you
have cosine of 1/3

00:04:21.420 --> 00:04:23.550
of 3 pi, that's cosine of pi.

00:04:23.550 --> 00:04:26.210
Cosine of pi is negative 1.

00:04:26.210 --> 00:04:32.140
So when x is equal to 3 pi,
we have cosine of 1/3 times

00:04:32.140 --> 00:04:34.350
3 pi is negative 1.

00:04:34.350 --> 00:04:36.335
So it's going to look
something like this.

00:04:41.540 --> 00:04:45.170
Trying my best attempt.

00:04:45.170 --> 00:04:47.020
to draw it.

00:04:47.020 --> 00:04:49.690
So it's going to look
something like this.

00:04:49.690 --> 00:04:52.750
So you see, to go from y
equals cosine of x to y

00:04:52.750 --> 00:04:57.210
equals cosine of 1/3 x, it
essentially stretched out

00:04:57.210 --> 00:05:00.190
to this function
by a factor of 3.

00:05:00.190 --> 00:05:03.050
You can see this period
is three times longer.

00:05:03.050 --> 00:05:06.520
The period here was 2 pi.

00:05:10.260 --> 00:05:12.830
All right, well there's
only one more transformation

00:05:12.830 --> 00:05:14.980
we need in order to
get to the function

00:05:14.980 --> 00:05:16.190
that they're asking us about.

00:05:16.190 --> 00:05:18.830
We just have to, instead of
having a cosine of 1/3 x,

00:05:18.830 --> 00:05:22.530
we just have to negative
2.5 cosine of 1/3 x.

00:05:22.530 --> 00:05:23.710
So let's try to draw that.

00:05:23.710 --> 00:05:29.260
So let me put my
axis here again.

00:05:29.260 --> 00:05:32.700
And let me label it.

00:05:32.700 --> 00:05:43.050
So that's 2 pi, 3 pi,
4 pi, 5 pi, and 6 pi.

00:05:43.050 --> 00:05:48.760
And our goal now is to draw
the graph of y is equal to--

00:05:48.760 --> 00:05:51.106
and we're just doing it over
between 0 and 6 pi here.

00:05:51.106 --> 00:05:52.730
We only did it between
0 and 2 pi here.

00:05:52.730 --> 00:05:58.460
Obviously they're all periodic,
they all keep going on and on.

00:05:58.460 --> 00:06:03.410
But now we want to graph y is
equal to negative 2.5 times

00:06:03.410 --> 00:06:08.290
cosine of 1/3 x.

00:06:08.290 --> 00:06:12.360
So given this change, we're now
multiplying by negative 2.5,

00:06:12.360 --> 00:06:14.040
what is going to
be-- well actually,

00:06:14.040 --> 00:06:15.350
let's think about a few things.

00:06:15.350 --> 00:06:18.150
What was the amplitude
in the first two graphs

00:06:18.150 --> 00:06:19.010
right over here?

00:06:26.824 --> 00:06:28.490
Well there's two ways
to think about it.

00:06:28.490 --> 00:06:30.940
You could say the amplitude
is half the difference

00:06:30.940 --> 00:06:33.090
between the minimum
and the maximum points.

00:06:33.090 --> 00:06:35.350
In either of these case,
the minimum is negative 1,

00:06:35.350 --> 00:06:36.940
maximum is 1.

00:06:36.940 --> 00:06:40.910
The difference is 2,
half of that is 1.

00:06:40.910 --> 00:06:43.040
Or you could just say
it's the absolute value

00:06:43.040 --> 00:06:45.620
of the coefficient here,
which is implicitly a 1.

00:06:45.620 --> 00:06:50.130
And the absolute value
of 1 is, once again, 1.

00:06:50.130 --> 00:06:54.810
What's going to be the amplitude
for this thing right over here?

00:06:54.810 --> 00:06:59.520
Well the amplitude is going to
be the absolute value of what's

00:06:59.520 --> 00:07:01.540
multiplying the cosine function.

00:07:01.540 --> 00:07:05.500
So the amplitude in this
case, do it in green,

00:07:05.500 --> 00:07:09.710
the amplitude is going to be
equal to the absolute value

00:07:09.710 --> 00:07:15.150
of negative 2.5,
which is equal to 2.5.

00:07:15.150 --> 00:07:20.620
So given that, how is
multiplying by negative 2.5

00:07:20.620 --> 00:07:26.990
going to transform this
graph right over here?

00:07:26.990 --> 00:07:28.200
Well let's think about it.

00:07:28.200 --> 00:07:31.700
If it was multiplying
by just a positive 2.5,

00:07:31.700 --> 00:07:33.140
you would stretch it out.

00:07:33.140 --> 00:07:37.020
At each point it would go
up by a factor of 2 and 1/2.

00:07:37.020 --> 00:07:39.565
But it's a negative
2.5, so at each point,

00:07:39.565 --> 00:07:41.190
you're going to
stretch it out and then

00:07:41.190 --> 00:07:43.580
you're going to flip
it over the x-axis.

00:07:43.580 --> 00:07:44.860
So let's do that.

00:07:44.860 --> 00:07:49.900
So when x was 0, you
got 1 in this case.

00:07:49.900 --> 00:07:52.550
But now we're going to
multiply that by negative 2.5,

00:07:52.550 --> 00:07:55.180
which means you're going
to get to negative 2.5.

00:07:55.180 --> 00:07:57.080
So let me draw negative
2.5 right over there.

00:07:57.080 --> 00:07:59.280
So that's negative 2.5.

00:07:59.280 --> 00:08:01.090
That'd be negative--
let me make it clear.

00:08:01.090 --> 00:08:04.460
This would be negative
3 right over here,

00:08:04.460 --> 00:08:06.310
this would be positive 3.

00:08:06.310 --> 00:08:09.291
So that number right over
there is negative 2.5.

00:08:09.291 --> 00:08:10.790
And let me draw a
dotted line there.

00:08:10.790 --> 00:08:13.390
It could serve to be useful.

00:08:13.390 --> 00:08:21.226
Now when cosine
of 1/3 x is 0, it

00:08:21.226 --> 00:08:22.850
doesn't matter what
you multiply it by,

00:08:22.850 --> 00:08:27.020
you're still going to
get 0 right over here.

00:08:27.020 --> 00:08:30.530
Now, when cosine of
1/3 x was negative 1,

00:08:30.530 --> 00:08:34.980
which was the case when x
is equal to 3 pi, what's

00:08:34.980 --> 00:08:38.030
going to happen over here?

00:08:38.030 --> 00:08:41.020
Well cosine of 1/3 x,
we see, is negative 1.

00:08:41.020 --> 00:08:45.330
Negative 1 times negative
2.5 is positive 2.5.

00:08:45.330 --> 00:08:48.300
So we're going to get to
positive 2.5, which is right--

00:08:48.300 --> 00:08:50.030
let me draw a dotted
line over here.

00:08:50.030 --> 00:08:54.340
We're going to get to positive
2.5, which is right over there.

00:08:54.340 --> 00:08:58.580
And then when cosine
of 1/3 x is equal to 0,

00:08:58.580 --> 00:09:02.700
doesn't matter what we
multiply it by, we get to 0.

00:09:02.700 --> 00:09:09.940
And then finally, when x
is at 6 pi, cosine of 1/3 x

00:09:09.940 --> 00:09:12.090
is equal to 1.

00:09:12.090 --> 00:09:15.460
What's that going to be when
you multiply it by negative 2.5?

00:09:15.460 --> 00:09:17.520
Well it's going to
be negative 2.5.

00:09:17.520 --> 00:09:19.720
So we're going to
get back over here.

00:09:19.720 --> 00:09:21.560
So we're ready to
draw our graph.

00:09:21.560 --> 00:09:24.320
It looks something-- let
me do that in magenta color

00:09:24.320 --> 00:09:26.620
since that's what the
color I wrote this in.

00:09:26.620 --> 00:09:27.800
It will look like this.

00:09:31.000 --> 00:09:33.640
I can draw it as a solid line.

00:09:33.640 --> 00:09:39.740
So it will look like that.

00:09:39.740 --> 00:09:41.050
So you saw what happened.

00:09:41.050 --> 00:09:44.930
By putting this 1/3 here,
it stretched out the graph.

00:09:44.930 --> 00:09:47.540
It increased the period
by a factor of 3.

00:09:47.540 --> 00:09:50.760
And then multiplying
it by negative 2.5--

00:09:50.760 --> 00:09:54.290
if you just multiply
it by 2.5, you

00:09:54.290 --> 00:10:00.640
would just multiply
that out a little bit.

00:10:00.640 --> 00:10:02.140
But now it's a
negative, so not only

00:10:02.140 --> 00:10:05.690
do you increase the amplitude,
but you flip it over.

00:10:05.690 --> 00:10:09.290
So it is, indeed, the case
that the amplitude here is 2.5.

00:10:09.290 --> 00:10:11.437
We vary 2.5 from
our middle position.

00:10:11.437 --> 00:10:13.770
Or you could say that the
difference between the minimum

00:10:13.770 --> 00:10:16.890
and the maximum is 5,
so half of that is 2.5.

00:10:16.890 --> 00:10:19.515
But it isn't just multiplying
this graph by 2.5.

00:10:19.515 --> 00:10:21.620
If you multiply
this graph by 2.5,

00:10:21.620 --> 00:10:24.770
you'd get something-- let
me be a little neater.

00:10:24.770 --> 00:10:32.160
You would get something that
looked something like that.

00:10:32.160 --> 00:10:34.460
But because we
had a negative, we

00:10:34.460 --> 00:10:36.480
had to flip it over the x-axis.

00:10:36.480 --> 00:10:37.370
And we got this here.

00:10:37.370 --> 00:10:39.810
So this amplitude
is 2.5, but it's

00:10:39.810 --> 00:10:43.440
a flipped over
version of this graph.


Example: Intersection of sine and cosine

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.430
We're asked at how many
points do the graph of y

00:00:03.430 --> 00:00:06.560
equals sine of theta and
y equal cosine of theta

00:00:06.560 --> 00:00:09.680
intersect for theta
between 0 and 2 pi.

00:00:09.680 --> 00:00:12.480
And 0 is less than or equal
to theta which is less than

00:00:12.480 --> 00:00:13.410
or equal to 2 pi.

00:00:13.410 --> 00:00:15.485
So we're going to
include 0 and 2 pi

00:00:15.485 --> 00:00:19.140
in the possible
values for theta.

00:00:19.140 --> 00:00:22.630
So to do this I've set up
a little chart for theta,

00:00:22.630 --> 00:00:24.450
cosine theta, and sine theta.

00:00:24.450 --> 00:00:26.590
And we can use this
and the unit circle

00:00:26.590 --> 00:00:29.830
to hopefully quickly
graph what the graphs of y

00:00:29.830 --> 00:00:32.560
equals sine theta and y
equals cosine theta are.

00:00:32.560 --> 00:00:35.770
And then we could think about
how many times they intersect

00:00:35.770 --> 00:00:38.630
and maybe where they
actually intersect.

00:00:38.630 --> 00:00:39.642
So let's get started.

00:00:39.642 --> 00:00:42.100
So first of all, just to be
clear, this is the unit circle.

00:00:42.100 --> 00:00:43.250
This is the x-axis.

00:00:43.250 --> 00:00:44.680
This is the y-axis.

00:00:44.680 --> 00:00:47.050
Over here we're going to
graph these two graphs.

00:00:47.050 --> 00:00:48.950
So this is going
to be the y-axis.

00:00:48.950 --> 00:00:51.360
And it's going to be a
function of theta, not

00:00:51.360 --> 00:00:53.960
x, on the horizontal axis.

00:00:53.960 --> 00:00:56.060
So first let's think
about what happens

00:00:56.060 --> 00:00:58.790
when theta is equal to 0.

00:00:58.790 --> 00:01:02.084
So when theta is equal to
0, you're at this point

00:01:02.084 --> 00:01:02.750
right over here.

