Determining the equation of a trigonometric function

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:01.800
Hello.

00:00:01.800 --> 00:00:04.820
I'm now going to use the actual
Khan Academy website to

00:00:04.820 --> 00:00:05.680
do some more problems.

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And this time we're going to
go the other way around.

00:00:07.490 --> 00:00:10.860
We're going to look at the
graph of a trigonometric

00:00:10.860 --> 00:00:13.090
function and we're going to try
to figure out the equation.

00:00:13.090 --> 00:00:15.950
So let's start with the problem
we have in front of us.

00:00:15.950 --> 00:00:18.990
So we have a-- well we don't
know if it's a sine curve or a

00:00:18.990 --> 00:00:21.680
cosine curve, but I guess it's
fair to say that it's

00:00:21.680 --> 00:00:23.390
one of the two.

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Actually, let's answer
that question first.

00:00:25.250 --> 00:00:26.020
What do you think this is?

00:00:26.020 --> 00:00:31.490
Do you think that this is a
sine curve or a cosine curve?

00:00:31.490 --> 00:00:32.290
What's the difference?

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Or what's the easiest
difference to differentiate

00:00:34.560 --> 00:00:35.840
between the two?

00:00:35.840 --> 00:00:37.870
Well, what was sine of 0?

00:00:37.870 --> 00:00:41.110
Let me get a little
chalkboard in here.

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Let me bring it right here.

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Let me make sure my
pen tool is correct.

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All right.

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What is sine of 0?

00:00:55.540 --> 00:00:58.420
And it could be 0
degrees or 0 radians.

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Well if you remember from a
couple of the other modules,

00:00:59.930 --> 00:01:03.570
or even if you want to use a
calculator, sine of 0 is 0.

00:01:03.570 --> 00:01:06.150
And what is cosine of 0?

00:01:06.150 --> 00:01:10.030
Well cosine of 0, if you
remember from the last modules

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or you want to use a
calculator-- although you

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shouldn't have to use a
calculator for cosine of 0--

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you might remember
that it was 1.

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So this graph that we have
here, it's definitely when x is

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0, when the x-axis is 0, the
function definitely isn't 0.

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In fact, in this
case it's 1 1/2.

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So this tells us that this
isn't a sine curve-- although

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later with shifts we'll learn
that it could be a shifted sine

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curve-- that this isn't a sine
curve, that this is

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a cosine curve.

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And then you might ask, well
Sal if this is a cosine curve

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why is f of 0 equal to 1
1/2, or 3/2, instead of 1?

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Because I just said here
that cosine of 0 is 1.

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Well that's because there must
be some type of a coefficient

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here, let's call it A, that is
changing the amplitude

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of this cosine curve.

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And if you remember from
the last module, what

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do you think this A is?

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Well that A is just literally
the amplitude of the curve.

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And what's the amplitude
of this curve?

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Well, the amplitude of this
curve, if we just see how much

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it moves above and below the
x-axis, well it's that 3/2,

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or that 1 1/2, we've
been talking about.

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See it moves up 3/2,
and it moves down 3/2.

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So let me just write that.

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So we know that this is 3/2
cosine of, well, something x.

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Right?

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We know f of x is equal to
3/2 cosine of something x.

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And we could use the formula
that we learned in the previous

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video, that it equals
2pi over the period x.

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So now we just have to look at
the graph and try to figure

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out what the period
of the graph is.

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Well how many radians does
it take for the graph to

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start repeating again?

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Let me click on the
hint button, and maybe

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this'll help us.

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When I click hint--
there; drew the period.

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And you could have figured
it out on your own.

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If you just go from any point
and then follow the curve back

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to the same point again, you'll
see how long it's period is.

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And the hint on the Khan
icon actually told us

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that the period is 4pi.

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And you could just start from
any point to any other point.

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You could have gone from this
point and then gone down,

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gone back up, come down.

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And then you would have seen
that this distance is also 4pi.

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So we know that the
period is 4pi.

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And then I could click hint
again and it'll tell us stuff

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that we already figured out.

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The amplitude, we already
figured out, was 1 1/2.

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But let's just use a period,
because we already knew

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what the amplitude is.

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So the period here we already
figured out was 4pi.

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So let's just write
that in our equation.

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So f of x is equal to 3/2
cosine of 2pi divided by

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the period-- the period
in this case is 4pi-- x.

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That equals 3/2
cosine of 1/2 x.

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Now if you ever forget this
formula, which frankly

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I always do forget it.

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I've actually never
memorized it.

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I just try to think about
what the period would be.

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The way I think about it is the
coefficient on the x-term,

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that's a measure of how
many cycles does the graph

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do within 2pi radians.

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Let me see if I can explain
that within the context

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of this problem.

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So this problem, if we start at
0 and then 2pi is here, how

00:04:40.720 --> 00:04:43.330
many cycles do we complete
by the time we get to 2pi?

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We start here, we go back
here, and then we're at 2pi.

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Well we only got halfway
done through a cycle.

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So that's the coefficient
on the x-term.

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And that's how I remember it.

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So I could just say, well
that's 3/2 cosine of

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how many cycles do I
complete in 2pi radians?

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Well I only complete
half a cycle.

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3/2 times the cosine of 1/2 x.

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So that's our f of x.

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Let's do another problem.

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All right.

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OK.

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This one's interesting.

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So the first thing, just by
inspection we can figure out

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what this amplitude is.

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This is pretty easy.

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Right?

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How much does it move above
and below the x-axis?

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Well it only goes 1, so we know
that the coefficient, or the

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multiplier times the sine
or the cosine function--

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whichever this is-- is 1.

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So let's write that down.

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Let's write down that the
amplitude is equal to 1.

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Now let's try to figure
out if this is a sine

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or a cosine function.

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In the last problem we
said sine of 0 is 0

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and cosine of 0 is 1.

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Well f of 0 of this function,
whichever it is, is 0.

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So we know this is a
non-shifted sine function.

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So there.

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We have another piece
of information.

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It's a sine function.

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So the last thing we have to
figure out, we can either

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figure out the period or we
could use the method that I

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just showed you where we say,
well how many times does it

00:06:23.500 --> 00:06:26.800
cycle within 2pi radians?

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So let's do it that way.

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And then we immediately
know the coefficient.

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Let's see.

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Well actually, this graph
board doesn't even get

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all the way to 2pi.

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But let's see.

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It goes one cycle, two cycles.

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And I did two cycles in
only pi radians, right?

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Because I'm only at
x equals pi here.

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So if I did two cycles in pi
radians, then we must be able

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to do four cycles
in 2pi radians.

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Or we could actually
start here.

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Actually, this is better.

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Right?

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Going from negative pi to pi.

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That's 2pi radians.

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So we finish one cycle,
two cycles, three

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cycles, four cycles.

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So then we know what the
coefficient on the x-term is.

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So we know that it
is sine of 4x.

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So the answer here is f of x is
equal to 1-- because that's the

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amplitude-- the amplitude
times sine of 4x.

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I think we have
time for one more.

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And I want you to-- don't just
mechanically do whatever

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I'm telling you.

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I want you to think about why
counting the number of cycles

00:07:25.420 --> 00:07:27.150
within 2pi radians, why
that makes sense to you.

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Then think back to
the unit circle.

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Or think back why that formula,
the 2pi divided by the period,

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is also the coefficient.

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Think about why
that makes sense.

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If you realize why it makes
sense you'll never have to

00:07:38.990 --> 00:07:41.310
memorize it, and then 20 years
later when you're doing it like

00:07:41.310 --> 00:07:44.520
I'm doing it right now
you won't be confused.

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You'll be able to
re-derive the formulas.

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Let's do one more.

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All right.

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So what's the amplitude here?

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Well, let's see.

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The amplitude is 1/2.

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So let me delete the old stuff
that I was writing before.

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For some reason
it's not deleting.

00:08:07.900 --> 00:08:09.220
OK.

00:08:09.220 --> 00:08:10.200
Hope I don't confuse you.

00:08:10.200 --> 00:08:13.980
So the amplitude, let's
just call it Amplitude,

00:08:13.980 --> 00:08:14.730
is equal to 1/2.

00:08:18.140 --> 00:08:24.040
And how many cycles does it
complete within 2pi radians?

00:08:24.040 --> 00:08:26.560
Let's see.

00:08:26.560 --> 00:08:30.180
If we start here it looks like
it completes only half a cycle.

00:08:30.180 --> 00:08:30.390
Right?

00:08:30.390 --> 00:08:35.770
Because it takes 4pi radians
to complete the entire cycle.

00:08:35.770 --> 00:08:38.820
So it only completes
half a cycle.

00:08:38.820 --> 00:08:41.450
So we could think of
it either two ways.

00:08:41.450 --> 00:08:45.730
We could say that the period is
equal to 4pi, because that's

00:08:45.730 --> 00:08:48.720
how long it takes to complete
one cycle, or we could say

00:08:48.720 --> 00:08:54.710
it can only complete half a
cycle within 2pi radians.

00:08:54.710 --> 00:08:56.130
The last thing we have to
figure out: Is it a sine

00:08:56.130 --> 00:08:57.340
or a cosine function?

00:08:57.340 --> 00:08:59.640
Well, f of 0 is 0.

00:08:59.640 --> 00:09:00.020
Right?

00:09:00.020 --> 00:09:02.250
So it's a non-shifted
sine function.

00:09:02.250 --> 00:09:03.500
So then we're done.

00:09:03.500 --> 00:09:09.160
We have f of x is equal to
1/2-- figured out it's a sine

00:09:09.160 --> 00:09:13.950
function-- sine of what?

00:09:13.950 --> 00:09:17.280
How many cycles did it
complete in 2pi radians?

00:09:17.280 --> 00:09:18.950
It only completed half a cycle.

00:09:18.950 --> 00:09:21.090
Let me not cover the problem.

00:09:21.090 --> 00:09:23.360
It only completes half a cycle.

00:09:23.360 --> 00:09:25.530
So it's 1/2 sine of 1/2 x.

00:09:25.530 --> 00:09:29.300
Or we could use the formula f
of x equals goes 1/2 sine of

00:09:29.300 --> 00:09:31.190
2pi divided by the period x.

00:09:31.190 --> 00:09:31.430
Right?

00:09:31.430 --> 00:09:37.840
Because 2pi divided by the
period is equal to 2pi over

00:09:37.840 --> 00:09:42.100
4pi, which also equals 1/2.

00:09:42.100 --> 00:09:44.210
I think that'll give you
a sense of how to do

00:09:44.210 --> 00:09:45.850
these problems now.

00:09:45.850 --> 00:09:47.740
And I encourage you to
practice them on the

00:09:47.740 --> 00:09:50.140
Khan Academy website.

00:09:50.140 --> 00:09:51.630
Have fun.


More trig graphs

WEBVTT
Kind: captions
Language: en

00:00:01.280 --> 00:00:03.830
So now we have this graph
of this-- what was

00:00:03.830 --> 00:00:04.865
clearly a trig function.

00:00:04.865 --> 00:00:09.300
And our task is to figure
out what the function is.

00:00:09.300 --> 00:00:11.040
So let's look at this.

00:00:11.040 --> 00:00:13.210
The first thing I do when I
look at something like this,

00:00:13.210 --> 00:00:16.460
I want to figure out its
period and its amplitude.

00:00:16.460 --> 00:00:17.380
So what's its amplitude?

00:00:17.380 --> 00:00:18.410
That's always an easy one.

00:00:18.410 --> 00:00:19.230
So the amplitude.

00:00:25.620 --> 00:00:28.240
Well, that's just how much does
it move up and down above

00:00:28.240 --> 00:00:29.390
and below the x-axis?

00:00:29.390 --> 00:00:33.760
Well, the amplitude here is how
much it moves up the x-axis.

00:00:33.760 --> 00:00:36.750
Well, it moves up 1/2 above
and below the x-axis.

