>> This is your Math Gal
Julie Harland.
Please visit my website
at YourMathGal.com where all
of my videos are organized
by topic.
We're going to learn how
to graph trig functions now,
so here are a couple
of examples of trig functions.
Y equals 3 sin of x
and y equals cosin of 2 x.
And [inaudible] are x
and y. The difference here is
that we're going to think
of the variables
as being real numbers.
So when I say sin
of x I'm taking like the sin
of a real number.
And we got
that from the circular
functions,
so remember the circular
functions here.
So we have all of these --
think of these radiants
as pi over 6.
But really what this means is,
if you take take a length pi
over 6 remember this is the
unit circle then the
coordinates here x y give you
the actual cosin
of that length.
So the cosin of pi
over 6 is squared to 3 over 2,
and the sin of pi
over 6 is 1 half.
So let's just figure
out what it is for the sin
of these particular
numbers here.
So if I wanted the sin of x,
in other words, I want the sin
of 0 -- let's see you go back
up to here.
0 is right here,
and you just look
at the y value at 0.
So the sin of 0 is 0.
But I'm talking
about the length 0.
How about if we skipped this
pi over 6 and see what it is
for pi over 4.
So now I'm talking
about this length from here
to here all the way
up to pi over 4.
I just look
at the y value is squared to 2
over 2, and that is
approximately .7.
All right.
How about if I keep going.
I want the sin
of this the y value is 1
so the sin of pi
over 2 is 1 etc. So you can go
through and do all of this.
Let's say you go over here
to 3 pi over 4.
The y value is squared
of 2 over 2.
Again, approximately .7.
At pi the y value is 0.
5 pi over 4 the y value is
negative square roots of 2
over 2 so it's approximately
negative .7.
At 3 pi over 2
down here we've got the
y value.
And when I say the y value I
mean the y coordinate
of that ordered pair is
negative 1.
And at 7 pi
over 4 we've got negative
square it to 2
over 2 it's approximately
negative .7.
And at 2 pi we're at 0 again.
So imagine you've gone all the
way around, you know,
this is a length of 2 pi.
And I've got 0 again.
Well, what if I want 9 pi
over 4 that means I
keep going.
So imagine you're --
it's a longer string.
If you want to think of it
that -- that number goes all
the way up to here.
I start repeating myself.
I'm just going to have
that squared to 2 over 2 again
which is approximately .7.
If I go up to, you know,
2 pi plus pi over 2
which is -- what is that --
9 pi over 2.
Is that right?
5 pi over 2 -- sorry.
Then I would have 1 --
and this one's
repeating itself.
So what's going to happen is
if you try to graph this is --
the values you get are just
going to repeat.
So that's something to keep
in mind when we're doing
our graphing.
So now let's go back
and try graphing.
Y equals sin
of x using
these approximations.
And we could put a lot more
ordered pairs in here.
Like I could have done the sin
of pi over 6, the sin of pi
over 3, but I just didn't have
enough room
to write all this in here.
But something
to notice is what's going
on with the y values it starts
at 0, right?
The sin of 0 is 0,
and it gets bigger
until it gets to 1, right?
See the y value is one that's
the top if you look
at the height here.
The top y coordinate is 1,
and then it starts going
down again to .7 to 0.
And then it goes negative all
the way down to negative 1.
So I could put any number I
wanted for x. In fact,
I could even put
in negative numbers.
I can go to the left, right,
starting at 1 0.
And so my --
I could put any number I want
in for x so there's no
restrictions here.
So the domain is going
to be all real numbers.
But notice that the range,
the values of y, they only go
in between negative 1 and 1.
If you look
at the picture here the y
coordinates go
in between negative 1 and 1.
There's nothing smaller
than negative 1;
nothing bigger
than positive 1.
So let's try graphing the most
basic trig function,
and this shouldn't say 6 it
should say sin
of y. Graph y equals sin of x.
So we noted
that the domain could be all
real numbers there are no
restrictions at all
for what you could put
in for x. But the range,
in other words the y values,
only went in between negative
1 and 1.
So therefore,
notice on my y axis I just
went ahead and put a mark
for negative 1 and a mark
for positive 1.
And what's different here
instead going 1, 2, 3, 4, 5,
6, 7, 8 since these are the
values I chose for x I went
from 0 to 9 pi
over 4 I'm going to do things
in terms of pi.
Now, I did pi over 4,
pi over 2.
If you want,
you could put these all
with the same
common denominator.
And you notice this could be
thought of as, you know,
pi over 4 --
2 pi over 4, 3 pi over 4,
5 pi over 4.
I mean, you know, 4 pi over 4,
5 pi over 4, 6 pi over 4,
7 pi over 4, 8 pi over 4,
9 pi over 4.
Some people find it easier
to just write them all
with a common denominator.
So basically it's hard
to write all those little
values here.
So hopefully you could see
this would be the pi
over 4 right in between 0
and pi over 2.
So let's graph these.
I've got 0 0.
All right.
So 0 0 would be right here.
All right.
Pi over 4 and about .7.
So here it's about .7.
It looks like it's
about right here.
Let's see pi
over 2 is up at 1.
3 pi over 4 is about .7 again.
[Silence]
Pi 0, 5 pi over 4 negative .7.
So that looks like it's
down about here.
3 pi over 2 is negative 1 --
about right here.
7 pi over 4 is negative .7.
2 pi is 0.
Then 9 pi over 4 was up at .7.
Remember we said that that's
when it started repeating
itself so you can guess the
next place over.
If I went up, it's going to be
up here at 1 again.
That's exactly what happens.
You know, you don't know
for sure what this looks like,
but hopefully it sort of looks
like a nice little curve
from here that you could see.
But you would need a lot more
ordered pairs
to tell for sure.
And -- but in fact,
it is a nice smooth curve.
It ends up being a nice
smooth curve.
So you could put
in a lot more values
and that's exactly
what calculates.