00:01:02.750 --> 00:01:04.166
Let me do it in a
different color.

00:01:04.166 --> 00:01:06.870
You're at this point right
over here on the unit circle.

00:01:06.870 --> 00:01:09.280
And what to coordinate is that?

00:01:09.280 --> 00:01:14.470
Well that's the point 1 comma 0.

00:01:14.470 --> 00:01:16.910
And so based on that,
what is cosine of theta

00:01:16.910 --> 00:01:19.040
when theta is equal to 0?

00:01:19.040 --> 00:01:24.690
Well cosine of theta is 1, and
sine of theta is going to be 0.

00:01:24.690 --> 00:01:30.529
This is the x-coordinate
at the point

00:01:30.529 --> 00:01:32.070
of intersection with
the unit circle.

00:01:32.070 --> 00:01:33.580
This is the y-coordinate.

00:01:33.580 --> 00:01:34.940
Let's keep going.

00:01:34.940 --> 00:01:36.130
What about pi over 2?

00:01:36.130 --> 00:01:37.580
So pi over 2.

00:01:37.580 --> 00:01:39.540
We are right over here.

00:01:39.540 --> 00:01:42.050
What is that coordinate?

00:01:42.050 --> 00:01:45.710
Well that's now x is 0, y is 1.

00:01:45.710 --> 00:01:49.180
So based on that,
cosine of theta is 0.

00:01:49.180 --> 00:01:51.420
And what is sine of theta?

00:01:51.420 --> 00:01:53.000
Well that's going to be 1.

00:01:53.000 --> 00:01:55.220
It's the y-coordinate
right over.

00:01:55.220 --> 00:01:58.710
Now let's go all the way to pi.

00:01:58.710 --> 00:02:00.750
We're at this point
in the unit circle.

00:02:00.750 --> 00:02:02.850
What is the coordinate?

00:02:02.850 --> 00:02:05.900
Well this is negative 1, 0.

00:02:05.900 --> 00:02:08.250
So what is cosine of theta?

00:02:08.250 --> 00:02:09.740
What's the x-coordinate here?

00:02:09.740 --> 00:02:11.260
Which is negative 1.

00:02:11.260 --> 00:02:16.120
And sine of theta is going to
be the y-coordinate, which is 0.

00:02:16.120 --> 00:02:17.720
Now let's keep going.

00:02:17.720 --> 00:02:20.780
Now we're down here
at 3 pi over 2.

00:02:20.780 --> 00:02:23.780
If we go all the way
around to 3 pi over 2,

00:02:23.780 --> 00:02:26.350
what is this coordinate?

00:02:26.350 --> 00:02:29.790
Well this is 0, negative 1.

00:02:29.790 --> 00:02:32.050
Cosine of theta is
the x-coordinate here.

00:02:32.050 --> 00:02:35.230
So cosine of theta
is going to be 0.

00:02:35.230 --> 00:02:37.900
And what is sine of
theta going to be?

00:02:37.900 --> 00:02:41.000
Well it's going
to be negative 1.

00:02:41.000 --> 00:02:43.100
And then finally
we go back to 2 pi,

00:02:43.100 --> 00:02:45.470
which is making a full
revolution around the circle.

00:02:45.470 --> 00:02:48.330
We went all the way around
and we're back to this point

00:02:48.330 --> 00:02:49.340
right over here.

00:02:49.340 --> 00:02:51.360
So the coordinate is
the exact same thing

00:02:51.360 --> 00:02:56.400
as when the angle
equals 0 radians.

00:02:56.400 --> 00:02:58.420
And so what is cosine of theta?

00:02:58.420 --> 00:02:59.590
Well that's 1.

00:02:59.590 --> 00:03:03.110
And sine of theta is 0.

00:03:03.110 --> 00:03:05.320
And from this we can make
a rough sketch of the graph

00:03:05.320 --> 00:03:07.420
and think about where
they might intersect.

00:03:07.420 --> 00:03:09.700
So first let's do
cosine of theta.

00:03:09.700 --> 00:03:13.400
When theta is 0-- and
let me mark this off.

00:03:13.400 --> 00:03:15.840
So this is going to be
when y is equal to 1.

00:03:15.840 --> 00:03:20.000
And this is when y is
equal to negative 1.

00:03:20.000 --> 00:03:23.431
So y equals cosine of theta.

00:03:23.431 --> 00:03:23.930
Let's see...

00:03:23.930 --> 00:03:25.060
theta equals 0.

00:03:25.060 --> 00:03:26.700
Cosine of theta equals 1.

00:03:26.700 --> 00:03:29.100
So cosine of theta
is equal to 1.

00:03:29.100 --> 00:03:33.590
When theta is equal to pi
2, cosine of theta is 0.

00:03:33.590 --> 00:03:38.750
When theta is equal to pi,
cosine of theta is negative 1.

00:03:38.750 --> 00:03:42.740
When theta is equal to 3
pi over 2, cosine of theta

00:03:42.740 --> 00:03:43.870
is equal to 0.

00:03:43.870 --> 00:03:45.410
That's this right over here.

00:03:45.410 --> 00:03:49.065
And then finally when theta is 2
pi, cosine of theta is 1 again.

00:03:51.780 --> 00:03:55.910
And the curve will look
something like this.

00:03:55.910 --> 00:03:58.170
My best attempt to draw it.

00:03:58.170 --> 00:04:01.140
Make it a nice smooth curve.

00:04:01.140 --> 00:04:05.450
So it's going to look
something like this.

00:04:05.450 --> 00:04:07.700
The look of these curves
should look somewhat familiar

00:04:07.700 --> 00:04:10.020
at this point.

00:04:10.020 --> 00:04:14.660
So this is the graph of y
is equal to cosine of theta.

00:04:14.660 --> 00:04:16.890
Now let's do the same
thing for sine theta.

00:04:16.890 --> 00:04:20.990
When theta is equal
to 0, sine theta is 0.

00:04:20.990 --> 00:04:25.600
When theta is pi over
2, sine of theta is 1.

00:04:25.600 --> 00:04:30.520
When theta is equal to
pi, sine of theta is 0.

00:04:30.520 --> 00:04:36.140
When theta is equal to 3 pi over
2, sine of theta is negative 1.

00:04:36.140 --> 00:04:40.620
When theta is equal to 2 pi,
sine of theta is equal to 0.

00:04:40.620 --> 00:04:42.360
And so the graph
of sine of theta

00:04:42.360 --> 00:04:47.870
is going to look
something like this.

00:04:47.870 --> 00:04:50.140
My best attempt at drawing it.

00:04:53.830 --> 00:04:56.560
So just visually, we can
think about the question.

00:04:56.560 --> 00:04:59.900
At how many points do the graphs
of y equals sine of theta and y

00:04:59.900 --> 00:05:03.190
equals cosine of theta intersect
for this range for theta?

00:05:03.190 --> 00:05:09.704
For theta being between 0 and 2
pi, including those two points.

00:05:09.704 --> 00:05:11.120
Well, you just
look at this graph.

00:05:11.120 --> 00:05:12.911
You see there's two
points of intersection.

00:05:12.911 --> 00:05:16.430
This point right over here and
this point right over here.

00:05:16.430 --> 00:05:19.430
Just between 0 and 2 pi.

00:05:19.430 --> 00:05:20.620
These are cyclical graphs.

00:05:20.620 --> 00:05:21.640
If we kept going,
they would keep

00:05:21.640 --> 00:05:22.848
intersecting with each other.

00:05:22.848 --> 00:05:27.340
But just over this 2
pi range for theta,

00:05:27.340 --> 00:05:30.730
you get two points
of intersection.

00:05:30.730 --> 00:05:32.230
Now let's think
about what they are,

00:05:32.230 --> 00:05:38.650
because they look to be pretty
close between 0 and pi over 2.

00:05:38.650 --> 00:05:41.622
And right between
pi and 3 pi over 2.

00:05:41.622 --> 00:05:43.830
So let's look at our unit
circle if we can figure out

00:05:43.830 --> 00:05:44.970
what those values are.

00:05:44.970 --> 00:05:48.830
It looks like this
is at pi over 4.

00:05:48.830 --> 00:05:50.050
So let's verify that.

00:05:50.050 --> 00:05:54.080
So let's think about what
these values are at pi over 4.

00:05:54.080 --> 00:05:58.230
So pi over 4 is that angle, or
that's the terminal side of it.

00:05:58.230 --> 00:05:59.410
So this is pi over 4.

00:05:59.410 --> 00:06:04.080
Pi over 4 is the exact same
thing as a 45 degree angle.

00:06:04.080 --> 00:06:06.400
So let's do pi over
4 right over here.

00:06:06.400 --> 00:06:09.020
So we have to figure out
what this point is what.

00:06:09.020 --> 00:06:10.450
What the coordinates are.

00:06:10.450 --> 00:06:13.695
So let's make this
a right triangle.

00:06:16.119 --> 00:06:18.160
And so what do we know
about this right triangle?

00:06:18.160 --> 00:06:19.830
And I'm going to draw
it right over here,

00:06:19.830 --> 00:06:20.913
to make it a little clear.

00:06:20.913 --> 00:06:23.240
This is a typical type
of right triangle.

00:06:23.240 --> 00:06:27.060
So it's good to get some
familiarity with it.

00:06:27.060 --> 00:06:29.510
So let me draw my best attempt.

00:06:29.510 --> 00:06:30.236
Alright.

00:06:30.236 --> 00:06:31.610
So we know it's
a right triangle.

00:06:31.610 --> 00:06:34.200
We know that this is 45 degrees.

00:06:34.200 --> 00:06:36.650
What is the length
of the hypotenuse?

00:06:36.650 --> 00:06:37.990
Well this is a unit circle.

00:06:37.990 --> 00:06:38.780
It has radius 1.

00:06:38.780 --> 00:06:41.610
So the length of the
hypotenuse here is 1.

00:06:41.610 --> 00:06:45.830
And what do we know about
this angle right over here?

00:06:45.830 --> 00:06:50.460
Well, we know that it
too must be 45 degrees,

00:06:50.460 --> 00:06:52.670
because all of these angles
have to add up to 180.

00:06:52.670 --> 00:06:55.580
And since these two
angles are the same,

00:06:55.580 --> 00:06:58.990
we know that these two sides
are going to be the same.

00:06:58.990 --> 00:07:01.260
And then we could use
the Pythagorean Theorem

00:07:01.260 --> 00:07:03.290
to think about the
length of those sides.

00:07:03.290 --> 00:07:05.150
So using the
Pythagorean Theorem,

00:07:05.150 --> 00:07:07.000
knowing that these
two sides are equal,

00:07:07.000 --> 00:07:10.720
what do we get for the
length of those sides?

00:07:10.720 --> 00:07:15.056
Well, if this has length a, well
then this also has length a.

00:07:15.056 --> 00:07:16.680
And we can use the
Pythagorean Theorem.

00:07:16.680 --> 00:07:20.207
And we could say a
squared plus a squared is

00:07:20.207 --> 00:07:21.540
equal to the hypotenuse squared.

00:07:21.540 --> 00:07:22.660
Is equal to 1.

00:07:22.660 --> 00:07:28.940
Or 2a squared is equal to
1a squared, is equal to 1/2.

00:07:28.940 --> 00:07:30.550
Take the principal
root of both sides.

00:07:30.550 --> 00:07:32.150
a is equal to the
square root of 1/2

00:07:32.150 --> 00:07:33.899
which is the square
root of 1, which is 1,

00:07:33.899 --> 00:07:35.610
over the square root of 2.

00:07:35.610 --> 00:07:37.730
We can rationalize
the denominator here

00:07:37.730 --> 00:07:39.580
by multiplying by
square root of 2

00:07:39.580 --> 00:07:42.380
over square root
of 2, which gives

00:07:42.380 --> 00:07:45.252
us a is equal to-- in the
numerator-- square root of 2.