00:00:36.750 --> 00:00:38.400
So the amplitude is 1/2.

00:00:38.400 --> 00:00:42.980
And keep in mind, the amplitude
is not this whole distance.

00:00:42.980 --> 00:00:44.830
It's not this.

00:00:44.830 --> 00:00:50.370
It's just how much it moves
above or below the x-axis.

00:00:50.370 --> 00:00:52.860
So in this case, the
amplitude is 1/2.

00:00:52.860 --> 00:00:56.040
And then we have to figure
out what the period is.

00:00:56.040 --> 00:01:00.510
The period is, how long does it
take-- how many radians does

00:01:00.510 --> 00:01:04.040
it go through for it to go
through 1 complete cycle?

00:01:04.040 --> 00:01:08.890
Well, if we start here, and we
were to follow the graph, it

00:01:08.890 --> 00:01:11.780
doesn't take until right here
until we complete

00:01:11.780 --> 00:01:12.430
1 cycle, right?

00:01:12.430 --> 00:01:13.900
Because here, we're still
going down, and now

00:01:13.900 --> 00:01:14.470
we're going below.

00:01:14.470 --> 00:01:15.450
So we're not repeating yet.

00:01:15.450 --> 00:01:17.370
And here's where we
start repeating again.

00:01:17.370 --> 00:01:19.240
And then we start
repeating again here.

00:01:19.240 --> 00:01:23.130
So every pi radians, we
start the cycle over again.

00:01:23.130 --> 00:01:25.010
It happens the same if you
go backwards into the

00:01:25.010 --> 00:01:26.540
negative radians.

00:01:26.540 --> 00:01:28.685
So the period is pi, right?

00:01:28.685 --> 00:01:30.420
The period is pi.

00:01:30.420 --> 00:01:31.650
And you could start
from any point.

00:01:31.650 --> 00:01:33.760
You could start
from this point.

00:01:33.760 --> 00:01:36.350
And if you go, follow the
graph, and then come back to

00:01:36.350 --> 00:01:40.070
the same point again, we
see once again that the

00:01:40.070 --> 00:01:43.540
period is pi radians.

00:01:43.540 --> 00:01:45.215
Now we have to figure out
if this is a sine or

00:01:45.215 --> 00:01:46.335
a cosine function.

00:01:46.335 --> 00:01:50.680
And for now, we'll not
think about shifting.

00:01:50.680 --> 00:01:53.930
So let's think about what
happens when-- you know,

00:01:53.930 --> 00:01:55.900
we want to know what
this function is.

00:01:55.900 --> 00:02:01.010
f of x is equal to
question mark.

00:02:01.010 --> 00:02:04.116
Well, we see that f of 0 is 0.

00:02:04.116 --> 00:02:08.800
f of 0 is equal to 0.

00:02:08.800 --> 00:02:09.880
What does that tell us?

00:02:09.880 --> 00:02:12.830
Is this a sine or a
cosine function?

00:02:12.830 --> 00:02:15.510
Well, what's cosine of 0?

00:02:15.510 --> 00:02:18.620
Cosine of 0 is 1.

00:02:18.620 --> 00:02:19.630
And what's sine of 0?

00:02:19.630 --> 00:02:21.390
Well, sine of 0 is 0.

00:02:21.390 --> 00:02:23.290
And this function is 0.

00:02:23.290 --> 00:02:26.100
So we know that this
is a sine function.

00:02:26.100 --> 00:02:31.050
So we know the formula is going
to take the form f of x, it's

00:02:31.050 --> 00:02:38.110
going to equal the amplitude
times sine of 2 pi

00:02:38.110 --> 00:02:40.900
over the period x.

00:02:40.900 --> 00:02:43.440
And if we just substitute these
numbers we just figured out, we

00:02:43.440 --> 00:02:56.155
know that f of x is equal to
1/2 sine of 2 pi over pi x.

00:02:56.155 --> 00:03:03.650
The pi's cancel out and
you get f of x is equal

00:03:03.650 --> 00:03:10.270
to 1/2 sine of 2x.

00:03:15.590 --> 00:03:17.020
Let's define another function.

00:03:17.020 --> 00:03:24.980
Let's define g of x is
equal to 1/2 cosine of 2x.

00:03:24.980 --> 00:03:26.370
What would have
this looked like?

00:03:26.370 --> 00:03:28.900
Or what would have-- yeah.

00:03:28.900 --> 00:03:30.083
The grammar's a
little difficult.

00:03:32.650 --> 00:03:34.660
I picked the wrong color,
because f of x is

00:03:34.660 --> 00:03:35.170
actually the pink one.

00:03:35.170 --> 00:03:37.080
This is the one we have now.

00:03:37.080 --> 00:03:38.470
So actually, let
me circle that.

00:03:38.470 --> 00:03:40.100
This is f of x.

00:03:40.100 --> 00:03:42.770
f of x is this one right here.

00:03:42.770 --> 00:03:46.500
And now, g of x,
I'm going to do--.

00:03:46.500 --> 00:03:52.910
So when x is 0, what is g of 0?

00:03:52.910 --> 00:03:54.800
Let's put 0 in here.

00:03:54.800 --> 00:03:56.230
So this whole term
will become 0.

00:03:56.230 --> 00:03:57.360
What's cosine of 0?

00:03:57.360 --> 00:03:59.000
It's 1.

00:03:59.000 --> 00:04:00.000
And 1 times 1/2.

00:04:00.000 --> 00:04:02.300
So g of 0 is 1/2.

00:04:02.300 --> 00:04:08.180
So we would start here, and
then we would have-- just

00:04:08.180 --> 00:04:11.220
like the sine function-- we
would have a period of pi.

00:04:11.220 --> 00:04:13.500
Because it has the same
coefficient here.

00:04:13.500 --> 00:04:15.070
So this'll just look like this.

00:04:20.150 --> 00:04:21.185
I think you get the point.

00:04:24.710 --> 00:04:27.150
It's just like the sine
function was just shifted

00:04:27.150 --> 00:04:28.000
to the left of it.

00:04:30.550 --> 00:04:32.920
Well, I'm getting confused
on this-- ignore this.

00:04:32.920 --> 00:04:35.120
But if you look at this side,
the important thing to realize

00:04:35.120 --> 00:04:41.610
is that it intersects the
y-axis at not 1, but 1/2.

00:04:41.610 --> 00:04:43.950
And the reason why it doesn't
intersect it at 1, even

00:04:43.950 --> 00:04:47.120
though cosine of 0 is 1, is
because we have this 1/2

00:04:47.120 --> 00:04:48.420
coefficient right here.

00:04:48.420 --> 00:04:49.610
I guess you can't call
that a coefficient.

00:04:49.610 --> 00:04:55.430
I guess it's a 1/2 times
the cosine function.

00:04:55.430 --> 00:04:58.760
Hopefully that gives you a
little bit more of a sense of,

00:04:58.760 --> 00:05:00.980
if you just looked at a graph,
being able to figure

00:05:00.980 --> 00:05:02.850
out its equation.

00:05:02.850 --> 00:05:05.590
And I'll actually do one more
video where we'll actually use

00:05:05.590 --> 00:05:08.900
the Khan Academy trig
graphing module to figure

00:05:08.900 --> 00:05:11.360
out a couple more.

00:05:11.360 --> 00:05:12.900
See you soon.


Graphing trig functions

WEBVTT
Kind: captions
Language: en

00:00:01.110 --> 00:00:04.020
In this presentation we're
going to learn how to graph

00:00:04.020 --> 00:00:06.160
trig functions without
having to kind of

00:00:06.160 --> 00:00:07.970
graph point by point.

00:00:07.970 --> 00:00:11.410
And hopefully after this
presentation you can also look

00:00:11.410 --> 00:00:15.120
at a trig function and be able
to figure out the actual

00:00:15.120 --> 00:00:17.740
analytic definition of
the function as well.

00:00:17.740 --> 00:00:18.440
So let's start.

00:00:18.440 --> 00:00:19.720
Let's say f of x.

00:00:19.720 --> 00:00:22.550
Let me make sure I'm using
all the right tools.

00:00:22.550 --> 00:00:45.200
So let's say that f of x is
equal to 2 sine of 1/2 x.

00:00:45.200 --> 00:00:47.890
So when we look at this, a
couple interesting things here.

00:00:47.890 --> 00:00:49.980
How is this different than just
the regular sine function?

00:00:49.980 --> 00:00:55.170
Well, here we're multiplying
the whole function by 2,

00:00:55.170 --> 00:00:57.780
and also the coefficient
on the x-term is 1/2.

00:00:57.780 --> 00:01:00.650
And if you've seen some of the
videos I've made, you'll know

00:01:00.650 --> 00:01:05.010
that this term affects the
amplitude and this term affects

00:01:05.010 --> 00:01:08.070
the period, or the inverse of
the period, which

00:01:08.070 --> 00:01:08.730
is the frequency.

00:01:08.730 --> 00:01:09.130
Either way.

00:01:09.130 --> 00:01:11.440
It depends whether you're
talking about one or the

00:01:11.440 --> 00:01:12.750
inverse of the other one.

00:01:12.750 --> 00:01:14.710
So let's start with
the amplitude.

00:01:14.710 --> 00:01:17.670
This 2 tells us that the
amplitude of this function

00:01:17.670 --> 00:01:19.890
is going to be 2.

00:01:19.890 --> 00:01:22.320
Because if it was just a 1
there the amplitude would be 1.

00:01:22.320 --> 00:01:24.000
So it's going to
be 2 times that.

00:01:24.000 --> 00:01:34.500
So let's draw a little dotted
line up here at y equals 2.

00:01:34.500 --> 00:01:37.230
And then another dotted line
at y equals negative 2.

00:01:45.780 --> 00:01:46.740
So we know this is
the amplitude.

00:01:46.740 --> 00:01:49.650
We know that the function is
going to somehow oscillate

00:01:49.650 --> 00:01:51.710
between these two points, but
we have to figure out how fast

00:01:51.710 --> 00:01:53.570
is it going to oscillate
between the two points,

00:01:53.570 --> 00:01:55.800
or what's its period.

00:01:55.800 --> 00:01:57.330
And I'll give you a
little formula here.

00:02:00.060 --> 00:02:06.390
The function is equal to the
amplitude times, let's say,

00:02:06.390 --> 00:02:08.890
sine, but it would also
work with cosine.

00:02:08.890 --> 00:02:16.030
The amplitude of the function
times sine of 2pi divided

00:02:16.030 --> 00:02:20.800
by the period of the
function, times x.

00:02:20.800 --> 00:02:22.850
This right here is a "p."

00:02:22.850 --> 00:02:26.340
So it might not be completely
obvious where this comes from.

00:02:26.340 --> 00:02:28.260
But what I want you to do is
maybe after this video or

00:02:28.260 --> 00:02:30.950
maybe in future videos we'll
experiment when we see what

00:02:30.950 --> 00:02:33.770
happens when we change this
coefficient on the x-term.

00:02:33.770 --> 00:02:36.520
And I think it'll start
to make sense to you why

00:02:36.520 --> 00:02:38.910
this equation holds.

00:02:38.910 --> 00:02:41.190
But let's just take this as
kind of an act of faith right

00:02:41.190 --> 00:02:45.140
now, that 2pi divided by the
period is the coefficient on x.

00:02:45.140 --> 00:02:52.670
So if we say that 2pi divided
by the period is equal to the

00:02:52.670 --> 00:02:56.100
coefficient, which is 1/2.

00:02:56.100 --> 00:02:57.440
I know this is extremely messy.

00:02:57.440 --> 00:02:59.040
And this is separate from this.

00:02:59.040 --> 00:03:01.760
So 2pi divided by the
period is equal to 1/2.

00:03:01.760 --> 00:03:09.150
Or we could say 1/2 the
period is equal to 2pi.