00:07:45.252 --> 00:07:46.710
And in the denominator,
square root

00:07:46.710 --> 00:07:49.390
of 2 times square
root of 2 is 2.

00:07:49.390 --> 00:07:51.960
So this length is
the square root of 2.

00:07:51.960 --> 00:07:53.970
And this length
is the same thing.

00:07:53.970 --> 00:07:57.700
So this length right over here
is square root of 2 over 2.

00:07:57.700 --> 00:07:59.570
And this height right
over here is also

00:07:59.570 --> 00:08:01.800
square root of 2 over 2.

00:08:01.800 --> 00:08:05.860
So based on that, what
is this coordinate point?

00:08:05.860 --> 00:08:09.370
Well, it's square
root of 2 over 2

00:08:09.370 --> 00:08:11.200
to the right in the
positive direction.

00:08:11.200 --> 00:08:13.520
So x is equal to square
root of 2 over 2.

00:08:13.520 --> 00:08:17.747
And y is square root of 2 over
2 in the upwards direction,

00:08:17.747 --> 00:08:20.080
the vertical direction, the
positive vertical direction.

00:08:20.080 --> 00:08:22.180
So it's also square
root of 2 over 2.

00:08:22.180 --> 00:08:25.410
Cosine of theta is
just the x-coordinate.

00:08:25.410 --> 00:08:27.410
So it's square root of 2 over 2.

00:08:27.410 --> 00:08:29.830
Sine of theta is just
the y-coordinate.

00:08:29.830 --> 00:08:33.620
So you see immediately that they
are indeed equal at that point.

00:08:33.620 --> 00:08:36.679
So at this point they're
both equal to square root

00:08:36.679 --> 00:08:41.549
of 2 over 2.

00:08:41.549 --> 00:08:43.409
Now what about this
point right over here,

00:08:43.409 --> 00:08:48.600
which looks right in
between pi and 3 pi over 2.

00:08:48.600 --> 00:08:51.660
So that's going to be-- so this
is pi, this is 3 pi over 2.

00:08:51.660 --> 00:08:56.040
It is right over here.

00:08:56.040 --> 00:08:58.540
So it's another
pi over 4 plus pi.

00:08:58.540 --> 00:09:03.500
So pi plus pi over 4 is the
same thing as 4 pi over 4

00:09:03.500 --> 00:09:04.840
plus pi over 4.

00:09:04.840 --> 00:09:07.635
So this is the
angle 5 pi over 4.

00:09:14.450 --> 00:09:16.610
So that's what we're
trying to figure out.

00:09:16.610 --> 00:09:19.810
What are the value of
these functions at theta

00:09:19.810 --> 00:09:21.350
equals 5 pi over 4?

00:09:21.350 --> 00:09:23.829
Well, there's multiple
ways to think about it.

00:09:23.829 --> 00:09:25.620
You could even use a
little bit of geometry

00:09:25.620 --> 00:09:28.240
to say, well if this
is a 45 degree angle,

00:09:28.240 --> 00:09:32.390
then this right over here
is also a 45 degree angle.

00:09:32.390 --> 00:09:34.940
You could say that the reference
angle in terms of degrees

00:09:34.940 --> 00:09:36.749
is 45 degrees.

00:09:36.749 --> 00:09:38.290
And we could do a
very similar thing.

00:09:38.290 --> 00:09:40.980
We can draw a right triangle.

00:09:40.980 --> 00:09:43.460
We know the hypotenuse is 1.

00:09:43.460 --> 00:09:47.180
We know that if this is a right
angle, this is 45 degrees.

00:09:47.180 --> 00:09:50.560
If that's 45 degrees, then
this is also 45 degrees.

00:09:50.560 --> 00:09:54.650
And we have a triangle
that's very similar.

00:09:54.650 --> 00:09:56.490
They're actually
congruent triangles.

00:09:56.490 --> 00:09:59.680
So hypotenuse is 1, 45, 90.

00:09:59.680 --> 00:10:01.820
We then know that the
length of this side

00:10:01.820 --> 00:10:03.340
is square root of 2 over 2.

00:10:03.340 --> 00:10:05.631
And the length of this side
is square root of 2 over 2.

00:10:05.631 --> 00:10:08.100
The exact same logic
we used over here.

00:10:08.100 --> 00:10:16.090
So based on that, what is
the coordinate of that point?

00:10:16.090 --> 00:10:17.690
Well, let's think
about the x value.

00:10:17.690 --> 00:10:21.099
It's square root of 2 over
2 in the negative direction.

00:10:21.099 --> 00:10:22.640
We have to go square
root of 2 over 2

00:10:22.640 --> 00:10:25.160
to the left of the origin.

00:10:25.160 --> 00:10:27.810
So it's negative square
root of 2 over 2.

00:10:27.810 --> 00:10:31.610
This point on the x-axis
is negative square root

00:10:31.610 --> 00:10:32.980
of 2 over 2.

00:10:32.980 --> 00:10:34.220
What about the y value?

00:10:34.220 --> 00:10:37.180
Well, we have to go square
root of 2 over 2 down,

00:10:37.180 --> 00:10:39.880
in the downward direction,
from the origin.

00:10:39.880 --> 00:10:43.540
So it's also negative
square root of 2 over 2.

00:10:43.540 --> 00:10:47.240
So cosine of theta is negative
square root of 2 over 2.

00:10:47.240 --> 00:10:49.560
And sine of theta is
also negative square root

00:10:49.560 --> 00:10:51.560
of 2 over 2.

00:10:51.560 --> 00:10:55.060
And so we see that we do indeed
have the same value for cosine

00:10:55.060 --> 00:10:57.040
of theta and sine of
theta right there.

00:10:57.040 --> 00:11:04.050
They are both equal to
the negative square root

00:11:04.050 --> 00:11:06.770
of 2 over 2.


Example: Graph of cosine

WEBVTT
Kind: captions
Language: en

00:00:00.529 --> 00:00:03.807
- [Instructor] In the graph
below, for what values

00:00:03.807 --> 00:00:07.696
of theta does cosine of theta equal one,

00:00:07.696 --> 00:00:09.995
and for what values of theta does cosine

00:00:09.995 --> 00:00:11.705
of theta equal negative one?

00:00:11.705 --> 00:00:14.178
And they've very nicely graphed it for us,

00:00:14.178 --> 00:00:16.986
the horizontal axis is the theta axis,

00:00:16.986 --> 00:00:19.256
and the vertical axis is the y-axis,

00:00:19.256 --> 00:00:23.395
and so this is the graph of y
is equal to cosine of theta.

00:00:23.395 --> 00:00:25.788
And it makes sense with
a unit circle definition,

00:00:25.788 --> 00:00:28.436
and I'll just make sure that
we're comfortable with that,

00:00:28.436 --> 00:00:31.348
because with our unit circle definition,

00:00:31.348 --> 00:00:34.789
so let me draw ourselves a unit circle,

00:00:34.789 --> 00:00:36.122
and I'm just going to
draw it very roughly,

00:00:36.122 --> 00:00:37.478
just so that we get the general idea

00:00:37.478 --> 00:00:39.048
of what's going on here.

00:00:39.048 --> 00:00:41.625
When theta is equal to
zero, we're at this point

00:00:41.625 --> 00:00:43.660
right over here on the unit circle.

00:00:43.660 --> 00:00:46.097
Well what's the
x-coordinate of that point?

00:00:46.097 --> 00:00:49.955
Well it's one, and you see
when theta's equal to zero,

00:00:49.955 --> 00:00:53.872
on this graph, cosine of
theta is equal to one.

00:00:55.261 --> 00:00:58.178
When theta is equal to pi over two,

00:00:59.089 --> 00:01:01.522
we're at this point on the unit circle,

00:01:01.522 --> 00:01:04.089
and the x-coordinate is what?

00:01:04.089 --> 00:01:06.647
Well, the x-coordinate there is zero.

00:01:06.647 --> 00:01:09.108
And you see once again,
when we're at pi over two,

00:01:09.108 --> 00:01:11.767
the x-coordinate is zero, so
this is completely consistent

00:01:11.767 --> 00:01:13.277
with our unit circle definition.

00:01:13.277 --> 00:01:15.673
As we move in the rightward direction,

00:01:15.673 --> 00:01:18.919
we're moving counterclockwise
around the unit circle,

00:01:18.919 --> 00:01:21.310
and as we move in the leftward direction,

00:01:21.310 --> 00:01:23.249
we're moving counter, sorry.

00:01:23.249 --> 00:01:24.605
If we move in the rightward direction,

00:01:24.605 --> 00:01:26.095
we're moving counterclockwise,

00:01:26.095 --> 00:01:27.719
and as we're moving in
the leftward direction

00:01:27.719 --> 00:01:29.567
along the axis in the negative angles,

00:01:29.567 --> 00:01:32.086
we're moving in the
clockwise, we're moving

00:01:32.086 --> 00:01:35.061
in the clockwise direction
around our unit circle.

00:01:35.061 --> 00:01:36.712
So let's answer their question.

00:01:36.712 --> 00:01:40.879
For what values of theta does
cosine of theta equal one?

00:01:42.688 --> 00:01:44.669
Well we can just read the
graph right over here.

00:01:44.669 --> 00:01:48.371
It equals one, so cosine
of theta equals one,

00:01:48.371 --> 00:01:50.871
cosine of theta equals one at,

00:01:51.884 --> 00:01:54.972
at theta is equal to, well,
we see it right over here.

00:01:54.972 --> 00:01:58.472
Theta is equal to zero, theta is equal to,

00:01:59.585 --> 00:02:02.730
well we've gotta go all
the way again to two pi,

00:02:02.730 --> 00:02:05.042
two pi, but then it just keeps going

00:02:05.042 --> 00:02:06.868
on and on, and it makes sense.

00:02:06.868 --> 00:02:08.943
Theta equaled, or sorry, cosine of theta,

00:02:08.943 --> 00:02:10.831
the x-coordinate on this
unit circle equaled one

00:02:10.831 --> 00:02:12.123
right when we were at zero angle,

00:02:12.123 --> 00:02:13.467
and we had to go all the
way around the circle

00:02:13.467 --> 00:02:15.684
to get back to that point, two pi radians.

00:02:15.684 --> 00:02:18.467
But then it'll be again when
we get to four pi radians,

00:02:18.467 --> 00:02:22.634
and then six pi radians,
so two pi, four pi, six pi,

00:02:25.078 --> 00:02:27.945
and I guess you could
see the pattern here.

00:02:27.945 --> 00:02:30.879
We're gonna keep hitting
cosine of theta equals one

00:02:30.879 --> 00:02:33.977
every two pi, so you could
really kind of view this

00:02:33.977 --> 00:02:36.283
as every multiple of two pi.

00:02:36.283 --> 00:02:40.283
Two pi n, where n is an
integer, n is integer...

00:02:42.589 --> 00:02:44.018
is an integer.

00:02:44.018 --> 00:02:46.486
And that applies also for negative values.

00:02:46.486 --> 00:02:48.606
If you're going the other way around,

00:02:48.606 --> 00:02:49.774
if we're going the other way around,

00:02:49.774 --> 00:02:53.120
we don't get back until
we get to negative two pi.

00:02:53.120 --> 00:02:55.119
Notice we were at zero,
and then the next time

00:02:55.119 --> 00:02:57.681
we're at one again is at negative two pi,

00:02:57.681 --> 00:02:59.349
and then negative four pi, and then over

00:02:59.349 --> 00:03:00.883
and over and over again.

00:03:00.883 --> 00:03:02.727
But this applies, if n
is an integer, n can be

00:03:02.727 --> 00:03:04.468
a negative number, and so we get

00:03:04.468 --> 00:03:06.885
to all of the negative values

00:03:07.815 --> 00:03:10.825
of theta where cosine of
theta is equal to one.

00:03:10.825 --> 00:03:12.193
Now let's think about when cosine

00:03:12.193 --> 00:03:14.608
of theta is equal to negative one.