00:03:09.150 --> 00:03:13.700
Or, the period is equal to 4pi.

00:03:13.700 --> 00:03:17.110
So we know the amplitude
is equal to 2 and the

00:03:17.110 --> 00:03:19.750
period is equal to 4pi.

00:03:19.750 --> 00:03:21.610
And once again, how did
we figure out that the

00:03:21.610 --> 00:03:23.050
period is equal to 4pi?

00:03:23.050 --> 00:03:26.670
We used this formula: 2pi
divided by the period is the

00:03:26.670 --> 00:03:27.590
coefficient on the x-term.

00:03:27.590 --> 00:03:31.020
So we set 2pi divided by the
period equal to 1/2, and then

00:03:31.020 --> 00:03:33.620
we solved that the
period is 4pi.

00:03:33.620 --> 00:03:34.650
So where do we start?

00:03:34.650 --> 00:03:40.580
Well, what is f of 0?

00:03:40.580 --> 00:03:43.330
Well, when x is equal to
0 this whole term is 0.

00:03:43.330 --> 00:03:45.210
So what's sine of 0?

00:03:45.210 --> 00:03:48.650
Sine of 0 is 0,
if you remember.

00:03:48.650 --> 00:03:49.870
I guess you could use a
calculator, but that's

00:03:49.870 --> 00:03:51.320
something you should remember.

00:03:51.320 --> 00:03:53.900
Or you could re-look at the
unit circle to remind yourself.

00:03:53.900 --> 00:03:55.990
Sine of 0 is 0.

00:03:55.990 --> 00:03:59.140
And then 0 times 2 is 0.

00:03:59.140 --> 00:04:02.640
So f of 0 is 0.

00:04:02.640 --> 00:04:02.920
Right?

00:04:02.920 --> 00:04:04.510
We'll draw it right there.

00:04:04.510 --> 00:04:07.290
And we know that it
has a period of 4pi.

00:04:07.290 --> 00:04:11.300
That means that the function
is going to repeat after 4pi.

00:04:11.300 --> 00:04:17.166
So if we go out it should
repeat back out here, at 4pi.

00:04:20.020 --> 00:04:21.930
And now we can just kind
of draw the function.

00:04:21.930 --> 00:04:24.594
And this will take a little bit
of practice, but-- actually I'm

00:04:24.594 --> 00:04:25.780
going to draw it, and then we
can explore it a little

00:04:25.780 --> 00:04:26.490
bit more as well.

00:04:26.490 --> 00:04:28.515
So the function's going
to look like this.

00:04:33.260 --> 00:04:33.750
Oh, boy.

00:04:33.750 --> 00:04:35.280
This is more difficult
than I thought.

00:04:35.280 --> 00:04:37.350
And it'll keep going in
this direction as well.

00:04:42.920 --> 00:04:51.980
And notice, the period here you
could do it from here to here.

00:04:51.980 --> 00:04:54.170
This distance is 4pi.

00:04:54.170 --> 00:04:56.490
That's how long it takes for
the function to repeat, or

00:04:56.490 --> 00:04:57.800
to go through one cycle.

00:04:57.800 --> 00:05:01.840
Or you could also, if you
want, you could measure this

00:05:01.840 --> 00:05:04.560
distance to this distance.

00:05:04.560 --> 00:05:06.230
This would also be 4pi.

00:05:06.230 --> 00:05:07.860
And that's the period
of the function.

00:05:07.860 --> 00:05:11.270
And then, of course, the
amplitude of the function,

00:05:11.270 --> 00:05:15.590
which is this right here, is 2.

00:05:15.590 --> 00:05:17.140
Here's the amplitude.

00:05:17.140 --> 00:05:22.590
And then the period of 4pi we
figured out from this equation.

00:05:22.590 --> 00:05:25.100
Another way we could have
thought about it, let's say

00:05:25.100 --> 00:05:29.710
that-- let me erase some of the
stuff-- let's say I didn't

00:05:29.710 --> 00:05:31.940
have this stuff right here.

00:05:31.940 --> 00:05:37.950
Let's say I didn't know
what the function was.

00:05:37.950 --> 00:05:40.090
Let me get rid of
all of this stuff.

00:05:40.090 --> 00:05:43.360
And all I saw was this
graph, and I asked you

00:05:43.360 --> 00:05:44.990
to go the other way.

00:05:44.990 --> 00:05:48.440
Using this graph, try to figure
out what the function is.

00:05:48.440 --> 00:05:51.910
Then we would just see,
how long does it take for

00:05:51.910 --> 00:05:52.750
the function to repeat?

00:05:52.750 --> 00:05:56.140
Well, it takes 4pi radians for
the function to repeat, so

00:05:56.140 --> 00:05:58.960
you'd be able to just visually
realize that the period

00:05:58.960 --> 00:06:00.620
of this function is 4pi.

00:06:00.620 --> 00:06:01.980
And then you would say,
well what's the amplitude?

00:06:01.980 --> 00:06:03.180
The amplitude is easy.

00:06:03.180 --> 00:06:06.320
You would just see how
high it goes up or down.

00:06:06.320 --> 00:06:09.170
And it goes up 2, right?

00:06:09.170 --> 00:06:11.030
When you're doing the amplitude
you don't do the whole swing,

00:06:11.030 --> 00:06:13.120
you just do how much it
swings in the positive

00:06:13.120 --> 00:06:14.120
or negative direction.

00:06:14.120 --> 00:06:17.840
So the amplitude is 2.

00:06:17.840 --> 00:06:19.680
I'm using the wrong color.

00:06:19.680 --> 00:06:20.350
The period is 4pi.

00:06:22.930 --> 00:06:24.430
And then your question
would be, well this is

00:06:24.430 --> 00:06:26.600
an oscillating, this is
a periodic function.

00:06:26.600 --> 00:06:30.080
Is it a sine or is it
a cosine function?

00:06:30.080 --> 00:06:34.610
Well, cosine function, assuming
we're not doing any shifting--

00:06:34.610 --> 00:06:38.070
and in a future module I will
shift along the x-axis-- but

00:06:38.070 --> 00:06:43.320
assuming we're not doing any
shifting, cosine of 0 is 1.

00:06:43.320 --> 00:06:44.500
Right?

00:06:44.500 --> 00:06:47.270
And sine of 0 is 0.

00:06:47.270 --> 00:06:49.900
And what's this function at 0?

00:06:49.900 --> 00:06:50.970
Well, it's 0.

00:06:50.970 --> 00:06:51.370
Right?

00:06:51.370 --> 00:06:53.670
So this is going to
be a sine function.

00:06:53.670 --> 00:06:56.290
So we would use
this formula here.

00:06:56.290 --> 00:07:00.120
f of x is equal to the
amplitude times the sine of 2pi

00:07:00.120 --> 00:07:02.190
divided by the period times x.

00:07:02.190 --> 00:07:07.780
So we would know that the
function is f of x is equal to

00:07:07.780 --> 00:07:19.690
the amplitude times sine of 2pi
over the period-- 4pi-- x.

00:07:19.690 --> 00:07:22.135
And, of course,
these cancel out.

00:07:22.135 --> 00:07:26.900
And then this cancels out and
becomes 2 sine of 1/2 x.

00:07:26.900 --> 00:07:30.070
I know this is a little
difficult to read.

00:07:30.070 --> 00:07:31.020
My apologies.

00:07:31.020 --> 00:07:31.960
And I'll ask a question.

00:07:31.960 --> 00:07:35.720
What would this
function look like?

00:07:35.720 --> 00:07:45.170
f of x equals 2
cosine of 1/2 x.

00:07:45.170 --> 00:07:47.530
Well, it's going to look the
same but we're going to

00:07:47.530 --> 00:07:50.930
start at a different point.

00:07:50.930 --> 00:07:52.430
What's cosine of 0?

00:07:52.430 --> 00:07:56.430
When x is equal to 0 this
whole term is equal to 0.

00:07:56.430 --> 00:07:59.550
Cosine of 0, we
learned before, is 1.

00:07:59.550 --> 00:08:01.500
So f of 0 is equal to 2.

00:08:01.500 --> 00:08:06.470
Let me write that. f
of 0 is equal to 2.

00:08:06.470 --> 00:08:07.600
Let me do this in a
different color.

00:08:07.600 --> 00:08:09.530
Let me draw the cosine function
in a different color.

00:08:09.530 --> 00:08:11.560
We would start here.

00:08:11.560 --> 00:08:14.270
f of 0 is equal to 2, but
everything else is the same.

00:08:14.270 --> 00:08:17.670
The amplitude is the same
and the period is the same.

00:08:17.670 --> 00:08:20.110
So now it's going
to look like this.

00:08:20.110 --> 00:08:21.600
I hope I don't mess this up.

00:08:21.600 --> 00:08:23.190
This is difficult.

00:08:23.190 --> 00:08:26.986
So now the function is
going to look like this.

00:08:26.986 --> 00:08:30.090
And you're going to go down
here, and you're going

00:08:30.090 --> 00:08:34.030
to rise up again here.

00:08:34.030 --> 00:08:38.580
And on this side you're
going to do the same thing.

00:08:38.580 --> 00:08:40.210
And keep going.

00:08:40.210 --> 00:08:42.510
So notice, the cosine
and the sine functions

00:08:42.510 --> 00:08:44.120
look awfully similar.

00:08:44.120 --> 00:08:47.780
And the way to differentiate
them is what they do-- well,

00:08:47.780 --> 00:08:48.570
what they do in general.

00:08:48.570 --> 00:08:54.020
But the easiest way is, what
happens when you input

00:08:54.020 --> 00:08:55.420
a 0 into the function?

00:08:55.420 --> 00:08:59.260
What happens at the y-axis, or
when x is equal to 0, or when

00:08:59.260 --> 00:09:02.040
the angle that you input
into it is equal 0?

00:09:02.040 --> 00:09:04.270
Unless we're doing shifting--
and don't worry about shifting

00:09:04.270 --> 00:09:07.610
for now, I'll do that in future
modules-- sine of 0 is 0

00:09:07.610 --> 00:09:09.910
while cosine of 0 would be 1.

00:09:09.910 --> 00:09:12.920
And since we're multiplying it
times this factor right here,

00:09:12.920 --> 00:09:16.120
times this number right
here, the 1 becomes a 2.

00:09:16.120 --> 00:09:18.900
And so this is the
graph of cosine of x.

00:09:18.900 --> 00:09:20.750
This is this graph
of sine of x.

00:09:20.750 --> 00:09:22.600
And this is a little bit of
a preview for shifting.

00:09:22.600 --> 00:09:25.760
Notice that the pink graph,
or cosine of x, is very

00:09:25.760 --> 00:09:27.210
similar to the green graph.

00:09:27.210 --> 00:09:33.590
And it's just shifted this
way by-- well, in this

00:09:33.590 --> 00:09:36.120
case it's shifted by pi.

00:09:36.120 --> 00:09:36.870
Right?

00:09:36.870 --> 00:09:38.790
And this actually has something
to do with the period

00:09:38.790 --> 00:09:39.770
of the coefficient.

00:09:39.770 --> 00:09:42.450
In general, cosine of x is
actually sine of x shifted

00:09:42.450 --> 00:09:44.810
to the left by pi/2.

00:09:44.810 --> 00:09:47.050
But I don't want to
confuse you too much.

00:09:47.050 --> 00:09:49.170
That's all the time I
have for this video.

00:09:49.170 --> 00:09:51.070
I will now do another video
with a couple of more

00:09:51.070 --> 00:09:53.030
examples like this.


Graphs of trig functions

WEBVTT
Kind: captions
Language: en

00:00:01.130 --> 00:00:01.950
Welcome.