00:03:14.608 --> 00:03:18.235
So cosine of theta is
equal to negative one

00:03:18.235 --> 00:03:21.326
at theta is equal to,
well we can just look

00:03:21.326 --> 00:03:23.476
at this graph right over here.

00:03:23.476 --> 00:03:26.136
Well when theta is equal to pi,

00:03:26.136 --> 00:03:29.747
when theta is equal to pi, and let's see,

00:03:29.747 --> 00:03:31.407
well, it kinda goes off this graph,

00:03:31.407 --> 00:03:34.455
but this graph would keep going like this,

00:03:34.455 --> 00:03:35.573
would keep going like this, and you'd see

00:03:35.573 --> 00:03:37.192
it would also be at three pi.

00:03:37.192 --> 00:03:39.562
And you can visualize it over here.

00:03:39.562 --> 00:03:41.517
Theta, cosine of theta
is equal to negative one

00:03:41.517 --> 00:03:45.362
when we're at this point
on the unit circle.

00:03:45.362 --> 00:03:48.146
So that happens when we get to pi radians,

00:03:48.146 --> 00:03:49.333
and then it won't happen again

00:03:49.333 --> 00:03:54.311
until we get to two pi, three
pi radians, three pi radians.

00:03:54.311 --> 00:03:57.135
And it won't happen again
until we go to two pi,

00:03:57.135 --> 00:03:59.277
until we add another two pi,

00:03:59.277 --> 00:04:01.280
until we make one entire revolution,

00:04:01.280 --> 00:04:05.767
so then that's going to
be five, five pi radians.

00:04:05.767 --> 00:04:08.431
And you can keep going on and on and on,

00:04:08.431 --> 00:04:10.349
and that's also true in
the negative direction,

00:04:10.349 --> 00:04:12.951
so if we take two pi away from this,

00:04:12.951 --> 00:04:15.353
so if we were here, and if
we go all the way around

00:04:15.353 --> 00:04:17.812
back to negative pi, it
should also be the case,

00:04:17.812 --> 00:04:20.413
and you actually see it
right over here on the graph.

00:04:20.413 --> 00:04:24.122
So you could think about this as two pi,

00:04:24.122 --> 00:04:25.539
two pi n plus pi,

00:04:27.863 --> 00:04:30.522
or you could view it as two n plus one,

00:04:30.522 --> 00:04:32.772
or two n plus one times pi,

00:04:34.356 --> 00:04:37.856
where pi is, sorry, where n is an integer.

00:04:41.003 --> 00:04:45.973
Let me right that a little
bit neater, n is integer.

00:04:45.973 --> 00:04:48.189
At every one of those points, cosine,

00:04:48.189 --> 00:04:50.895
or for every one of these
thetas, cosine of theta is going

00:04:50.895 --> 00:04:53.672
to keep hitting negative
one over and over again.

00:04:53.672 --> 00:04:56.875
And you see it, it goes
it goes from one bottom,

00:04:56.875 --> 00:04:58.914
where you can kind of
valley to the next valley,

00:04:58.914 --> 00:05:02.215
it takes two pi to get to the next valley,

00:05:02.215 --> 00:05:04.500
two pi to get to the next valley.

00:05:04.500 --> 00:05:07.168
And that was also the
same thing for the peaks.

00:05:07.168 --> 00:05:10.010
It took two pi to go
from the top of one hill

00:05:10.010 --> 00:05:14.460
to the top of the next,
and then two pi again

00:05:14.460 --> 00:05:17.293
to the top of the hill after that.


Example: Graph, domain, and range of sine function

WEBVTT
Kind: captions
Language: en

00:00:00.775 --> 00:00:03.438
- [Narrator] We're asked
what are the domain

00:00:03.438 --> 00:00:06.298
and range of the sine function?

00:00:06.298 --> 00:00:09.322
So to think about that,
let's actually draw

00:00:09.322 --> 00:00:12.579
the sine function out,
and what I have here,

00:00:12.579 --> 00:00:15.198
on the left hand side right
over here I've got a unit

00:00:15.198 --> 00:00:18.373
circle, and I can, let me
truncate this a little bit.

00:00:18.373 --> 00:00:19.960
I don't need that space right there,

00:00:19.960 --> 00:00:21.578
so let me clear that out.

00:00:21.578 --> 00:00:24.161
So I have a unit circle
on the left hand side

00:00:24.161 --> 00:00:26.817
right over here, and I'm
gonna use that to figure out

00:00:26.817 --> 00:00:30.076
the values of sine of
theta for a given theta.

00:00:30.076 --> 00:00:34.076
So, on the unit circle
this is X, and this is Y,

00:00:35.060 --> 00:00:37.733
or you could even do this, as the, well,

00:00:37.733 --> 00:00:39.424
we can just use X or Y,
and so for a given theta

00:00:39.424 --> 00:00:42.649
we can see where that angle
entered to the terminal side

00:00:42.649 --> 00:00:45.857
of the angle intersects the
unit circle, and then those,

00:00:45.857 --> 00:00:47.625
the Y coordinate of that point

00:00:47.625 --> 00:00:49.746
is going to be sine of theta.

00:00:49.746 --> 00:00:52.088
And over here I'm going to graph.

00:00:52.088 --> 00:00:54.758
Still Y in the vertical axis,

00:00:54.758 --> 00:00:59.086
but I'm gonna graph the graph
of Y is equal to sine of theta.

00:00:59.086 --> 00:01:03.253
Y is equal to sine of theta,
and on the horizontal axis

00:01:04.490 --> 00:01:07.215
I'm not gonna graph X,
but I'm gonna graph theta.

00:01:07.215 --> 00:01:09.657
You can do theta as the
independent variable here,

00:01:09.657 --> 00:01:12.520
and it's gonna be theta is
going to be in radians.

00:01:12.520 --> 00:01:16.585
So we're essentially going
to pick a bunch of thetas

00:01:16.585 --> 00:01:19.406
and then come up with
what sine of theta is,

00:01:19.406 --> 00:01:20.976
and then graph it.

00:01:20.976 --> 00:01:23.629
So let's set up a little
bit of a table here.

00:01:23.629 --> 00:01:26.562
Let's set up a little bit of a table.

00:01:26.562 --> 00:01:28.827
And so, over here I have theta,

00:01:28.827 --> 00:01:30.723
and over here we're going

00:01:30.723 --> 00:01:34.102
to figure out what sine of theta is,

00:01:34.102 --> 00:01:36.743
and we could do a bunch of theta values.

00:01:36.743 --> 00:01:40.910
We could start, we could start,
let's say we start at zero.

00:01:42.372 --> 00:01:45.328
Let's say we start at
theta is equal to zero.

00:01:45.328 --> 00:01:47.064
What is sine of theta gonna be?

00:01:47.064 --> 00:01:48.640
Well, when the angle is zero

00:01:48.640 --> 00:01:51.421
we intersect the unit
circle right over there.

00:01:51.421 --> 00:01:53.815
The Y coordinate of this is still zero.

00:01:53.815 --> 00:01:57.823
This is the point, this is
the point one comma zero.

00:01:57.823 --> 00:02:00.580
The Y coordinate is zero,
so sine of theta is zero.

00:02:00.580 --> 00:02:04.724
We could say sine of
zero is equal to zero.

00:02:04.724 --> 00:02:07.985
Sine of zero is equal to zero.

00:02:07.985 --> 00:02:11.568
Now let's try theta is
equal to pi over two.

00:02:13.402 --> 00:02:15.331
Theta is equal to pi over two.

00:02:15.331 --> 00:02:18.041
I'm just doing the ones that
are really easy to figure out.

00:02:18.041 --> 00:02:19.447
So if theta is equal to pi over two,

00:02:19.447 --> 00:02:21.571
that's the same thing
as a 90 degree angle.

00:02:21.571 --> 00:02:24.466
So, the terminal side is going
to be right along the Y axis

00:02:24.466 --> 00:02:28.999
just like that, and where it
intersects, where it intersects

00:02:28.999 --> 00:02:33.346
the unit circle is right over
here, and what point is that?

00:02:33.346 --> 00:02:36.429
Well that's the point zero comma one.

00:02:37.513 --> 00:02:40.096
So what is sine of pi over two?

00:02:42.384 --> 00:02:44.430
Well sine of pi over two is just

00:02:44.430 --> 00:02:46.044
the Y coordinate right over here.

00:02:46.044 --> 00:02:46.877
It is one.

00:02:46.877 --> 00:02:48.884
Sine of pi over two is one.

00:02:48.884 --> 00:02:51.407
Let's keep going and you might
see a little pattern here.

00:02:51.407 --> 00:02:54.070
We're just going more and
more around the circle.

00:02:54.070 --> 00:02:56.318
So let's think about what's, what,

00:02:56.318 --> 00:02:58.240
what happens when theta is equal to pi.

00:02:58.240 --> 00:03:01.990
When theta is equal to
pi, what is sine of pi?

00:03:03.211 --> 00:03:06.666
Well, we intersect the unit
circle right over there.

00:03:06.666 --> 00:03:09.921
That coordinate is negative one, zero.

00:03:09.921 --> 00:03:11.951
Sine is the Y coordinate,

00:03:11.951 --> 00:03:14.567
so this right over here is sine of pi.

00:03:14.567 --> 00:03:16.924
Sine of pi is zero.

00:03:16.924 --> 00:03:19.382
Let's go to three pi over two.

00:03:19.382 --> 00:03:23.480
Three pi over two, well now
we've gone three quarters

00:03:23.480 --> 00:03:26.563
of the way around, around the circle.

00:03:28.415 --> 00:03:32.028
We intersect the terminal
side of the angle intersects

00:03:32.028 --> 00:03:33.762
the unit circle right over here,

00:03:33.762 --> 00:03:38.558
and so based on that what is
sine of three pi over two?

00:03:38.558 --> 00:03:41.732
Well, this point right over
here is the point negative,

00:03:41.732 --> 00:03:46.723
we gotta be careful, is
zero, is zero negative one.

00:03:46.723 --> 00:03:49.774
The sine of theta is the
same thing as a Y coordinate

00:03:49.774 --> 00:03:52.661
if the Y coordinate is
the sine of theta, so theta,

00:03:52.661 --> 00:03:54.671
when theta's pi over two sine of theta,

00:03:54.671 --> 00:03:56.427
or when theta's three pi over two

00:03:56.427 --> 00:03:59.594
sine of theta is equal to negative one.

00:04:00.759 --> 00:04:02.957
And let's come full circle.

00:04:02.957 --> 00:04:05.046
Let's come full circle here.

00:04:05.046 --> 00:04:08.664
So let's go all the way
to theta equaling two pi.

00:04:08.664 --> 00:04:11.454
Let me do a color, hey, I'll
just use the yellow here.

00:04:11.454 --> 00:04:13.242
What happens when theta is equal to two pi?

00:04:13.242 --> 00:04:16.281
Well then we've gone all
the way around the circle,

00:04:16.281 --> 00:04:18.801
and we are back to where we started,

00:04:18.801 --> 00:04:21.483
and the Y coordinate is zero,

00:04:21.483 --> 00:04:24.333
so sine of two pi is once again zero,

00:04:24.333 --> 00:04:25.882
and if we were to keep going around,

00:04:25.882 --> 00:04:28.480
we're gonna start seeing as we
keep incrementing the angle,

00:04:28.480 --> 00:04:30.930
we're gonna start seeing the
same pattern emerge again.

00:04:30.930 --> 00:04:32.740
Well, let's try to graph this.

00:04:32.740 --> 00:04:36.907
So when theta is equal to
zero, sine of theta is zero.

00:04:38.069 --> 00:04:42.668
When theta is equal to pi
over two, when theta is equal

00:04:42.668 --> 00:04:46.751
to pi over two, pi over
two, sine of theta is one.

00:04:47.861 --> 00:04:49.751
So, we'll use the same scale.

00:04:49.751 --> 00:04:53.584
So sine of theta, sine
of theta is equal to one.

00:04:55.017 --> 00:04:56.707
This is, I'll just make this,

00:04:56.707 --> 00:04:59.047
this is one on this
axis, and on that axis.

00:04:59.047 --> 00:05:01.607
So we can maybe see a little
bit of a parallel here.

00:05:01.607 --> 00:05:05.849
When theta is equal to
pi, sine of theta is zero.

00:05:05.849 --> 00:05:09.491
So when theta is equal to
pi, sine of theta is zero.

00:05:09.491 --> 00:05:11.887
So we go back right over there.

00:05:11.887 --> 00:05:14.401
When theta is equal to three pi over two,

00:05:14.401 --> 00:05:18.250
so that would be right over
here, three pi over two,

00:05:18.250 --> 00:05:20.667
sine of theta is negative one.

00:05:21.725 --> 00:05:23.241
So this is negative one over here.

00:05:23.241 --> 00:05:25.648
I'll do the same scale over here.

00:05:25.648 --> 00:05:26.828
I'll make this negative.

00:05:26.828 --> 00:05:28.205
I'll make, let me make down a little bit,

00:05:28.205 --> 00:05:30.376
I'll make this negative one, and so,

00:05:30.376 --> 00:05:32.793
sine of theta is negative one.

00:05:33.905 --> 00:05:38.538
And then, when theta is two
pi, sine of theta is zero.

00:05:38.538 --> 00:05:42.705
So when theta is two pi, two
pi, sine of theta is zero.

00:05:44.598 --> 00:05:46.020
And so we can connect the dots.

00:05:46.020 --> 00:05:47.371
You could try other points in between

00:05:47.371 --> 00:05:49.500
and you get something, you get a graph

00:05:49.500 --> 00:05:52.331
that looks something like this.

00:05:52.331 --> 00:05:54.311
It looks something like this.

00:05:54.311 --> 00:05:56.687
My best attempt at drawing it freehand.

00:05:56.687 --> 00:06:00.020
It looks something, something like this.

00:06:00.877 --> 00:06:02.721
There's a reason why curves
that look like this are called

00:06:02.721 --> 00:06:06.338
sinusoids, because they're
the graph of a sine function.

00:06:06.338 --> 00:06:08.584
So this like, just like this,

00:06:08.584 --> 00:06:09.928
but that's not the entire graph.

00:06:09.928 --> 00:06:11.036
We could keep going.

00:06:11.036 --> 00:06:13.462
We could go, we could
add another pi over two.

00:06:13.462 --> 00:06:16.232
If you added another pi over
two, so if you go to two pi,

00:06:16.232 --> 00:06:18.253
and then you add another pi over two.

00:06:18.253 --> 00:06:19.837
So you could view this
as two and a half pi,

00:06:19.837 --> 00:06:21.392
or however you wanna think about it,

00:06:21.392 --> 00:06:23.048
then you're gonna go back over here.

00:06:23.048 --> 00:06:24.298
So then you're gonna get back

00:06:24.298 --> 00:06:26.161
to sine of theta being equal to one.

00:06:26.161 --> 00:06:28.569
So you're gonna go back to
this point right over here,

00:06:28.569 --> 00:06:29.899
and you could keep going.

00:06:29.899 --> 00:06:31.702
You go another pi over
two, you're gonna go back

00:06:31.702 --> 00:06:33.881
to this point, and you're
gonna be over here,

00:06:33.881 --> 00:06:37.357
and so the curve, the curve
or the function sine of theta

00:06:37.357 --> 00:06:40.255
is really defined for any theta value,

00:06:40.255 --> 00:06:43.255
any real theta value that you choose.

00:06:44.095 --> 00:06:45.569
So any theta value.

00:06:45.569 --> 00:06:46.952
Well, what about negatives?

00:06:46.952 --> 00:06:48.818
I mean obviously I agree,
as you keep increasing theta

00:06:48.818 --> 00:06:50.888
like this we just keep going
around and around the circle

00:06:50.888 --> 00:06:52.465
and this pattern kind of emerges,

00:06:52.465 --> 00:06:54.395
but what happens when we go
in the negative direction?

00:06:54.395 --> 00:06:55.654
Well, let's try it out.

00:06:55.654 --> 00:06:58.365
What happens if we were to
take, if we were to take

00:06:58.365 --> 00:07:00.229
negative pi over two?

00:07:00.229 --> 00:07:01.575
So let me do that.

00:07:01.575 --> 00:07:05.742
So negative pi over two, well,
that's going right over here,

00:07:08.205 --> 00:07:11.283
and so we intersect the unit
circle right over there.

00:07:11.283 --> 00:07:13.555
The Y coordinate is negative one.

00:07:13.555 --> 00:07:16.721
So sine of negative pi
over two is negative one,

00:07:16.721 --> 00:07:18.950
and we see that it just continues.

00:07:18.950 --> 00:07:19.931
It just continues.

00:07:19.931 --> 00:07:23.201
So sine of theta is defined
for any positive, negative,

00:07:23.201 --> 00:07:26.198
or any theta, positive or negative,

00:07:26.198 --> 00:07:28.703
non negative, zero, anything.

00:07:28.703 --> 00:07:31.023
So it's defined for anything.

00:07:31.023 --> 00:07:33.345
So, let's go back to the question.

00:07:33.345 --> 00:07:36.648
So I could just keep drawing
this function on and on and on.

00:07:36.648 --> 00:07:38.212
So let's go back to the question.

00:07:38.212 --> 00:07:41.462
What is the domain, what is the domain?

00:07:43.121 --> 00:07:47.121
What is the domain of
sine of the sine function?

00:07:49.656 --> 00:07:52.831
And just as a reminder, the
domain are all of the inputs

00:07:52.831 --> 00:07:54.516
over which the function is defined,

00:07:54.516 --> 00:07:56.879
or all of the valid
inputs into the function

00:07:56.879 --> 00:07:59.691
that the function will actually
spit out a valid answer.

00:07:59.691 --> 00:08:02.906
So what is the domain
of the sine function?

00:08:02.906 --> 00:08:04.735
Well, we already saw.

00:08:04.735 --> 00:08:07.076
We can put in any theta here.

00:08:07.076 --> 00:08:10.746
So you could say the
domain, the domain is all,

00:08:10.746 --> 00:08:14.496
all real numbers, all
real, all real numbers.

00:08:19.673 --> 00:08:21.560
Now, what about the range?

00:08:21.560 --> 00:08:23.310
What about the range?

00:08:26.163 --> 00:08:29.348
Well just as a review,
the range is sometimes

00:08:29.348 --> 00:08:32.358
in more technical math
classes called the image.

00:08:32.358 --> 00:08:34.797
It's the set of all the values that

00:08:34.797 --> 00:08:37.243
the function can actually take on.

00:08:37.243 --> 00:08:38.633
Well what is that set?

00:08:38.633 --> 00:08:39.642
What is the range here?

00:08:39.642 --> 00:08:41.570
What is all the values that Y equals

00:08:41.570 --> 00:08:44.204
sine of theta could actually take on?

00:08:44.204 --> 00:08:48.022
Well, we see that it keeps
going between positive one,

00:08:48.022 --> 00:08:52.379
it keeps going between positive
one and then to negative one

00:08:52.379 --> 00:08:54.312
and then back to positive
one and then negative one.

00:08:54.312 --> 00:08:57.755
It takes on all the values in between.

00:08:57.755 --> 00:09:01.977
So you see that sine of
theta, sine of theta is always

00:09:01.977 --> 00:09:04.267
going to be less than or equal to one,

00:09:04.267 --> 00:09:05.678
and it's always going to be greater than

00:09:05.678 --> 00:09:07.599
or equal to negative one.

00:09:07.599 --> 00:09:10.137
So you could say that
the range of sine of theta

00:09:10.137 --> 00:09:13.147
is the set of all numbers
between negative one

00:09:13.147 --> 00:09:17.036
and positive one, and it
includes negative one and one,

00:09:17.036 --> 00:09:21.203
and that's why we put brackets
here instead of parentheses.


Example: Amplitude and period

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.880
We're asked to determine the
amplitude and the period of y

00:00:04.880 --> 00:00:08.835
equals negative
1/2 cosine of 3x.

00:00:08.835 --> 00:00:10.210
So the first thing
we have to ask

00:00:10.210 --> 00:00:16.300
ourselves is, what does
amplitude even refer to?

00:00:16.300 --> 00:00:19.220
Well the amplitude of
a periodic function

00:00:19.220 --> 00:00:23.180
is just half the difference
between the minimum and maximum

00:00:23.180 --> 00:00:24.360
values it takes on.

00:00:24.360 --> 00:00:27.040
So if I were to draw a
periodic function like this,

00:00:27.040 --> 00:00:29.106
and it would just go back
and forth between two--

00:00:29.106 --> 00:00:30.605
let me draw it a
little bit neater--

00:00:30.605 --> 00:00:34.630
it goes back and forth
between two values like that.

00:00:34.630 --> 00:00:37.260
So between that
value and that value.

00:00:37.260 --> 00:00:39.890
You take the difference between
the two, and half of that

00:00:39.890 --> 00:00:40.939
is the amplitude.

00:00:40.939 --> 00:00:42.730
Another way of thinking
about the amplitude

00:00:42.730 --> 00:00:47.760
is how much does it sway
from its middle position.

00:00:47.760 --> 00:00:49.700
Right over here,
we have y equals

00:00:49.700 --> 00:00:52.810
negative 1/2 cosine of 3x.

00:00:52.810 --> 00:00:57.790
So what is going to be
the amplitude of this?

00:00:57.790 --> 00:00:59.930
Well, the easy way
to think about it

00:00:59.930 --> 00:01:04.040
is just what is multiplying
the cosine function.

00:01:04.040 --> 00:01:07.140
And you could do the same thing
if it was a sine function.

00:01:07.140 --> 00:01:09.800
We have negative
1/2 multiplying it.

00:01:09.800 --> 00:01:15.280
So the amplitude
in this situation

00:01:15.280 --> 00:01:22.450
is going to be the absolute
value of negative 1/2,

00:01:22.450 --> 00:01:25.350
which is equal to 1/2.

00:01:25.350 --> 00:01:28.970
And you might say, well, why
do I not care about the sign?

00:01:28.970 --> 00:01:31.740
Why do I take the
absolute value of it?

00:01:31.740 --> 00:01:34.310
Well, the negative just
flips the function around.

00:01:34.310 --> 00:01:35.920
It's not going to
change how much it

00:01:35.920 --> 00:01:38.734
sways between its minimum
and maximum position.

00:01:38.734 --> 00:01:40.400
The other thing is,
well, how is it just

00:01:40.400 --> 00:01:42.840
simply the absolute
value of this thing?

00:01:42.840 --> 00:01:44.540
And to realize the
y, you just have

00:01:44.540 --> 00:01:47.630
to remember that a cosine
function or a sine function

00:01:47.630 --> 00:01:50.070
varies between positive
1 and negative 1,

00:01:50.070 --> 00:01:51.610
if it's just a simple function.

00:01:51.610 --> 00:01:55.070
So this is just multiplying
that positive 1 or negative 1.

00:01:55.070 --> 00:01:57.414
And so if normally
the amplitude,

00:01:57.414 --> 00:01:59.080
if you didn't have
any coefficient here,

00:01:59.080 --> 00:02:01.460
if the coefficient was
positive or negative 1,

00:02:01.460 --> 00:02:03.440
the amplitude would just be 1.

00:02:03.440 --> 00:02:04.950
Now, you're changing
it or you're

00:02:04.950 --> 00:02:06.510
multiplying it by this amount.

00:02:06.510 --> 00:02:09.240
So the amplitude is 1/2.

00:02:09.240 --> 00:02:11.450
Now let's think
about the period.