00:00:01.950 --> 00:00:05.350
Well what I want to do now is
actually I'm going to use this

00:00:05.350 --> 00:00:09.920
graphing application to explore
the trigonometric functions or

00:00:09.920 --> 00:00:11.450
explore the graphs of them.

00:00:11.450 --> 00:00:14.480
And just to start off or just
to let you, this application

00:00:14.480 --> 00:00:17.010
I'm using is from gcalc.net.

00:00:17.010 --> 00:00:18.940
it's G C A L C .net.

00:00:18.940 --> 00:00:21.030
It's not mine, but I want to
give them credit because

00:00:21.030 --> 00:00:24.090
that's what I'm using and
I hope they don't mind.

00:00:24.090 --> 00:00:26.440
So let's start off just
graphing some functions.

00:00:26.440 --> 00:00:28.750
Let's start off with
the sine function.

00:00:28.750 --> 00:00:30.820
So let's say sine of x.

00:00:30.820 --> 00:00:32.420
I hope you all can see it.

00:00:32.420 --> 00:00:34.020
I'm typing it in up here.

00:00:34.020 --> 00:00:35.580
So sine x, let me graph that.

00:00:35.580 --> 00:00:35.890
Look at that.

00:00:35.890 --> 00:00:37.260
Like how nice that looks.

00:00:37.260 --> 00:00:38.530
So let's interpret this.

00:00:38.530 --> 00:00:43.250
So it's oscillating
between-- well, let's

00:00:43.250 --> 00:00:44.280
just go point by point.

00:00:44.280 --> 00:00:45.780
I guess that's the
easiest way to do it.

00:00:45.780 --> 00:00:50.900
So when x is equal to 0, what
is the value of this function?

00:00:50.900 --> 00:00:53.720
Well, if we look here, the
value of the function is--

00:00:53.720 --> 00:00:56.510
let me actually trace it.

00:00:56.510 --> 00:00:59.370
When x equals 0 and it has
it written down at the

00:00:59.370 --> 00:01:00.320
bottom of this grey area.

00:01:00.320 --> 00:01:05.450
So when x is 0, y is also 0.

00:01:05.450 --> 00:01:08.060
And if you're remember when we
looked for the definitions

00:01:08.060 --> 00:01:10.040
in the unit circle
that's what we got.

00:01:10.040 --> 00:01:15.080
That the sine of
0 radians is 0.

00:01:15.080 --> 00:01:19.060
And now as we move on, or
move along the curve, I

00:01:19.060 --> 00:01:21.620
have the trace function on.

00:01:21.620 --> 00:01:24.820
This is when x is equal-- if we
look in the grey area at the

00:01:24.820 --> 00:01:26.650
bottom left it says 1.57.

00:01:26.650 --> 00:01:28.180
But what is that?

00:01:28.180 --> 00:01:33.240
If you're familiar with
the-- 1.57 is more

00:01:33.240 --> 00:01:34.580
commonly known as what?

00:01:34.580 --> 00:01:36.510
It's 1/2 of what famous number?

00:01:36.510 --> 00:01:36.840
Right.

00:01:36.840 --> 00:01:38.010
It's half of pi.

00:01:38.010 --> 00:01:39.910
We're at pi over 2.

00:01:39.910 --> 00:01:42.020
And if you want to convert
pi over 2 to degrees

00:01:42.020 --> 00:01:43.860
that's 90 degrees.

00:01:43.860 --> 00:01:47.620
So when we're at the angle of
pi over 2 radians the sine

00:01:47.620 --> 00:01:49.520
function is equal to 1.

00:01:49.520 --> 00:01:51.410
And if you go back to some
previous modules you'll

00:01:51.410 --> 00:01:55.060
remember that that's exactly
what the sine function was

00:01:55.060 --> 00:01:57.270
equal to when we looked
at the unit circle.

00:01:57.270 --> 00:02:00.045
Because we were essentially
at the point 1 comma 0.

00:02:00.045 --> 00:02:02.550
I hope it's not confusing that
I keep referring to the unit

00:02:02.550 --> 00:02:03.470
circle that you can't see.

00:02:03.470 --> 00:02:05.460
But we'll keep going.

00:02:05.460 --> 00:02:08.380
But one thing I want to
introduce here is the concept

00:02:08.380 --> 00:02:12.840
of the period or the frequency
of the sine function.

00:02:12.840 --> 00:02:16.280
It's pretty obvious to you at
this point that the function

00:02:16.280 --> 00:02:18.440
keeps repeating itself.

00:02:18.440 --> 00:02:21.910
It goes from 0, moves up to
1, goes back to 0, goes

00:02:21.910 --> 00:02:22.850
down to negative 1.

00:02:22.850 --> 00:02:26.500
Then goes back to 0 and
then repeats again.

00:02:26.500 --> 00:02:30.500
So the period of this periodic
function because that's what we

00:02:30.500 --> 00:02:33.100
call a function that keeps
repeating, the period of this

00:02:33.100 --> 00:02:38.100
periodic function is this
distance from here to here.

00:02:38.100 --> 00:02:38.760
And what's that?

00:02:38.760 --> 00:02:41.450
Well, that's 2 pi radians.

00:02:41.450 --> 00:02:42.770
And does that make sense?

00:02:42.770 --> 00:02:47.660
Well sure, because 2 pi radians
is one complete revolution

00:02:47.660 --> 00:02:48.990
around the unit circle.

00:02:48.990 --> 00:02:50.650
And then it repeats again.

00:02:50.650 --> 00:02:52.130
And then it goes the other way.

00:02:52.130 --> 00:02:54.450
You go 2 pi radians
backwards and things

00:02:54.450 --> 00:02:55.850
start repeating again.

00:02:55.850 --> 00:02:58.580
Pretty interesting, right?

00:02:58.580 --> 00:03:00.200
Oh, and another thing.

00:03:00.200 --> 00:03:02.380
What two numbers does
it oscillate between?

00:03:02.380 --> 00:03:05.970
It oscillates between
positive 1 and negative 1.

00:03:05.970 --> 00:03:09.450
And that makes sense because in
the unit circle you can never

00:03:09.450 --> 00:03:12.750
get to a point on the perimeter
of the unit circle that's

00:03:12.750 --> 00:03:16.700
larger than positive 1 or
less than negative 1.

00:03:16.700 --> 00:03:19.150
And that's why the sine of x
keeps oscillating between

00:03:19.150 --> 00:03:20.990
these two points.

00:03:20.990 --> 00:03:22.760
Let's do the cosine of x.

00:03:22.760 --> 00:03:26.030
Actually, I'm going to
leave the sine of x there.

00:03:26.030 --> 00:03:26.520
Interesting.

00:03:26.520 --> 00:03:31.095
It looks almost the same,
but it looks shifted.

00:03:31.095 --> 00:03:33.670
It actually looks shifted
to the left about

00:03:33.670 --> 00:03:35.360
pi over 2 radians.

00:03:35.360 --> 00:03:37.160
And that's actually the case.

00:03:37.160 --> 00:03:39.550
So let's first think about why.

00:03:39.550 --> 00:03:41.970
We figured out before that
sine-- actually, it looks

00:03:41.970 --> 00:03:44.830
like this program is still
tracing the sine function.

00:03:44.830 --> 00:03:47.630
That sine of 0 was 0.

00:03:47.630 --> 00:03:51.340
But if you look at the green
function, the cosine of

00:03:51.340 --> 00:03:53.580
0 radians is actually 1.

00:03:53.580 --> 00:03:59.470
Let me see if I can-- no.

00:03:59.470 --> 00:04:01.150
I don't know how to trace
the cosine function, so

00:04:01.150 --> 00:04:02.130
I'll just do it here.

00:04:02.130 --> 00:04:04.930
The cosine of 0 is 1.

00:04:04.930 --> 00:04:06.110
And why does that make sense?

00:04:06.110 --> 00:04:08.450
Well, the cosine is
the x-coordinate on

00:04:08.450 --> 00:04:09.150
the unit circle.

00:04:12.530 --> 00:04:16.900
When you have 0 radians or 0
degrees, you're at the point 1

00:04:16.900 --> 00:04:19.080
comma 0 on the unit circle.

00:04:19.080 --> 00:04:23.270
So 1 is the cosine or is
the x-coordinate and

00:04:23.270 --> 00:04:25.380
0 is the sine value.

00:04:25.380 --> 00:04:28.620
And if any of this is
confusing, review the video

00:04:28.620 --> 00:04:33.410
where I use the unit circle to
solve the various values of the

00:04:33.410 --> 00:04:35.880
trig functions and then
this should make sense.

00:04:35.880 --> 00:04:38.840
And notice that this has
a period similar to

00:04:38.840 --> 00:04:39.650
the sine function.

00:04:39.650 --> 00:04:43.100
It starts at 1, goes down
to negative 1, and then

00:04:43.100 --> 00:04:45.000
comes back to positive 1.

00:04:45.000 --> 00:04:50.560
And it takes 2 pi radians
to complete that cycle.

00:04:50.560 --> 00:04:54.210
And just like the sine function
it's oscillating between 1 and

00:04:54.210 --> 00:04:59.370
negative 1 because on the unit
circle you can't get to a point

00:04:59.370 --> 00:05:01.850
on the perimeter that's
higher than that.

00:05:01.850 --> 00:05:04.450
And now to really hit
the point home let's do

00:05:04.450 --> 00:05:05.290
the tangent function.

00:05:05.290 --> 00:05:08.280
I think this one
might surprise you.

00:05:08.280 --> 00:05:09.470
Well, look at that.

00:05:09.470 --> 00:05:11.860
So the blue line is
the tangent function.

00:05:11.860 --> 00:05:14.280
And why does it do
this crazy thing?

00:05:14.280 --> 00:05:16.620
Well, if you remember, the
tangent function is equal to

00:05:16.620 --> 00:05:19.625
the y over the x on the
perimeter of the unit circle.

00:05:23.130 --> 00:05:29.350
Or since the y is the sine and
cosine is the x, it also equals

00:05:29.350 --> 00:05:31.260
the sine over the cosine.

00:05:31.260 --> 00:05:34.240
So here, tangent is 0
whatever sine is 0

00:05:34.240 --> 00:05:35.050
because that makes sense.

00:05:35.050 --> 00:05:39.025
Because tangent is equal
to sine over cosine.

00:05:39.025 --> 00:05:42.590
So it makes sense that when
sine is 0, tangent is 0.

00:05:42.590 --> 00:05:48.770
But then, as the sine function
becomes greater and the cosine

00:05:48.770 --> 00:05:52.100
function becomes less, the
numerator in the tangent

00:05:52.100 --> 00:05:54.420
function becomes greater
because the numerator is sine.

00:05:54.420 --> 00:05:56.690
So we get larger and larger
values, all the way to the

00:05:56.690 --> 00:06:02.225
point where the denominator of
the tangent function, which is

00:06:02.225 --> 00:06:03.970
the cosine function-- I think
this is probably the most

00:06:03.970 --> 00:06:05.870
confusing module I've ever
said because I can't really

00:06:05.870 --> 00:06:07.910
write these things down.

00:06:07.910 --> 00:06:10.020
The denominator goes to 0.

00:06:10.020 --> 00:06:11.000
The cosine right here.

00:06:11.000 --> 00:06:14.070
And then tan spikes and it
actually approaches infinity.

00:06:14.070 --> 00:06:17.210
And if you look back at the
unit circle, that actually

00:06:17.210 --> 00:06:18.750
might make a little
bit of sense.

00:06:18.750 --> 00:06:21.980
But like the other functions,
actually the tangent

00:06:21.980 --> 00:06:26.210
function has a period of pi
instead of pi over 2.

00:06:26.210 --> 00:06:27.130
Instead of 2 pi.

00:06:27.130 --> 00:06:29.050
And I'll leave that as
an exercise for you

00:06:29.050 --> 00:06:30.630
to think about.