00:02:14.190 --> 00:02:16.710
So the first thing
I want to ask you

00:02:16.710 --> 00:02:21.760
is, what does the period
of a cyclical function--

00:02:21.760 --> 00:02:23.880
or periodic function,
I should say--

00:02:23.880 --> 00:02:27.530
what does the period of
a periodic function even

00:02:27.530 --> 00:02:29.800
refer to?

00:02:29.800 --> 00:02:34.730
Well let me draw some axes on
this function right over here.

00:02:34.730 --> 00:02:39.840
Let's say that this right
over here is the y-axis.

00:02:39.840 --> 00:02:40.776
That's the y-axis.

00:02:40.776 --> 00:02:42.650
And let's just say, for
the sake of argument,

00:02:42.650 --> 00:02:46.170
this is the x-axis
right over here.

00:02:46.170 --> 00:02:48.800
So the period of a
periodic function

00:02:48.800 --> 00:02:52.030
is the length of the
smallest interval that

00:02:52.030 --> 00:02:55.220
contains exactly one copy
of the repeating pattern

00:02:55.220 --> 00:02:58.307
of that periodic function.

00:02:58.307 --> 00:02:59.390
So what do they mean here?

00:02:59.390 --> 00:03:00.370
Well, what's repeating?

00:03:00.370 --> 00:03:04.980
So we go down and then
up just like that.

00:03:04.980 --> 00:03:11.340
Then we go down
and then we go up.

00:03:11.340 --> 00:03:13.910
So in this case, the length
of the smallest interval

00:03:13.910 --> 00:03:16.560
that contains exactly one
copy of the repeating pattern.

00:03:16.560 --> 00:03:20.250
This could be one of the
smallest repeating patterns.

00:03:20.250 --> 00:03:25.710
And so this length between here
and here would be one period.

00:03:25.710 --> 00:03:29.340
Then we could go between here
and here is another period.

00:03:29.340 --> 00:03:31.430
And there's multiple--
this isn't the only pattern

00:03:31.430 --> 00:03:32.070
that you could pick.

00:03:32.070 --> 00:03:34.570
You could say, well, I'm going
to define my pattern starting

00:03:34.570 --> 00:03:40.430
here going up and then
going down like that.

00:03:40.430 --> 00:03:42.547
So you could say that's
my smallest length.

00:03:42.547 --> 00:03:44.130
And then you would
see that, OK, well,

00:03:44.130 --> 00:03:45.629
if you go in the
negative direction,

00:03:45.629 --> 00:03:48.130
the next repeating
version of that pattern

00:03:48.130 --> 00:03:49.660
is right over there.

00:03:49.660 --> 00:03:52.690
But either way you're going
to get the same length

00:03:52.690 --> 00:03:57.690
that it takes to
repeat that pattern.

00:03:57.690 --> 00:04:00.290
So given that,
what is the period

00:04:00.290 --> 00:04:03.750
of this function
right over here?

00:04:03.750 --> 00:04:08.600
Well, to figure out the
period, we just take 2 pi

00:04:08.600 --> 00:04:10.920
and divide it by
the absolute value

00:04:10.920 --> 00:04:13.020
of the coefficient
right over here.

00:04:13.020 --> 00:04:17.029
So we divide it by the absolute
value of 3, which is just 3.

00:04:17.029 --> 00:04:21.540
So we get 2 pi over 3.

00:04:21.540 --> 00:04:24.650
Now we need to think
about why does this work?

00:04:24.650 --> 00:04:26.890
Well, if you think about
just a traditional cosine

00:04:26.890 --> 00:04:32.130
function, a traditional cosine
function or a traditional sine

00:04:32.130 --> 00:04:34.890
function, it has
a period of 2 pi.

00:04:34.890 --> 00:04:39.290
If you think about
the unit circle,

00:04:39.290 --> 00:04:42.380
2 pi, if you start at
0, 2 pi radians later,

00:04:42.380 --> 00:04:44.020
you're back to
where you started.

00:04:44.020 --> 00:04:45.750
2 pi radians,
another 2 pi, you're

00:04:45.750 --> 00:04:47.730
back to where you started.

00:04:47.730 --> 00:04:49.230
If you go in the
negative direction,

00:04:49.230 --> 00:04:51.800
you go negative 2 pi, you're
back to where you started.

00:04:51.800 --> 00:04:54.460
For any angle here,
if you go 2 pi,

00:04:54.460 --> 00:04:56.540
you're back to where
you were before.

00:04:56.540 --> 00:04:59.630
You go negative 2 pi, you're
back to where you were before.

00:04:59.630 --> 00:05:06.672
So the periods for
these are all 2 pi.

00:05:06.672 --> 00:05:08.130
And the reason why
this makes sense

00:05:08.130 --> 00:05:11.400
is that this coefficient
makes you get to 2 pi

00:05:11.400 --> 00:05:13.200
or negative-- in
this case 2 pi--

00:05:13.200 --> 00:05:16.790
it's going to make you get
to 2 pi all that much faster.

00:05:16.790 --> 00:05:19.306
And so it gets-- your period
is going to be a lower number.

00:05:19.306 --> 00:05:20.180
It takes less length.

00:05:20.180 --> 00:05:22.805
You're going to get to 2
pi three times as fast.

00:05:22.805 --> 00:05:24.305
Now you might say,
well, why are you

00:05:24.305 --> 00:05:26.410
taking the absolute value here?

00:05:26.410 --> 00:05:28.320
Well, if this was
a negative number,

00:05:28.320 --> 00:05:31.250
it would get you to negative
2 pi all that much faster.

00:05:31.250 --> 00:05:34.114
But either way, you're going
to be completing one cycle.

00:05:34.114 --> 00:05:35.530
So with that out
of the way, let's

00:05:35.530 --> 00:05:37.100
visualize these two things.

00:05:37.100 --> 00:05:45.790
Let's actually draw
negative 1/2 cosine of 3x.

00:05:45.790 --> 00:05:49.870
So let me draw my axes here.

00:05:49.870 --> 00:05:51.220
My best attempt.

00:05:51.220 --> 00:05:52.750
So this is my y-axis.

00:05:52.750 --> 00:05:54.790
This is my x-axis.

00:05:54.790 --> 00:05:59.530
And then let me draw some--
So this is 0 right over here.

00:05:59.530 --> 00:06:00.940
x is equal to 0.

00:06:00.940 --> 00:06:03.410
And let me draw x is
equal to positive 1/2.

00:06:03.410 --> 00:06:05.400
I'll draw it right over here.

00:06:05.400 --> 00:06:07.820
So x is equal to positive 1/2.

00:06:07.820 --> 00:06:10.460
And we haven't shifted this
function up or down any.

00:06:10.460 --> 00:06:13.640
Then, if we wanted to, we
could add a constant out here,

00:06:13.640 --> 00:06:15.550
outside of the cosine function.

00:06:15.550 --> 00:06:22.060
But this is positive 1/2, or we
could just write that as 1/2.

00:06:22.060 --> 00:06:26.440
And then down here, let's say
that this is negative 1/2.

00:06:26.440 --> 00:06:29.442
And so let me draw that bound.

00:06:29.442 --> 00:06:30.900
I'm just drawing
these dotted lines

00:06:30.900 --> 00:06:33.170
so it'll become
easy for me to draw.

00:06:33.170 --> 00:06:38.160
And what happens when this is 0?

00:06:38.160 --> 00:06:40.535
Well cosine of 0 is 1.

00:06:40.535 --> 00:06:44.500
But we're going to multiply
it by negative 1/2.

00:06:44.500 --> 00:06:49.420
So it's going to be negative
1/2 right over here.

00:06:49.420 --> 00:06:51.360
And then it's going
to start going up.

00:06:55.516 --> 00:06:56.890
It can only go in
that direction.

00:06:56.890 --> 00:06:57.950
It's bounded.

00:06:57.950 --> 00:07:02.610
It's going to start going
up, then it'll come back down

00:07:02.610 --> 00:07:07.650
and then it will get back
to that original point

00:07:07.650 --> 00:07:08.980
right over here.

00:07:08.980 --> 00:07:11.310
And the question is,
what is this distance?

00:07:11.310 --> 00:07:13.090
What is this length?

00:07:13.090 --> 00:07:15.050
What is this length going to be?

00:07:15.050 --> 00:07:16.490
Well, we know what
its period is.

00:07:16.490 --> 00:07:17.830
It's 2 pi over 3.

00:07:17.830 --> 00:07:20.200
It's going to get to
this point three times

00:07:20.200 --> 00:07:22.840
as fast as a traditional
cosine function.

00:07:22.840 --> 00:07:26.500
So this is going
to be 2 pi over 3.

00:07:26.500 --> 00:07:30.840
And then if you give
it another 2 pi over 3,

00:07:30.840 --> 00:07:32.920
it's going to get back
to that same point again.

00:07:32.920 --> 00:07:37.200
So if you go another 2 pi
over 3, so in this case,

00:07:37.200 --> 00:07:44.277
you've now gone 4 pi over 3,
you've completed another cycle.

00:07:44.277 --> 00:07:46.110
So that length right
over there is a period.

00:07:46.110 --> 00:07:49.640
And then you could
also do the same thing

00:07:49.640 --> 00:07:50.933
in the negative direction.

00:07:50.933 --> 00:07:57.540
So this right over here would be
negative, negative 2 pi over 3.

00:07:57.540 --> 00:08:01.092
And to visualize the amplitude,
you see that it can go 1/2.

00:08:01.092 --> 00:08:02.800
Well, there's two ways
to think about it.

00:08:02.800 --> 00:08:05.310
The difference between the
maximum and the minimum point

00:08:05.310 --> 00:08:05.810
is 1.

00:08:05.810 --> 00:08:07.630
Half of that is 1/2.

00:08:07.630 --> 00:08:12.200
Or you could say that it's
going 1/2 in magnitude,

00:08:12.200 --> 00:08:15.570
or it's swaying 1/2 away
from its middle position

00:08:15.570 --> 00:08:20.330
in the positive or the
negative direction.


Example: Calculator to evaluate inverse trig function

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:04.260
Use your calculator to find
the angle to the nearest

00:00:04.260 --> 00:00:10.214
degree, whose tangent
value is 0.875.

00:00:10.214 --> 00:00:11.880
So what do you think
that's going to be?

00:00:11.880 --> 00:00:18.430
The angle to the nearest degree,
whose tangent value is 0.875.

00:00:18.430 --> 00:00:21.400
Well, the first thing
is to be careful.

00:00:21.400 --> 00:00:28.350
They are not asking for
the tangent of 0.875.

00:00:28.350 --> 00:00:31.020
They're not saying
that 0.875 is an angle

00:00:31.020 --> 00:00:33.850
and we want to find
the tangent of it.

00:00:33.850 --> 00:00:35.510
That is not what they're asking.

00:00:35.510 --> 00:00:37.070
They're asking
something different.

00:00:37.070 --> 00:00:39.650
They're saying that
there's some angle.

00:00:39.650 --> 00:00:41.440
Maybe we could call it theta.

00:00:41.440 --> 00:00:42.910
There's some angle
that if we were

00:00:42.910 --> 00:00:44.880
to take the tangent
of that angle,

00:00:44.880 --> 00:00:50.920
then we are going to get 0.875.

00:00:50.920 --> 00:00:54.820
So how do we figure
out this mystery angle

00:00:54.820 --> 00:00:57.470
using our calculator?

00:00:57.470 --> 00:00:59.270
Well, the way we
do it is a function

00:00:59.270 --> 00:01:01.420
called the inverse
tangent function.

00:01:01.420 --> 00:01:04.640
Or sometimes it's called
the arc tangent function.

00:01:04.640 --> 00:01:06.250
And essentially,
if you apply that,

00:01:06.250 --> 00:01:07.958
and if you just view
this as an equation,

00:01:07.958 --> 00:01:09.730
you take the inverse
tangent of something.

00:01:09.730 --> 00:01:13.820
So if you take the
inverse tangent of this.