00:06:30.630 --> 00:06:32.450
But with that drawn
out, I'm now going to

00:06:32.450 --> 00:06:33.650
explore something else.

00:06:33.650 --> 00:06:35.380
Let me reset this.

00:06:35.380 --> 00:06:38.070
Yes, I really want to reset.

00:06:38.070 --> 00:06:40.860
I drew the sine
function before.

00:06:40.860 --> 00:06:46.650
Let me draw the sine
of let's say, 2x.

00:06:46.650 --> 00:06:49.000
Whoops, that's not right.

00:06:49.000 --> 00:06:54.820
sine of 2-- maybe I need to
put some parentheses in.

00:06:54.820 --> 00:06:56.090
Oh, there we go.

00:06:56.090 --> 00:06:57.760
Actually, let me reset it.

00:06:57.760 --> 00:06:58.980
Yes, I want to reset.

00:06:58.980 --> 00:07:01.400
So first I'll draw the
sine of x and then I'll

00:07:01.400 --> 00:07:04.310
draw the sine of 2x.

00:07:07.250 --> 00:07:09.970
So what's the first thing you
notice about the difference

00:07:09.970 --> 00:07:10.570
between these two?

00:07:10.570 --> 00:07:14.760
The brown one is sine of x and
the green one is sine of 2x.

00:07:17.680 --> 00:07:20.860
They both oscillate between the
same two numbers and just so

00:07:20.860 --> 00:07:25.600
you know, the height of the
oscillation is called

00:07:25.600 --> 00:07:27.630
the amplitude of this
periodic function.

00:07:27.630 --> 00:07:31.410
So in both cases, the amplitude
is 1 because they oscillate

00:07:31.410 --> 00:07:32.950
from 1 to negative 1.

00:07:32.950 --> 00:07:38.550
So the amplitude is 1, but
their period is different.

00:07:38.550 --> 00:07:45.860
Sine of x takes 2 pi radians to
complete one circle-- one cycle

00:07:45.860 --> 00:07:51.910
while sine of 2x only takes pi
radians to complete one cycle.

00:07:51.910 --> 00:07:55.572
So it actually completes
it twice as fast.

00:07:55.572 --> 00:07:59.480
And I want you to sit and think
about why sine of 2x has 1/2

00:07:59.480 --> 00:08:02.390
the period of sine of x.

00:08:02.390 --> 00:08:03.820
And you can probably
guess what happens if

00:08:03.820 --> 00:08:07.420
I type in sine of 3x.

00:08:07.420 --> 00:08:10.030
Actually, let's do sine of 4x.

00:08:10.030 --> 00:08:12.770
It should have 1/2 the
period of sine of 2x then.

00:08:12.770 --> 00:08:14.530
And it does, even though
this is probably a

00:08:14.530 --> 00:08:18.950
very confusing graph.

00:08:18.950 --> 00:08:19.920
So let's explore.

00:08:19.920 --> 00:08:21.550
So I think you understand
what the coefficient

00:08:21.550 --> 00:08:22.650
on the x term does.

00:08:22.650 --> 00:08:25.680
When you have a larger
coefficient it kind of

00:08:25.680 --> 00:08:27.570
speeds up the cycles.

00:08:27.570 --> 00:08:30.876
And let's explore a
little bit more.

00:08:30.876 --> 00:08:33.670
Let's start off with
sine of x again.

00:08:33.670 --> 00:08:36.220
And now, instead of making the
coefficient larger, let's

00:08:36.220 --> 00:08:37.380
make the coefficient less.

00:08:37.380 --> 00:08:41.090
Let's make it sine of 0.5x.

00:08:41.090 --> 00:08:41.600
Look at that.

00:08:41.600 --> 00:08:44.960
Now, all of a sudden, it
takes 4 pi radians to

00:08:44.960 --> 00:08:46.030
complete one cycle.

00:08:46.030 --> 00:08:47.830
And I want you to think
about why that is.

00:08:47.830 --> 00:08:50.480
Because we're now slowing
down how fast it cycles

00:08:50.480 --> 00:08:51.275
through the angles.

00:08:54.820 --> 00:08:57.500
Now I want to start playing
with the amplitude.

00:08:57.500 --> 00:09:02.310
So we had sine of x, what do
you think will happen if I put

00:09:02.310 --> 00:09:09.330
in this 2 times sine of x?

00:09:09.330 --> 00:09:12.840
So here, the period
is the same.

00:09:12.840 --> 00:09:16.540
It's still 2 pi, but notice
that it oscillates between 2

00:09:16.540 --> 00:09:20.840
and negative 2 instead of
between 1 and negative 1.

00:09:20.840 --> 00:09:24.120
So whatever the coefficient, or
whatever the number is in front

00:09:24.120 --> 00:09:27.690
of the sine or the cosine
function, that actually

00:09:27.690 --> 00:09:28.885
affects its amplitude.

00:09:28.885 --> 00:09:34.030
And similarly, we can
look at 0.5 sine-- let's

00:09:34.030 --> 00:09:38.970
say 0.5 sine of 2x.

00:09:38.970 --> 00:09:39.480
Interesting.

00:09:39.480 --> 00:09:44.340
So now it only goes up to
0.5 and down to minus 0.5.

00:09:44.340 --> 00:09:48.780
So it's amplitude
is 1/2 or 0.5.

00:09:48.780 --> 00:09:52.290
And it also oscillates twice
as fast as the sine function

00:09:52.290 --> 00:09:56.300
because it was 0.5 sine of 2z.

00:09:56.300 --> 00:09:58.840
I think that's all
the time I have now.

00:09:58.840 --> 00:10:01.020
I have a feeling this might
have confused you more than

00:10:01.020 --> 00:10:04.240
helped, but I'll still put
the video up just in case

00:10:04.240 --> 00:10:05.230
it's helpful for someone.

00:10:05.230 --> 00:10:07.950
But in the future I might
actually record another video

00:10:07.950 --> 00:10:09.980
where I can actually write
things down so it doesn't

00:10:09.980 --> 00:10:10.950
confuse you as much.

00:10:10.950 --> 00:10:13.080
So if it confused you
I apologize, but I

00:10:13.080 --> 00:10:14.940
hope it was helpful.

00:10:14.940 --> 00:10:16.470
See you later.


Graph of the sine function

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:02.150
Hello.

00:00:02.150 --> 00:00:06.100
In the last presentation we
kind of re-defined the sine,

00:00:06.100 --> 00:00:09.140
the cosine, and the tangent
functions in a broader way

00:00:09.140 --> 00:00:15.580
where we said if we have a unit
circle and our theta is, or our

00:00:15.580 --> 00:00:19.795
angle, is -- let me use the
right tool -- let's say, and

00:00:19.795 --> 00:00:24.950
our angle is the angle between,
say, the x-axis and a radius in

00:00:24.950 --> 00:00:26.570
the unit circle, and
this is our radius.

00:00:30.525 --> 00:00:33.870
the coordinate of the point
where this radius intersects

00:00:33.870 --> 00:00:39.070
the unit circle is x comma y.

00:00:39.070 --> 00:00:45.490
Our new definition of the trig
functions was that sine of

00:00:45.490 --> 00:00:49.000
theta is equal to the
y-coordinate, right, this is

00:00:49.000 --> 00:00:50.920
y-coordinate where it
intersects the unit circle.

00:00:50.920 --> 00:00:52.400
And remember, this
is the unit circle.

00:00:52.400 --> 00:00:56.260
It's not just any circle, which
means it has a radius of 1.

00:00:56.260 --> 00:01:03.990
Cosine of theta is equal to the
x-coordinate of this point.

00:01:03.990 --> 00:01:06.360
This is the x-coordinate.

00:01:06.360 --> 00:01:12.550
And tangent of theta
equaled opposite over

00:01:12.550 --> 00:01:14.050
adjacent or y over x.

00:01:17.190 --> 00:01:19.665
That's interesting because
that's also equal to sine of

00:01:19.665 --> 00:01:22.060
theta over cosine of theta.

00:01:22.060 --> 00:01:23.000
I'll just do that.

00:01:23.000 --> 00:01:26.000
I wasn't even planning on
covering that, but just it

00:01:26.000 --> 00:01:29.370
leaves you something
to think about.

00:01:29.370 --> 00:01:32.850
So with that said, let's take a
look or let's try to see how

00:01:32.850 --> 00:01:34.940
this defines these functions.

00:01:34.940 --> 00:01:37.120
And I guess a good place to
start is just with the sine

00:01:37.120 --> 00:01:39.450
function and we can
try to graph it.

00:01:39.450 --> 00:01:42.080
So let's write, let's do a
little table like we always do

00:01:42.080 --> 00:01:45.470
when we define a function.

00:01:45.470 --> 00:01:50.190
Let's put in values of
theta, and let's figure

00:01:50.190 --> 00:01:53.690
out what sine of theta is.

00:01:53.690 --> 00:01:57.370
So when theta is equal to 0
radians, what is sine of theta?

00:01:57.370 --> 00:02:05.430
So when theta's 0, right, then
the radius between it -- this

00:02:05.430 --> 00:02:07.600
is the radius and this is
the point where the radius

00:02:07.600 --> 00:02:09.210
intersects the unit circle.

00:02:09.210 --> 00:02:12.720
And this point has a
coordinate 1 comma 0, right?

00:02:15.340 --> 00:02:19.120
And so if where it intersects
the unit circle is at 1 comma

00:02:19.120 --> 00:02:22.460
0, then sine of theta is
just the y-coordinate.

00:02:22.460 --> 00:02:25.620
So sine of theta is 0.

00:02:25.620 --> 00:02:31.570
If we said what is sine
of theta when theta is

00:02:31.570 --> 00:02:32.870
equal to pi over 2.

00:02:36.840 --> 00:02:39.370
So now our radius is this
radius and we intersect the

00:02:39.370 --> 00:02:45.180
unit circle right here
at the point 0 comma 1.

00:02:45.180 --> 00:02:47.010
And what's the y-coordinate
at 0 comma 1?

00:02:47.010 --> 00:02:49.900
Well it's 1.

00:02:49.900 --> 00:02:54.910
What happens when we have
theta is equal to pi radians?

00:02:54.910 --> 00:02:59.350
So at pi radians we intersect
the unit circle right here.

00:02:59.350 --> 00:03:00.700
We're at pi radian.

00:03:00.700 --> 00:03:03.600
This is the angle, pi.

00:03:03.600 --> 00:03:08.520
We intersect with unit circle
at negative 1 comma 0.

00:03:08.520 --> 00:03:10.950
Because once again, this
is the unit circle.

00:03:10.950 --> 00:03:13.790
So at negative 1 comma 0,
what's the y-coordinate?

00:03:13.790 --> 00:03:16.410
Well, it's 0.

00:03:16.410 --> 00:03:18.710
So sine of pi is equal to 0.

00:03:18.710 --> 00:03:20.750
Let's just keep going
around the circle.

00:03:20.750 --> 00:03:32.185
When we have the angle, when
theta is equal to 3 pi over 4

00:03:32.185 --> 00:03:37.370
-- no, sorry, 3 pi over 2.

00:03:37.370 --> 00:03:40.220
Because this is pi and
this is another half pi.

00:03:40.220 --> 00:03:42.310
So this is 3 pi over 2, sorry.

00:03:42.310 --> 00:03:46.720
So when theta is equal to 3 pi
over 2, what is sine of theta?

00:03:46.720 --> 00:03:49.430
Well, now we intersect the
unit circle down here at the

00:03:49.430 --> 00:03:51.920
point 0 comma negative 1.

00:03:51.920 --> 00:03:56.040
So now sine of theta is
equal to negative 1.

00:03:56.040 --> 00:03:59.320
Then if we go all the way
around the circle to 2

00:03:59.320 --> 00:04:03.680
pi radians, we're back
at this point again.