00:01:13.820 --> 00:01:15.910
And you take the
inverse tangent--

00:01:15.910 --> 00:01:17.560
let me clear this
out a little bit.

00:01:22.710 --> 00:01:25.150
I'm just taking the inverse
tangent of both sides.

00:01:25.150 --> 00:01:27.360
0.875.

00:01:27.360 --> 00:01:29.610
On the left hand side you're
just left with the angle.

00:01:29.610 --> 00:01:32.430
The inverse tangent of tangent
is just your original angle.

00:01:32.430 --> 00:01:36.130
And on the right hand side
you get the inverse tangent

00:01:36.130 --> 00:01:39.660
of 0.875.

00:01:39.660 --> 00:01:41.150
And if you're in
degree mode this

00:01:41.150 --> 00:01:43.467
is going to return
an angle in degrees.

00:01:43.467 --> 00:01:45.550
And if you're in radians
mode it will return angle

00:01:45.550 --> 00:01:47.280
in terms of radians.

00:01:47.280 --> 00:01:51.350
But let's figure this out
using our calculators.

00:01:51.350 --> 00:01:54.900
So we could use the
traditional calculator,

00:01:54.900 --> 00:01:56.600
or more traditional calculator.

00:01:56.600 --> 00:01:58.290
Let me clear everything out.

00:01:58.290 --> 00:01:59.977
Now all I see is a tangent.

00:01:59.977 --> 00:02:01.310
I don't see the inverse tangent.

00:02:01.310 --> 00:02:03.684
In order to get that you have
to press the inverse button

00:02:03.684 --> 00:02:04.647
right over here.

00:02:04.647 --> 00:02:06.230
And this is not the
reciprocal button.

00:02:06.230 --> 00:02:07.650
This is the reciprocal button.

00:02:07.650 --> 00:02:09.108
This right over
here is the inverse

00:02:09.108 --> 00:02:10.508
to get the inverse functions.

00:02:10.508 --> 00:02:13.070
So you press inverse and notice
you saw that this changed.

00:02:13.070 --> 00:02:15.595
This would not actually happened
on an actual calculator.

00:02:15.595 --> 00:02:16.470
But you get the idea.

00:02:16.470 --> 00:02:18.553
The actual calculator might
have something up here

00:02:18.553 --> 00:02:20.860
to show that you can
access it that way.

00:02:20.860 --> 00:02:26.340
And so now we could say 0.875.

00:02:26.340 --> 00:02:28.510
And we're going to take
the inverse tangent of it.

00:02:28.510 --> 00:02:30.420
Inverse tangent.

00:02:30.420 --> 00:02:32.090
So take the inverse
tangent of it.

00:02:32.090 --> 00:02:33.840
We already know that
we're in degree mode.

00:02:33.840 --> 00:02:35.631
And we want to round
to the nearest degree.

00:02:35.631 --> 00:02:38.020
So it's going to be, if we
round to the nearest degree--

00:02:38.020 --> 00:02:40.610
we get 41 degrees.

00:02:40.610 --> 00:02:43.885
So theta is approximately,
it's not exactly--

00:02:43.885 --> 00:02:46.930
we're rounding-- 41 degrees.

00:02:46.930 --> 00:02:51.420
Let's do the exact same thing
with the graphing calculator.

00:02:51.420 --> 00:02:53.810
So let me turn it on.

00:02:53.810 --> 00:02:55.440
And let me clear it.

00:02:55.440 --> 00:02:58.592
And now we want to take
the inverse tangent.

00:02:58.592 --> 00:03:01.050
If you have this calculator it
will look exactly like this.

00:03:01.050 --> 00:03:02.540
You see this tangent.

00:03:02.540 --> 00:03:03.440
Inverse tangent.

00:03:03.440 --> 00:03:04.200
Tan.

00:03:04.200 --> 00:03:06.250
They have it written as
to the negative 1 power.

00:03:06.250 --> 00:03:06.860
It looks like that.

00:03:06.860 --> 00:03:08.693
But it's not really to
the negative 1 power.

00:03:08.693 --> 00:03:10.160
It's really inverse tangent.

00:03:10.160 --> 00:03:11.284
It's right there in yellow.

00:03:11.284 --> 00:03:14.230
To access that you press
second and then that function.

00:03:14.230 --> 00:03:19.210
Inverse tangent of 0.875.

00:03:19.210 --> 00:03:21.370
We get the exact same value.

00:03:21.370 --> 00:03:23.780
It's roughly equal
to 41 degrees if we

00:03:23.780 --> 00:03:26.992
round to the nearest degree.


Example: Converting radians to degrees

WEBVTT
Kind: captions
Language: en

00:00:00.773 --> 00:00:07.481
We're asked to convert pi radians and negative pi/3 radians to degrees.

00:00:07.481 --> 00:00:08.855
And the first question I'll ask you:

00:00:08.855 --> 00:00:13.613
If you do one revolution, You have an angle that went all the way around once.

00:00:13.613 --> 00:00:16.867
How many radians is that?

00:00:16.867 --> 00:00:23.816
Well we know that it is 2 pi radians.

00:00:23.816 --> 00:00:28.101
Now that exact same angle if we were to measure it in degrees,

00:00:28.101 --> 00:00:29.488
How many degrees is that?

00:00:29.488 --> 00:00:38.189
Well if you were doing degrees, it would be one full revolution. That is equal to 360 degrees

00:00:38.189 --> 00:00:39.714
Now, can we simplify this?

00:00:39.714 --> 00:00:42.701
That's a bore to write this little, superscript circle

00:00:42.701 --> 00:00:44.251
That's literally the units of the question.

00:00:44.251 --> 00:00:46.468
Sometimes it doesn't look like a unit but it is a unit.

00:00:46.468 --> 00:00:48.521
You could literally write degrees instead of that little symbol.

00:00:48.521 --> 00:00:55.337
Now can we simplify this a little bit? Well sure,

00:00:55.337 --> 00:00:57.783
Both two pi and 360 are divisible by two

00:00:57.783 --> 00:01:00.990
so lets divide things by two, and if we do that, what do we get?

00:01:00.990 --> 00:01:03.871
Or what are pi radians equal to?

00:01:03.871 --> 00:01:09.731
Well on the left side here we're just left with pi radians,

00:01:09.731 --> 00:01:18.200
and on the righthand side here, 360 divided by two is 180.

00:01:18.200 --> 00:01:21.169
And we have still the units which are degrees.

00:01:21.169 --> 00:01:24.241
So we get pi radians are equal to 180 degrees.

00:01:24.241 --> 00:01:26.372
Which actually answers the first part of our question.

00:01:26.372 --> 00:01:28.336
We wanted to convert pi radians, well we just figured out!

00:01:28.336 --> 00:01:37.039
Pi radians are equal to 180 degrees.

00:01:37.039 --> 00:01:41.398
If you want to think about it, pi radians are halfway around the circle

00:01:41.398 --> 00:01:47.442
Halfway around the circle like that, and it is the same thing as 180 degrees.

00:01:47.442 --> 00:01:49.384
So now lets think about the second part.

00:01:49.384 --> 00:01:52.884
We want to convertnegative pi over three radians.

00:01:52.884 --> 00:01:55.720
--Switch to a new color-- so negative pi over three,

00:01:55.720 --> 00:02:03.519
so how do we convert that?

00:02:03.519 --> 00:02:07.810
So what do we get based on this information right over here.

00:02:07.810 --> 00:02:09.449
Well, to figure this out we need to know how many degrees there are per radian.

00:02:09.449 --> 00:02:21.017
We need to multiply this by degrees -- I'm going to write the word out instead of the circle here --

00:02:21.017 --> 00:02:26.799
It would be really hard to visualize that, degrees per radian

00:02:26.799 --> 00:02:29.437
So how many degrees are there per radian?

00:02:29.437 --> 00:02:34.971
well we know that for 180 degrees we have pi radians.

00:02:34.971 --> 00:02:43.971
Or you can say there are 180 over pi degrees per radian.

00:02:43.971 --> 00:02:44.803
This is going to work out:

00:02:44.803 --> 00:02:46.593
We have however manyradians we have

00:02:46.593 --> 00:02:49.082
times the number of degrees per radian.

00:02:49.082 --> 00:02:54.969
So of course the units are going to work out. The radians cancel out, the pi also cancels.

00:02:54.969 --> 00:03:03.702
And you are left with 180 divided by 3, leaving us with

00:03:03.702 --> 00:03:05.796
what is that?

00:03:05.796 --> 00:03:06.863
Negative 60, and we don't want to forget the units

00:03:06.863 --> 00:03:10.863
We could write them out, the only unit left is degrees. WE could write out the word degrees or just put that symbol there.


Example: Converting degrees to radians

WEBVTT
Kind: captions
Language: en

00:00:00.709 --> 00:00:04.184
- [Instructor] We're asked
to convert 150 degrees

00:00:04.184 --> 00:00:07.101
and negative 45 degrees to radians.

00:00:08.389 --> 00:00:10.139
Let's think about the relationship

00:00:10.139 --> 00:00:13.545
between degrees and
radians, and to do that,

00:00:13.545 --> 00:00:15.361
let me just draw a little circle here.

00:00:15.361 --> 00:00:16.907
So that's the center of the circle,

00:00:16.907 --> 00:00:19.939
and then do my best shot, best attempt

00:00:19.939 --> 00:00:24.206
to freehand draw a
reasonable-looking circle.

00:00:24.206 --> 00:00:27.373
That's not, I've done worse than that.

00:00:28.506 --> 00:00:32.155
Alright, now, if we were to go in degrees,

00:00:32.155 --> 00:00:34.688
if we were to go one
time around the circle

00:00:34.688 --> 00:00:37.688
like that, how many degrees is that?

00:00:38.822 --> 00:00:42.072
We know that that would be 360 degrees.

00:00:44.738 --> 00:00:47.388
If we did the same thing,
how many radians is that,

00:00:47.388 --> 00:00:51.488
if we were to go all the
way around the circle?

00:00:51.488 --> 00:00:53.465
We just have to remember,
when we're measuring

00:00:53.465 --> 00:00:55.221
in terms of radians, we're really talking

00:00:55.221 --> 00:00:57.739
about the arc that subtends that angle.

00:00:57.739 --> 00:00:59.538
So if you go all the way around,

00:00:59.538 --> 00:01:01.655
you're really talking about the arc length

00:01:01.655 --> 00:01:03.704
of the entire circle, or essentially

00:01:03.704 --> 00:01:05.137
the circumference of the circle.

00:01:05.137 --> 00:01:07.238
And you're essentially
saying, how many radius's

00:01:07.238 --> 00:01:09.738
this is, or radii, or how many

00:01:11.712 --> 00:01:13.522
radii is the circumference of the circle.

00:01:13.522 --> 00:01:18.021
You know a circumference
of a circle is two pi

00:01:18.021 --> 00:01:19.963
times the radius, or you could say

00:01:19.963 --> 00:01:22.171
that the length of the circumference

00:01:22.171 --> 00:01:24.671
of the circle is two pi radii.

00:01:28.562 --> 00:01:30.597
If you wanna know the
exact length, you just have

00:01:30.597 --> 00:01:33.590
to get the length of the radius
and multiply it by two pi.

00:01:33.590 --> 00:01:36.787
That just comes from the, really, actually

00:01:36.787 --> 00:01:39.421
the definition of pi, but
it comes from what we know

00:01:39.421 --> 00:01:43.554
as the formula for the
circumference of a circle.

00:01:43.554 --> 00:01:45.025
If we were to go all the way around this,

00:01:45.025 --> 00:01:47.358
this is also two pi radians.

00:01:55.003 --> 00:01:57.836
That tells us that two pi radians,

00:02:00.205 --> 00:02:03.069
as an angle measure, is
the exact same thing,

00:02:03.069 --> 00:02:05.902
and I'm gonna write it out, as 360

00:02:07.352 --> 00:02:08.352
360 degrees.