00:04:03.680 --> 00:04:09.300
So sine of theta, so we're
at 2 pi, sine of theta

00:04:09.300 --> 00:04:11.960
is now 0 once again.

00:04:11.960 --> 00:04:15.210
So let's graph these points out
and then we'll try to figure

00:04:15.210 --> 00:04:16.970
out what the points in between
look like, and I'll show

00:04:16.970 --> 00:04:19.050
you the graph of a
sine function is.

00:04:24.200 --> 00:04:25.800
So let's draw the x-axis.

00:04:28.310 --> 00:04:29.450
This is my x-axis.

00:04:32.350 --> 00:04:36.595
And let's draw the y-axis.

00:04:41.180 --> 00:04:43.700
Not as clean as I
wanted to draw it.

00:04:43.700 --> 00:04:44.735
This is y.

00:04:48.320 --> 00:04:49.410
And that's x.

00:04:49.410 --> 00:04:51.360
But in this case instead of
saying that's the x-axis, let's

00:04:51.360 --> 00:04:54.910
call that the theta axis,
because we defined theta as the

00:04:54.910 --> 00:04:59.370
input or our domains
in terms of theta.

00:04:59.370 --> 00:05:02.170
So this is the theta axis.

00:05:02.170 --> 00:05:04.290
Now we're going to
graph sine of theta.

00:05:04.290 --> 00:05:07.565
So when we said when
theta equaled 0, sine

00:05:07.565 --> 00:05:09.030
of theta is equal to 0.

00:05:09.030 --> 00:05:12.260
So that's this point
right here, 0 comma 0.

00:05:12.260 --> 00:05:24.760
When theta is equal to pi over
2, sine of theta is equal to 1.

00:05:24.760 --> 00:05:30.680
So this is the point pi
over 2 comma 1, right?

00:05:30.680 --> 00:05:32.270
That's just this 1.

00:05:32.270 --> 00:05:40.610
When theta is equal to pi,
sine of theta is 0 again.

00:05:40.610 --> 00:05:44.660
So this is the
point pi comma 0.

00:05:44.660 --> 00:05:56.540
And when theta equaled 3 pi
over 2, what was sine of theta?

00:05:56.540 --> 00:05:58.420
I equaled negative 1.

00:05:58.420 --> 00:06:00.730
Interesting.

00:06:00.730 --> 00:06:04.330
Then when we got to 2 pi --
when we got to theta equal

00:06:04.330 --> 00:06:07.700
to 2 pi, sine of theta,
again, equaled 0.

00:06:07.700 --> 00:06:11.570
So we know that these
points are on the graph

00:06:11.570 --> 00:06:12.680
of sine of theta.

00:06:12.680 --> 00:06:15.720
And if you actually tried the
points in between, and as

00:06:15.720 --> 00:06:18.480
an exercise it might be
interesting for you to do so.

00:06:18.480 --> 00:06:20.140
You could actually figure out
a lot of the points using

00:06:20.140 --> 00:06:23.650
30-60-90 triangles or using
the Pythagorean Theorem.

00:06:23.650 --> 00:06:25.930
But you actually get a curve
that looks something -- let me

00:06:25.930 --> 00:06:30.630
use a nicer color than this
kind of drab grey -- you

00:06:30.630 --> 00:06:35.360
get a graph that looks
something like this.

00:06:39.870 --> 00:06:43.430
And you've probably
seen that before.

00:06:43.430 --> 00:06:45.990
The term for this function
is actually a sine wave.

00:06:45.990 --> 00:06:47.870
It looks like something
that's oscillating or

00:06:47.870 --> 00:06:49.140
that's moving up and down.

00:06:49.140 --> 00:06:53.390
And actually if you were to put
in thetas that were less than

00:06:53.390 --> 00:06:58.470
0, the sine wave will
keep going into the

00:06:58.470 --> 00:06:59.520
negative theta axis.

00:06:59.520 --> 00:07:01.220
It keeps going forever
in both directions.

00:07:01.220 --> 00:07:05.940
It keeps oscillating between
1 and negative 1 and

00:07:05.940 --> 00:07:07.960
the points in between.

00:07:07.960 --> 00:07:11.180
So that's the graph of
the sign function.

00:07:11.180 --> 00:07:13.390
In the next module I'll
actually do the graph of the

00:07:13.390 --> 00:07:15.350
cosine function, or actually I
might just show you the graph

00:07:15.350 --> 00:07:16.840
of the cosine function.

00:07:16.840 --> 00:07:20.580
Then I'll show you how they
relate and how these can

00:07:20.580 --> 00:07:27.480
describe any kind of, or many
types of oscillatory things in

00:07:27.480 --> 00:07:31.150
the world and how it relates to
frequency and amplitude.

00:07:31.150 --> 00:07:32.740
So I'll see you in
the next module.

00:07:32.740 --> 00:07:36.270
And just for fun you might want
to sit down with a piece of

00:07:36.270 --> 00:07:39.920
paper and try to graph the
cosine function or the

00:07:39.920 --> 00:07:41.680
tangent function as well.

00:07:41.680 --> 00:07:43.160
Have fun.


Unit Circle Definition of Trig Functions

WEBVTT
Kind: captions
Language: en

00:00:01.280 --> 00:00:02.210
Hello.

00:00:02.210 --> 00:00:06.150
Well, welcome to the next
presentation in the

00:00:06.150 --> 00:00:08.260
trigonometry modules.

00:00:08.260 --> 00:00:10.340
Just to start off a little
bit, let's review what

00:00:10.340 --> 00:00:13.070
we've done so far.

00:00:13.070 --> 00:00:15.320
In the last couple modules, we
learned the definitions-- or at

00:00:15.320 --> 00:00:17.720
least, I guess, we could call
it a partial definition-- of

00:00:17.720 --> 00:00:19.880
the sine, the cosine, and
the tangent functions.

00:00:19.880 --> 00:00:24.250
And the mnemonic we used to
memorize that was sohcahtoa.

00:00:24.250 --> 00:00:26.190
Let me write that down.

00:00:26.190 --> 00:00:26.620
Sohcahtoa.

00:00:32.490 --> 00:00:35.320
And what that told us is, let's
say we had a right triangle.

00:00:35.320 --> 00:00:36.705
Let me draw a right triangle.

00:00:39.720 --> 00:00:40.850
This is a right angle here.

00:00:40.850 --> 00:00:42.355
This is the hypotenuse.

00:00:42.355 --> 00:00:44.680
Let me label the hypotenuse, h.

00:00:44.680 --> 00:00:49.570
Let me label this-- and so we
want to figure out, we want to

00:00:49.570 --> 00:00:50.730
use this angle right here.

00:00:50.730 --> 00:00:52.110
Theta, we'll call this theta.

00:00:52.110 --> 00:00:52.780
Whatever.

00:00:52.780 --> 00:00:57.650
Then this is the adjacent side,
and this is the opposite side.

00:00:57.650 --> 00:00:58.770
And that's an o.

00:00:58.770 --> 00:01:02.310
So soh tells us that sine
is equal to opposite

00:01:02.310 --> 00:01:03.880
over hypotenuse.

00:01:03.880 --> 00:01:07.600
Cosine is equal to
adjacent over hypotenuse.

00:01:07.600 --> 00:01:11.250
And tangent is equal to
opposite over adjacent.

00:01:11.250 --> 00:01:14.310
And I think, by this point--
and especially if you did some

00:01:14.310 --> 00:01:17.590
of the exercises on the Khan
Academy-- that should be second

00:01:17.590 --> 00:01:19.470
nature and should make
a lot of sense to you.

00:01:19.470 --> 00:01:23.460
But this definition, using a
right triangle like this,

00:01:23.460 --> 00:01:25.140
actually breaks down
at certain points.

00:01:25.140 --> 00:01:26.390
Actually, at a lot of points.

00:01:26.390 --> 00:01:29.640
For example, what happens
as this angle right here

00:01:29.640 --> 00:01:31.160
approaches 90 degrees?

00:01:31.160 --> 00:01:34.280
You can't have two 90
degree angles in a right

00:01:34.280 --> 00:01:35.330
triangle, can you?

00:01:35.330 --> 00:01:37.620
Then it would be like a
rectangle or something.

00:01:37.620 --> 00:01:39.340
But you could actually probably
figure out what happens as

00:01:39.340 --> 00:01:40.510
it approaches 90 degrees.

00:01:40.510 --> 00:01:42.980
But the definition, this
definition, breaks

00:01:42.980 --> 00:01:44.340
down for that.

00:01:44.340 --> 00:01:46.280
Also, what happens if
this angle is negative?

00:01:46.280 --> 00:01:49.655
Or what happens if this angle
is more than 90 degrees?

00:01:49.655 --> 00:01:52.600
Or what happens if
it's 800 degrees?

00:01:52.600 --> 00:01:55.600
Or you know, 8 pi radians?

00:01:55.600 --> 00:01:57.830
Not that 800 and 8 pi
radians are the same thing.

00:01:57.830 --> 00:02:00.180
But obviously, this definition
starts to break down.

00:02:00.180 --> 00:02:02.100
Because we couldn't even
draw a right triangle that

00:02:02.100 --> 00:02:03.890
has those properties.

00:02:03.890 --> 00:02:07.630
So now I'm going to introduce
you to an extension

00:02:07.630 --> 00:02:08.500
of this definition.

00:02:08.500 --> 00:02:12.920
It's really the same thing, but
it allows the sine, the cosine,

00:02:12.920 --> 00:02:16.970
and the tangent functions to be
defined for angles greater than

00:02:16.970 --> 00:02:21.530
or equal to pi over 2, or 90
degrees, or less than 0.

00:02:21.530 --> 00:02:22.900
So let's draw a unit circle.

00:02:22.900 --> 00:02:25.290
So this is just the coordinate
axis, and here is a

00:02:25.290 --> 00:02:27.700
circle of radius 1.

00:02:27.700 --> 00:02:30.540
And let's make-- let me see.

00:02:30.540 --> 00:02:32.920
Let me make sure I'm using
the correct pen tool.

00:02:32.920 --> 00:02:33.670
OK.

00:02:33.670 --> 00:02:39.930
So let's call this right
here-- so this is theta.

00:02:39.930 --> 00:02:42.220
This is an angle, right?

00:02:42.220 --> 00:02:45.640
Between the x-axis and this
line I just drew here.

00:02:45.640 --> 00:02:47.010
And this is a radius, right?

00:02:47.010 --> 00:02:49.590
And we said that this
has a radius 1.

00:02:49.590 --> 00:02:51.600
So the length of this
line is 1, right?

00:02:51.600 --> 00:02:54.020
Because it just goes
from the origin to the

00:02:54.020 --> 00:02:54.820
outside of the circle.

00:02:54.820 --> 00:02:56.940
So it has a radius of 1.

00:02:56.940 --> 00:02:59.890
And now I'm going to draw
a right triangle again.

00:02:59.890 --> 00:03:02.500
Let me just drop a
line from here.

00:03:02.500 --> 00:03:05.920
So there I have a
right triangle.

00:03:05.920 --> 00:03:07.950
So if we use the old
definition we learned before.

00:03:07.950 --> 00:03:10.010
Let's just focus
on sine for now.

00:03:10.010 --> 00:03:17.920
So sine is equal to
opposite over hypotenuse.

00:03:17.920 --> 00:03:20.280
Let's apply that to this
right triangle right here.

00:03:20.280 --> 00:03:22.030
This is the right angle.

00:03:22.030 --> 00:03:24.170
So what's the opposite
angle of this?

00:03:24.170 --> 00:03:25.840
What's the opposite
side from this angle?

00:03:29.200 --> 00:03:32.090
I'm going to change to yellow.

00:03:32.090 --> 00:03:32.960
It's this side, right?

00:03:32.960 --> 00:03:34.630
This is the opposite side.

00:03:34.630 --> 00:03:37.040
And what's the hypotenuse?

00:03:37.040 --> 00:03:42.500
The hypotenuse is just
this radius, right?

00:03:42.500 --> 00:03:45.440
And let's just say that this
point, where it intersects

00:03:45.440 --> 00:03:54.550
the circle-- let's call this
point right here x comma y.

00:03:54.550 --> 00:04:00.450
So what's the height of
this opposite side?

00:04:00.450 --> 00:04:01.300
Well, it's y, right?

00:04:01.300 --> 00:04:05.130
Because it's just the
height of that point.

00:04:05.130 --> 00:04:06.570
This is of height y.

00:04:06.570 --> 00:04:13.170
So sine of this angle right
here, sine of theta, is

00:04:13.170 --> 00:04:15.400
going to equal the opposite
side-- which is this yellow

00:04:15.400 --> 00:04:18.340
side, which is just the
y-coordinate-- is going to

00:04:18.340 --> 00:04:21.490
equal y over the hypotenuse.

00:04:21.490 --> 00:04:23.930
The hypotenuse is
this pink side here.

00:04:23.930 --> 00:04:25.350
And what's the length
of the hypotenuse?

00:04:25.350 --> 00:04:27.750
Well, it's the radius
of this unit circle.

00:04:27.750 --> 00:04:29.600
So it's 1.

00:04:29.600 --> 00:04:31.100
And y divided by 1?

00:04:31.100 --> 00:04:31.860
Well, that's just y.

00:04:31.860 --> 00:04:40.040
So we see that sine of
theta is equal to y.

00:04:40.040 --> 00:04:42.520
Let's do the same thing
for cosine of theta.

00:04:42.520 --> 00:04:49.700
Well, we know that cosine
is equal to adjacent

00:04:49.700 --> 00:04:51.430
over hypotenuse.

00:04:51.430 --> 00:04:55.410
Well, what's the
adjacent side here?

00:04:55.410 --> 00:04:56.470
I'm running out of colors.

00:04:56.470 --> 00:05:00.200
The adjacent side is this
bottom side, right here.

00:05:00.200 --> 00:05:02.390
So that would equal-- so
if I said-- I'm running

00:05:02.390 --> 00:05:03.760
out of space, too.

00:05:03.760 --> 00:05:09.460
Cosine of theta would equal
this gray side-- which is

00:05:09.460 --> 00:05:11.250
the adjacent side--
and what is that?

00:05:11.250 --> 00:05:14.350
What is this length?

00:05:14.350 --> 00:05:16.610
What is the length
of this side?

00:05:16.610 --> 00:05:18.430
Well, it's just x, right?

00:05:18.430 --> 00:05:23.070
If this is the point x, y then
this distance here is x and we

00:05:23.070 --> 00:05:25.250
already learned this distance--
or we already observed--

00:05:25.250 --> 00:05:26.740
that this distance is y.

00:05:26.740 --> 00:05:29.100
So this distance being
just x, we know that the

00:05:29.100 --> 00:05:31.280
adjacent length is x.

00:05:31.280 --> 00:05:34.980
So we say cosine of theta is
equal to x over the hypotenuse.

00:05:34.980 --> 00:05:37.360
And once again, the
hypotenuse is 1.

00:05:37.360 --> 00:05:43.330
So cosine of theta
is equal to x.

00:05:43.330 --> 00:05:44.120
I know what you're thinking.

00:05:44.120 --> 00:05:45.820
Sal, that's very nice and cute.

00:05:45.820 --> 00:05:48.280
Cosine of theta equals x,
sine of theta equals y.

00:05:48.280 --> 00:05:49.760
But how is this really
different from what we

00:05:49.760 --> 00:05:51.670
were doing before?

00:05:51.670 --> 00:05:56.160
Well, if I define it this way,
now all of a sudden when the

00:05:56.160 --> 00:06:02.940
angle becomes 90 degrees,
now I can actually define

00:06:02.940 --> 00:06:04.980
what sine of theta is.

00:06:04.980 --> 00:06:07.540
Sine of theta now is just y.

00:06:07.540 --> 00:06:10.190
Is just the y-coordinate,
which is 1.

00:06:10.190 --> 00:06:13.400
If theta is equal to-- I'm
going to make sure it's very

00:06:13.400 --> 00:06:15.610
messy right here-- if theta
is equal to 90 degrees,

00:06:15.610 --> 00:06:18.000
or pi over 2 radians.

00:06:18.000 --> 00:06:20.680
This is pi over 2.

00:06:20.680 --> 00:06:22.870
This angle right here.

00:06:22.870 --> 00:06:26.590
And similar, cosine
of pi over 2 is 0.

00:06:26.590 --> 00:06:31.650
Because the x-coordinate
right here is 0.

00:06:31.650 --> 00:06:33.060
Let me do it with a
couple more examples.

00:06:33.060 --> 00:06:35.050
Oh, I'm forgetting the
tangent function.

00:06:35.050 --> 00:06:36.740
And you could probably
figure out now, what is the

00:06:36.740 --> 00:06:39.650
definition now we can use
for the tangent function?

00:06:39.650 --> 00:06:41.600
Well, going back-- let's
use this green theta here.

00:06:41.600 --> 00:06:43.490
Because it's kind
of a normal angle.

00:06:43.490 --> 00:06:45.830
So in this green angle
here, tangent is

00:06:45.830 --> 00:06:47.680
opposite over adjacent.

00:06:47.680 --> 00:06:54.040
So tangent now, we
can define as y/x.

00:06:54.040 --> 00:06:57.760
And remember, these y's and x's
that we're using are the point

00:06:57.760 --> 00:07:01.780
on the unit circle where the
angle that's defined by this--

00:07:01.780 --> 00:07:06.440
by whatever-- where the radius
that is subtended by this

00:07:06.440 --> 00:07:09.150
angle, or I guess the arc,
intersects-- actually,

00:07:09.150 --> 00:07:10.940
I'm getting confused
with terminology.

00:07:10.940 --> 00:07:13.810
It's where this line
intersects the circumference.

00:07:13.810 --> 00:07:17.800
The coordinate of that-- the
sine of theta is equal to y.

00:07:17.800 --> 00:07:19.750
The cosine of theta
is equal to x.

00:07:19.750 --> 00:07:22.740
And the tangent of
theta is equal to y/x.

00:07:22.740 --> 00:07:27.020
Let's do a couple of examples
and hopefully this'll make a

00:07:27.020 --> 00:07:28.110
little bit more sense to you.

00:07:31.550 --> 00:07:35.246
Let me try to really fast
draw a new unit circle.

00:07:37.780 --> 00:07:39.030
So that's my unit circle.

00:07:42.810 --> 00:07:44.960
And here's the coordinate axis.

00:07:44.960 --> 00:07:46.530
It's one of them.

00:07:46.530 --> 00:07:48.000
And here is the other one.

00:07:58.180 --> 00:08:04.010
So if we use the angle-- let's
use the angle pi over 2, right?

00:08:04.010 --> 00:08:06.450
Theta equals pi over 2.

00:08:06.450 --> 00:08:09.670
Well, pi over 2 is right here.

00:08:09.670 --> 00:08:12.860
It's a 90 degree angle, if
you wanted to use degrees.

00:08:12.860 --> 00:08:15.300
And now, we just figure
out where it intersects

00:08:15.300 --> 00:08:15.840
the unit circle.

00:08:15.840 --> 00:08:17.715
And once again, this is
a unit circle, so it

00:08:17.715 --> 00:08:20.300
has a radius of 1.

00:08:20.300 --> 00:08:29.210
So we can see that sine of pi
over 2 equals the y-coordinate

00:08:29.210 --> 00:08:32.080
where it intersects
the unit circle.

00:08:32.080 --> 00:08:34.630
So that's just 1.

00:08:34.630 --> 00:08:36.090
What's cosine of pi over 2?

00:08:39.340 --> 00:08:41.090
Well, it's just the
x-coordinate, where you

00:08:41.090 --> 00:08:42.220
intersect the unit circle.

00:08:42.220 --> 00:08:46.060
And the x-coordinate here is 0.

00:08:46.060 --> 00:08:47.900
And what's the tangent
of pi over 2?

00:08:47.900 --> 00:08:49.700
This is interesting.

00:08:49.700 --> 00:08:52.770
The tangent of pi over 2.

00:08:52.770 --> 00:08:55.450
Well, the tangent we
defined now as y/x.

00:08:55.450 --> 00:08:58.910
So the y-coordinate, this
is the point 0, 1, right?

00:08:58.910 --> 00:09:01.610
The y-coordinate is 1.

00:09:01.610 --> 00:09:02.460
So it equals 1/0.

00:09:05.400 --> 00:09:06.750
So this is undefined.

00:09:06.750 --> 00:09:09.800
So still, we don't have a
tangent function that can

00:09:09.800 --> 00:09:11.640
define itself at
certain points.

00:09:11.640 --> 00:09:14.660
But in the next module, we're
actually going to graph this.

00:09:14.660 --> 00:09:17.470
And you'll see that it
approaches infinity.

00:09:17.470 --> 00:09:21.580
And similarly, we could try to
find the functions for when

00:09:21.580 --> 00:09:24.100
theta equals pi, right?

00:09:24.100 --> 00:09:27.060
That's like 180 degrees.

00:09:27.060 --> 00:09:28.570
That's this point right here.

00:09:28.570 --> 00:09:31.450
So sine of pi.

00:09:31.450 --> 00:09:33.820
What's the y-coordinate
at this point?

00:09:33.820 --> 00:09:37.720
Well, this point is
negative 1 comma 0.

00:09:37.720 --> 00:09:39.880
So the y-coordinate is 0.

00:09:39.880 --> 00:09:40.940
What's the x-coordinate?

00:09:40.940 --> 00:09:43.520
Cosine of pi.

00:09:43.520 --> 00:09:45.260
That's negative 1.

00:09:45.260 --> 00:09:49.570
And of course, what's the
tangent of pi radians?

00:09:49.570 --> 00:09:51.220
It's y/x.

00:09:51.220 --> 00:09:54.400
So it's 0 over negative
1, which equals 0.

00:09:54.400 --> 00:09:55.760
Hopefully this makes sense.

00:09:55.760 --> 00:09:58.730
Now in the next module I'll
actually graph these points.

00:09:58.730 --> 00:10:04.450
And you'll see how it all comes
together and why it is useful

00:10:04.450 --> 00:10:09.010
to define the sine, the cosine,
the tangent functions this way.

00:10:09.010 --> 00:10:10.160
See you soon.

00:10:10.160 --> 00:10:11.460
Bye.


Using Trig Functions Part II

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:01.580
Welcome back.

00:00:01.580 --> 00:00:05.340
I just want to do a couple more
of these problems, just so you,

00:00:05.340 --> 00:00:09.440
I guess, see a few couple more
problems and get some practice.

00:00:09.440 --> 00:00:12.040
So let's start with
another problem.

00:00:12.040 --> 00:00:16.440
As always, let me
draw my triangle.

00:00:16.440 --> 00:00:18.270
I always like to draw
it a little different

00:00:18.270 --> 00:00:21.240
to confuse you.

00:00:21.240 --> 00:00:34.410
Let's say that this angle
up here is 0.36 radians.

00:00:34.410 --> 00:00:38.990
An angle is 0.36 radians, and
then this side right here, this

00:00:38.990 --> 00:00:44.390
top side-- let me do it in a
different color-- this top side

00:00:44.390 --> 00:00:53.190
in pink is square root of
73 units long or whatever.

00:00:53.190 --> 00:00:54.800
It could be inches or
feet or whatever.

00:00:54.800 --> 00:00:57.690
But it's square root
of 73 units long.

00:00:57.690 --> 00:01:05.060
So my question is, what is this
side here, this green side?