00:02:10.503 --> 00:02:12.752
And then we can take
all of this relationship

00:02:12.752 --> 00:02:14.236
and manipulate it in different ways.

00:02:14.236 --> 00:02:15.535
If we wanna simplify a little bit,

00:02:15.535 --> 00:02:18.869
we can divide both sides
of this equation by two,

00:02:18.869 --> 00:02:21.135
in which case, you are left with,

00:02:21.135 --> 00:02:22.585
if you divide both sides by two,

00:02:22.585 --> 00:02:24.918
you are left with pi radians

00:02:27.569 --> 00:02:29.569
is equal to 180 degrees.

00:02:39.668 --> 00:02:42.437
How can we use this relationship now

00:02:42.437 --> 00:02:45.270
to figure out what 150 degrees is?

00:02:49.017 --> 00:02:50.197
Well, this relationship, we could write it

00:02:50.197 --> 00:02:51.101
in different ways.

00:02:51.101 --> 00:02:53.551
We could divide both sides by 180 degrees,

00:02:53.551 --> 00:02:55.801
and we could get pi radians

00:02:58.751 --> 00:03:00.084
over 180 degrees

00:03:03.634 --> 00:03:05.901
is equal to one, which is just another way

00:03:05.901 --> 00:03:07.834
of saying that there are pi radians

00:03:07.834 --> 00:03:10.366
for every 180 degrees, or you could say,

00:03:10.366 --> 00:03:12.949
pi over 180 radians per degree.

00:03:14.300 --> 00:03:15.266
The other option, you could divide

00:03:15.266 --> 00:03:17.789
both sides of this by pi radians.

00:03:17.789 --> 00:03:20.882
You could say, you would
get on the left hand side

00:03:20.882 --> 00:03:23.400
you'd get one, and you would also get,

00:03:23.400 --> 00:03:25.717
on the right hand side,
you would get 180 degrees

00:03:25.717 --> 00:03:27.467
for every pi radians.

00:03:34.625 --> 00:03:37.005
Or you could interpret this as 180

00:03:37.005 --> 00:03:39.255
over pi degrees per radian.

00:03:40.888 --> 00:03:44.692
How would we figure out, how
would we do what they asked us?

00:03:44.692 --> 00:03:47.005
Let's convert 150 degrees to radians.

00:03:47.005 --> 00:03:48.338
Let me write the word out.

00:03:48.338 --> 00:03:49.671
So, 150 degrees.

00:03:54.238 --> 00:03:56.056
Well, we wanna convert this to radians,

00:03:56.056 --> 00:04:00.223
so we really care about
how many radians there are

00:04:01.131 --> 00:04:03.578
per degree, actually, let
me do that in that color.

00:04:03.578 --> 00:04:04.528
(typing)

00:04:04.528 --> 00:04:07.028
We care about how many radians

00:04:10.288 --> 00:04:12.038
there are per degree.

00:04:13.772 --> 00:04:16.355
We'll do that same green color.

00:04:17.397 --> 00:04:18.314
Per degree.

00:04:19.421 --> 00:04:20.755
How many radians are there per degree?

00:04:20.755 --> 00:04:23.071
Well, we already know, there's pi radians

00:04:23.071 --> 00:04:26.772
for every 180 degrees, or there are pi...

00:04:26.772 --> 00:04:28.688
Let me do that yellow color.

00:04:28.688 --> 00:04:30.438
There are pi over 180

00:04:34.188 --> 00:04:36.320
radians per degree.

00:04:36.320 --> 00:04:37.971
And so, if we multiply,
and this all works out

00:04:37.971 --> 00:04:39.704
because you have degrees in the numerator,

00:04:39.704 --> 00:04:43.437
degrees in the denominator,
these cancel out,

00:04:43.437 --> 00:04:45.986
and so you are left with 150 times pi

00:04:45.986 --> 00:04:48.470
divided by 180 radians.

00:04:48.470 --> 00:04:49.970
So what do we get?

00:04:51.219 --> 00:04:53.869
This becomes, let me just rewrite it.

00:04:53.869 --> 00:04:54.952
150 times pi.

00:04:58.569 --> 00:05:00.319
All of that over 180,

00:05:01.994 --> 00:05:05.827
so this is equal to, and
we get it in radians.

00:05:07.370 --> 00:05:09.469
And so, if we simplify it, let's see,

00:05:09.469 --> 00:05:11.120
we can divide the numerator
and the denominator

00:05:11.120 --> 00:05:13.120
both by, looks like, 30.

00:05:13.970 --> 00:05:17.785
So if you divide the
numerator by 30, you get five.

00:05:17.785 --> 00:05:21.735
You divide the denominator
by 30, you get six.

00:05:21.735 --> 00:05:24.752
So you get five pi over six radians,

00:05:24.752 --> 00:05:29.368
or 5/6 pi radians, depending
how you wanna do it.

00:05:29.368 --> 00:05:32.569
Now let's do the same thing
for negative 45 degrees.

00:05:32.569 --> 00:05:34.785
What do you get for negative 45 degrees

00:05:34.785 --> 00:05:38.704
if you were to convert that to radians?

00:05:38.704 --> 00:05:41.104
Same exact process.

00:05:41.104 --> 00:05:44.136
You have negative, and I'll
do this one a little quicker.

00:05:44.136 --> 00:05:46.419
Negative 45 degrees.

00:05:46.419 --> 00:05:48.154
I'll write down the word.

00:05:48.154 --> 00:05:50.154
Times, times pi radians,

00:05:52.838 --> 00:05:55.588
pi radians for every 180 degrees.

00:06:00.154 --> 00:06:02.918
The degrees cancel out, and you're left

00:06:02.918 --> 00:06:06.001
with negative 45 pi over 180 radians.

00:06:07.668 --> 00:06:10.501
So this is equal to negative 45 pi

00:06:14.384 --> 00:06:15.217
over 180,

00:06:17.384 --> 00:06:18.801
over 180 radians.

00:06:23.601 --> 00:06:25.539
How can we simplify this?

00:06:25.539 --> 00:06:27.169
Well it looks like
they're both, at minimum,

00:06:27.169 --> 00:06:30.586
divisible by nine, nine times five is 45,

00:06:31.968 --> 00:06:34.134
this is nine times 20,
so actually it's gonna be

00:06:34.134 --> 00:06:37.551
divisible by more than just, let's see...

00:06:39.409 --> 00:06:41.088
Actually, they're both divisible by 45.

00:06:41.088 --> 00:06:42.388
What am I doing?

00:06:42.388 --> 00:06:46.221
If you divide the numerator
by 45, you get one.

00:06:46.221 --> 00:06:48.590
You divide the denominator by 45,

00:06:48.590 --> 00:06:51.323
45 goes into 180 four times.

00:06:51.323 --> 00:06:54.505
You're left with negative
pi over four radians.

00:06:54.505 --> 00:06:56.038
This is equal to negative pi

00:06:56.038 --> 00:06:57.538
over four radians.

00:06:58.769 --> 00:07:00.102
And we are done.


Example: Radian measure and arc length

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:07.400
In the circle below,
central angle GOH

00:00:07.400 --> 00:00:10.540
has a measure of 3/2 radians.

00:00:10.540 --> 00:00:14.000
What is the length of arc GH?

00:00:14.000 --> 00:00:15.719
So let's define
some variables that

00:00:15.719 --> 00:00:17.760
will help us manipulate
them a little bit easier.

00:00:17.760 --> 00:00:20.520
So where is central angle GOH?

00:00:20.520 --> 00:00:23.120
It's this angle right over here.

00:00:23.120 --> 00:00:24.580
Let's call that theta.

00:00:24.580 --> 00:00:36.750
So let's say that theta is equal
to the measure of angle GOH.

00:00:36.750 --> 00:00:40.880
Now they're asking us,
what's the length of arc GH?

00:00:40.880 --> 00:00:44.110
Well, arc GH is this
right over here.

00:00:44.110 --> 00:00:45.950
And let's call that s.

00:00:45.950 --> 00:00:54.520
So let's let s be equal
to the length of arc GH.

00:00:54.520 --> 00:00:56.490
And they also give
us the radius is

00:00:56.490 --> 00:00:58.340
6 centimeters for this circle.

00:00:58.340 --> 00:00:59.930
But let's just call
that r for now.

00:00:59.930 --> 00:01:03.280
So the radius is equal
to 6 centimeters.

00:01:03.280 --> 00:01:05.112
So the first thing
that we should

00:01:05.112 --> 00:01:07.570
think about when we're dealing
with arc lengths and radians

00:01:07.570 --> 00:01:12.880
is how do angles measured in
radians relate to arc length?

00:01:12.880 --> 00:01:15.960
In particular, how does an
angle measured in radians

00:01:15.960 --> 00:01:20.800
relate to the length of the
arc that subtends that angle?

00:01:20.800 --> 00:01:23.670
And in this case, the length
of this arc right over here,

00:01:23.670 --> 00:01:29.470
arc GH, subtends angle theta,
subtends central angle GOH.

00:01:29.470 --> 00:01:32.010
So what's the relationship?

00:01:32.010 --> 00:01:35.120
Well, the definition
of radian measure

00:01:35.120 --> 00:01:38.060
says that an angle
measured in radians

00:01:38.060 --> 00:01:42.450
is the ratio between
the length of arc

00:01:42.450 --> 00:01:51.210
that subtends that angle, so s,
and the radius of the circle.

00:01:51.210 --> 00:01:53.830
If this was a unit circle,
then it would just literally

00:01:53.830 --> 00:01:56.420
be equal to that arc length.

00:01:56.420 --> 00:01:59.530
If it's not a unit circle--
or, I guess, in all cases,

00:01:59.530 --> 00:02:01.750
unit circle or
not-- it's the ratio

00:02:01.750 --> 00:02:03.400
between the length
of the arc that

00:02:03.400 --> 00:02:08.130
subtends the angle
and the radius.

00:02:08.130 --> 00:02:13.380
Now given this definition,
really, of what a radian is,

00:02:13.380 --> 00:02:15.710
let's see what we know
and what we don't know.

00:02:15.710 --> 00:02:19.800
And we can manipulate
this however we see fit.

00:02:19.800 --> 00:02:25.570
You could see this written as if
you multiply both sides of this

00:02:25.570 --> 00:02:29.020
by r, you get r
theta is equal to s.

00:02:29.020 --> 00:02:30.760
Or another way of
writing it is you

00:02:30.760 --> 00:02:39.170
could get s is equal to r theta.

00:02:39.170 --> 00:02:43.040
And we could have solved this
equation or this equation.

00:02:43.040 --> 00:02:44.817
But for this problem,
this one's actually

00:02:44.817 --> 00:02:46.650
going to be a little
bit useful because this

00:02:46.650 --> 00:02:48.540
is the unknown,
the length of GH.

00:02:48.540 --> 00:02:50.840
We want to know what this is.

00:02:50.840 --> 00:02:53.130
This is the unknown
that they're asking for.

00:02:53.130 --> 00:02:55.010
And this explicitly
solves for the unknown.

00:02:55.010 --> 00:02:57.030
So this has essentially
solved it for us.

00:02:57.030 --> 00:02:58.420
So what else do we know?

00:02:58.420 --> 00:02:59.700
Well, we know the radius.

00:02:59.700 --> 00:03:01.870
We know the radius
is 6 centimeters.

00:03:01.870 --> 00:03:04.860
We multiply that times
the angle in radians,

00:03:04.860 --> 00:03:07.260
which is 3/2 radians.

00:03:07.260 --> 00:03:11.470
And we are left with, so s
is equal to 6 centimeters

00:03:11.470 --> 00:03:13.390
times 3/2.

00:03:13.390 --> 00:03:16.720
And so we are left with, this
is going to be equal to 6 times

00:03:16.720 --> 00:03:20.910
3 is 18 divided by 2 is 9.

00:03:20.910 --> 00:03:25.060
It's going to be equal
to 9 centimeters.

00:03:25.060 --> 00:03:28.455
And we are done.