00:01:05.060 --> 00:01:08.350
Well, how did we do this
type of problem last time?

00:01:08.350 --> 00:01:11.600
We should figure out what
sides we're dealing with.

00:01:11.600 --> 00:01:14.040
Are we dealing with the
adjacent and the opposite, the

00:01:14.040 --> 00:01:16.460
adjacent and the hypotenuse,
the opposite and the

00:01:16.460 --> 00:01:20.050
hypotenuse, and then we'll know
which trig functions we

00:01:20.050 --> 00:01:21.370
should be dealing with.

00:01:21.370 --> 00:01:25.960
So we know this side, this pink
side-- and this pink side, it

00:01:25.960 --> 00:01:28.090
should be almost second
nature by now, is the

00:01:28.090 --> 00:01:28.450
hypotenuse, right?

00:01:28.450 --> 00:01:31.050
It's the longest side, and it's
opposite the right angle.

00:01:31.050 --> 00:01:34.600
So we know the hypotenuse,
and what are we solving for?

00:01:34.600 --> 00:01:36.480
Well, this is the angle we
know, so we're solving

00:01:36.480 --> 00:01:38.000
for the opposite side.

00:01:38.000 --> 00:01:40.520
So we're solving for the
opposite side, and we know the

00:01:40.520 --> 00:01:45.000
hypotenuse, so what trig
function will we probably use?

00:01:45.000 --> 00:01:47.360
Let's write out our mnemonic.

00:01:47.360 --> 00:01:52.540
SOH CAH TOA.

00:01:52.540 --> 00:01:54.120
So what did I say before?

00:01:54.120 --> 00:01:56.600
We're solving for the opposite
sign, and we know the

00:01:56.600 --> 00:02:01.245
hypotenuse, so we are going to
use the opposite and

00:02:01.245 --> 00:02:02.470
the hypotenuse.

00:02:02.470 --> 00:02:04.600
So which of these is that?

00:02:04.600 --> 00:02:06.200
The opposite and the
hypotenuse, the o

00:02:06.200 --> 00:02:07.420
and the h, right?

00:02:07.420 --> 00:02:10.460
It's SOH, right?

00:02:10.460 --> 00:02:13.250
And SOH says that sine is
equal to the opposite

00:02:13.250 --> 00:02:15.180
over the hypotenuse.

00:02:15.180 --> 00:02:20.050
Sine of an angle-- in this
case, it's this angle-- sine of

00:02:20.050 --> 00:02:24.900
0.36 radians-- remember, this
is radians we're dealing with,

00:02:24.900 --> 00:02:28.910
not degrees-- is equal
to the opposite side.

00:02:28.910 --> 00:02:31.730
And the opposite side is this
green side right here, so I'll

00:02:31.730 --> 00:02:34.920
just write opposite instead of
writing o because o would look

00:02:34.920 --> 00:02:38.720
like a 0-- is equal to the
opposite side over the

00:02:38.720 --> 00:02:40.470
hypotenuse, right?

00:02:40.470 --> 00:02:41.920
This is just sine is equal to
opposite over hypotenuse.

00:02:41.920 --> 00:02:43.130
Well, what's the
hypotenuse length?

00:02:43.130 --> 00:02:44.240
Well, it's the
square root of 73.

00:02:47.600 --> 00:02:48.970
These are p's, by the way.

00:02:48.970 --> 00:02:52.850
I know they don't look
like p's, but opposite.

00:02:52.850 --> 00:02:58.350
So the opposite side-- we're
just multiplying both sides by

00:02:58.350 --> 00:03:03.680
the square root of 73-- is
equal to the square root of 73

00:03:03.680 --> 00:03:11.200
times the sine of 0.36 radians.

00:03:11.200 --> 00:03:14.450
Now, once again, I don't know
what the sign of 0.36 radians

00:03:14.450 --> 00:03:17.440
is in my head, but I'll
tell you the answer.

00:03:17.440 --> 00:03:23.000
The sine of 0.36 radians is
equal to-- I'm just rewriting

00:03:23.000 --> 00:03:26.630
this-- the sine of 0.36
radians, if you looked it up on

00:03:26.630 --> 00:03:32.550
a table or if you used your
calculator in radian mode, is 3

00:03:32.550 --> 00:03:36.870
square roots of 73 over 73.

00:03:36.870 --> 00:03:38.560
And, of course, your calculator
is going to give you

00:03:38.560 --> 00:03:41.420
something-- it'll give
you some decimal number.

00:03:41.420 --> 00:03:43.390
I won't write it this way.

00:03:43.390 --> 00:03:46.470
So just remember, this is this.

00:03:46.470 --> 00:03:47.490
And I just looked that up.

00:03:47.490 --> 00:03:49.270
There's no magic there.

00:03:49.270 --> 00:03:52.740
And the square root-- or you
could use a calculator.

00:03:52.740 --> 00:03:55.020
On the Khan Academy, when you
do problems, it'll actually

00:03:55.020 --> 00:03:58.270
tell you what it is, so you
don't have to use a calculator.

00:03:58.270 --> 00:03:59.350
So now we just simplify.

00:03:59.350 --> 00:04:03.660
Square root of 73 times square
root of 73 is 73 over 73 is

00:04:03.660 --> 00:04:05.760
equal to 1, so these
all cancel out.

00:04:05.760 --> 00:04:09.340
And we get the answer of 3.

00:04:09.340 --> 00:04:13.510
So this side right here is 3.

00:04:13.510 --> 00:04:15.900
And just out of curiosity, if
you wanted to solve for this

00:04:15.900 --> 00:04:18.460
side, there's two ways
we could do it, right?

00:04:18.460 --> 00:04:21.680
We could use the Pythagorean
theorem, because, you know, a

00:04:21.680 --> 00:04:24.520
squared plus b squared is
equal to c squared, or we

00:04:24.520 --> 00:04:26.510
could use trigonometry.

00:04:26.510 --> 00:04:28.400
I'll let you guess
what trig function--

00:04:28.400 --> 00:04:29.490
actually, let's do that.

00:04:29.490 --> 00:04:33.330
Let's figure out that side
using trigonometry, and then

00:04:33.330 --> 00:04:37.200
let's figure out that side
using the Pythagorean theorem,

00:04:37.200 --> 00:04:41.600
just to show that everything
fits together in math.

00:04:41.600 --> 00:04:44.660
So I wrote that 3 there, so I
can erase all of this stuff.

00:04:51.610 --> 00:04:53.050
Let me erase it.

00:04:58.110 --> 00:04:59.840
I shouldn't have
erased the SOHCAHTOA.

00:04:59.840 --> 00:05:02.570
Actually, we should have
that memorized by now.

00:05:02.570 --> 00:05:02.920
SOHCAHTOA.

00:05:07.240 --> 00:05:10.560
All right, so let's figure out
what this orange side is here.

00:05:10.560 --> 00:05:13.150
And if you think about it, we
could do it a bunch of ways.

00:05:13.150 --> 00:05:15.110
We could say, well, this is
the adjacent side, right?

00:05:15.110 --> 00:05:16.990
Because we know the opposite
and we know the hypotenuse.

00:05:16.990 --> 00:05:22.020
So we could either use-- we
know the opposite, so we could

00:05:22.020 --> 00:05:26.260
say what trig function uses the
opposite and the adjacent?

00:05:26.260 --> 00:05:28.540
Well, that's tangent
function, right?

00:05:28.540 --> 00:05:35.970
So we could say tangent
of 0.36-- let's call

00:05:35.970 --> 00:05:37.110
this side A, right?

00:05:37.110 --> 00:05:38.720
A for adjacent.

00:05:38.720 --> 00:05:44.210
Tangent of 0.36 is equal
to the opposite, 3, over

00:05:44.210 --> 00:05:46.770
the adjacent, over A.

00:05:46.770 --> 00:05:49.720
Is there another trigonometry
way we could think about this?

00:05:49.720 --> 00:05:51.580
Well, we also know
the hypotenuse.

00:05:51.580 --> 00:05:55.730
What trig function uses the
hypotenuse and the adjacent?

00:05:55.730 --> 00:05:59.940
Well, if you remember,
SOH CAH TOA.

00:05:59.940 --> 00:06:03.730
CAH, cosine is adjacent
over hypotenuse.

00:06:03.730 --> 00:06:11.260
So we could say cosine of
0.36 is equal to adjacent

00:06:11.260 --> 00:06:16.040
over square root of 73.

00:06:16.040 --> 00:06:18.630
And I'll just write SOHCAHTOA
here, just so you can

00:06:18.630 --> 00:06:21.900
confirm what I'm doing.

00:06:21.900 --> 00:06:27.660
The TOA says the tangent is
equal to the opposite, 3, over

00:06:27.660 --> 00:06:32.840
the adjacent, and CAH tells us
the cosine is equal to the

00:06:32.840 --> 00:06:36.810
adjacent over the hypotenuse.

00:06:36.810 --> 00:06:38.300
So we could solve
either one of these.

00:06:38.300 --> 00:06:41.820
If we use the second formula,
we would get the adjacent side

00:06:41.820 --> 00:06:48.980
is equal to the square root of
73 times the cosine of 0.36,

00:06:48.980 --> 00:06:51.610
and then if you use your
calculator in radian mode, or

00:06:51.610 --> 00:07:01.650
I'll just tell you, that cosine
of 0.36 radians is equal to 8

00:07:01.650 --> 00:07:06.810
square roots of 73 over 73.

00:07:06.810 --> 00:07:09.300
And you can confirm that by
getting a decimal number and

00:07:09.300 --> 00:07:11.780
then-- making sure once again
your calculator is not in

00:07:11.780 --> 00:07:12.880
degree mode, but
in radian mode.

00:07:12.880 --> 00:07:14.370
I think that's actually the
default mode in a lot of

00:07:14.370 --> 00:07:15.960
calculators-- and
solving for this.

00:07:15.960 --> 00:07:19.720
But once again, the 73, this
square root of 73 times this

00:07:19.720 --> 00:07:23.100
square root of 73, is equal to
73, and then divided by 73.

00:07:23.100 --> 00:07:24.720
These all cancel out.

00:07:24.720 --> 00:07:26.320
How convenient, huh?

00:07:26.320 --> 00:07:28.460
And you get 8.

00:07:28.460 --> 00:07:31.220
So the adjacent side
is equal to 8.

00:07:31.220 --> 00:07:33.500
And so if we'd solved for A
here using the tangent

00:07:33.500 --> 00:07:36.300
function, we should've also
gotten the adjacent

00:07:36.300 --> 00:07:37.570
side is equal to 8.

00:07:37.570 --> 00:07:42.510
And just to show you that
everything works out from other

00:07:42.510 --> 00:07:49.250
concepts, let me show you this
using the Pythagorean theorem.

00:07:53.990 --> 00:08:03.100
So 8 squared plus 3
squared should equal

00:08:03.100 --> 00:08:03.970
the hypotenuse squared.

00:08:03.970 --> 00:08:08.100
The square root of 73 squared.

00:08:08.100 --> 00:08:13.530
Well, 8 squared is 64 plus 9
should equal-- what's the

00:08:13.530 --> 00:08:14.820
square root of 73 squared?

00:08:14.820 --> 00:08:16.690
Right, it's 73.

00:08:16.690 --> 00:08:19.250
And, of course,
64 plus 9 is 73.

00:08:19.250 --> 00:08:22.600
And sure enough, that
equals 73, so it works.

00:08:22.600 --> 00:08:26.470
Isn't that interesting how math
just kind of fits together?

00:08:26.470 --> 00:08:29.180
I think at this point you're
ready to try the modules, the

00:08:29.180 --> 00:08:32.790
Trigonometry II modules, and I
guess let me know if you

00:08:32.790 --> 00:08:35.510
have any problems, or if you
want to see more videos.

00:08:35.510 --> 00:08:37.010
Have fun!