>> This is your math gal,
Julie Harland.
Please visit my website
at yourmathgal.com,
where all of my videos are
organized by topic.
All right,
this is the first video on how
to graph the tangent function,
and we're going
to do the most basic one,
y equals the tangent of x.
And we're going to start
by looking at the unit circle.
So here's the unit circle,
where I've got angles
in radians and also
in degrees,
and those special ordered
pairs here
for these special angles.
And remember,
you could say this as going
from 0 to 360,
but you could also think
of these as negative angles,
so I'm going to consider,
instead of thinking this is 11
pi over 6,
remember that's co-terminal
with negative pi over 6.
And 7 pi over 4 is co-terminal
with negative pi over 4,
and 5 pi over 3 is co-terminal
with negative pi over 3,
and 3 pi over 2 is co-terminal
with negative pi over 2.
Since we're going
to be graphing,
I like to think of this.
I want to see what's going
on between negative pi over 2
and pi over 2.
Well, on the unit circle,
as I've labeled it,
with these ordered pairs here,
I have the cosine
and the sine.
Remember on the unit circle,
the first coordinate is the
cosine and the second
coordinate is the sine.
So I don't have the tangent
written here.
And there are three ways you
can get the tangent.
You can simply put y over x,
so for instance,
you can take this ordered
pair, since the tangent's y
over x and I'm just going
to write down what
that is in green.
That's 0 over 1, which is 0.
So the tangent
of 0 would be 0, right,
because this is 0 degrees
right here or 0 radians.
Well, of course it gets more
complicated
when I have 1 half divided
by squared of 3 over 2,
but you could do that,
as well.
So another way
of doing it is getting
out your calculator,
and figuring out the tangent
of 30 degrees.
Make sure your calculator's
in degree mode if you do that.
Or you could put it
in radian mode
and compute the tangent
of pi over 6.
All right,
so I got out my calculator,
and the tangent of pi
over 6 is approximately .6.
All right,
so this is rounding,
so I'm rounding,
[writing on board] all right,
to find the tangent.
Okay, so in green,
I could go around
and find the tangent
of all these.
Pi over 4 you should know.
That's the easy one.
That's 1, the tangent of pi
over 4 or 45 degrees.
Or look at it this way,
square root of 2,
over 2 divided by square root
of 2 over 2,
that's the same number
over itself,
so the tangent here is going
to be 1, etcetera.
And then you're going
to have some negative ones
down here because the y is
negative and the x is
positive, etcetera.
So I've gone ahead
and done this for you
for all these special angles.
So in purple,
you see the tangents
of those special angles.
Okay, so let's
notice something.
Why is this undefined
at negative pi over 2
down here?
Because negative 1 divided
by 0, you'd have a 0
in the denominator.
So remember the reason,
up here at pi over 2
or negative pi over 2,
its undefined
because it's negative --
this is a positive 1
over 0, right?
And when you divide
by 0 its undefined.
Let's look at the values.
Remember from All Students
Take Calculus,
the tangent is positive
in the first
and the third quadrant
as you note.
Okay? But let's see
what's happening.
At negative pi over 2
down here,
tangent is undefined.
And then it's still negative
if I get to negative pi
over 3, and then it's still
negative and it's a smaller
negative number.
We're at negative 1,
negative .6.
And I mean the absolute value
is smaller.
Until we get to 0,
and then it starts
getting positive.
Positive .6, positive 1,
positive 1.7.
And if you check something
like 89 degrees,
it's going
to be a huge number.
Now, what happens
if we keep going to the left
in quadrant 3?
We have negative 1.7.
Negative 1.
Negative .60.
You know what,
it starts repeating
like it did
between negative pi over 2
to pi over 2.
So the interesting thing
about the tangent function,
the tangent has a pi.
The tangent function has a
period of pi.
It starts repeating
after pi units.
Unlike the sine and cosine,
which repeats
after 2 pi units,
and so has a period of 2 pi.
All right, I have a chart here
for all the values I have
on that previous page,
and let's just see what
happens if we graph
these values.
You could of course not tell
for sure what it looks
like with just acute values.
You would have to put
in maybe 100,
and that's what the graphing
calculator does.
But let's check out about
where it's undefined.
At negative pi over 2 and pi
over 2, its undefined.
That's right here.
So I'm going
to have vertical asymptotes
right here.
[ Writing on Board ]
And negative -- at x equals --
so the vertical asymptotes,
all right,
I can just take one of those.
It doesn't matter if it's pi
over 2, or negative pi over 2.
So let's just take the
positive one, pi over 2.
But you know there's going
to be another one over here.
Pi units because it's going
to repeat, right?
So I'm going
to have plus how far
over is that, is pi units.
It's a period.
So after one period,
you're going
to get another asymptote.
So if you have an asymptotes
of negative pi over 2,
you add pi,
you get another one and get pi
over 2, you add pi again,
you're going to be over here
at 3 pi over 2.
So this is pi k
where k is an integer.
So the asymptote --
or I could have said start
at negative pi over 2.
Just pick any asymptote you
feel like, and you have
to remember that it repeats.
There's a lot
of vertical asymptotes.
And we talked
about the period.
The period was only pi.
Sort of repeating.
We don't know the range yet,
so let's see
if we could figure that out.
So I'm going
to plot these points.
Negative pi over 3,
negative 1.7.
So I didn't put
where negative pi over 3 is.
I didn't put
that many markings here,
but it's about right here.
And then pi over 6 is
about right here.
Okay? So if I had broken this
up into 6 pieces,
so if I mark that,
it's about negative 1.7.
About right here.
And at negative pi over 4,
its negative 1.
And negative pi
over 6 is negative .6.
And so you can't really tell
what's going on,
so you'll have
to take a little bit of a leap
of faith unless you want
to put a lot of points
in yourself.
What happens is,
you get this interesting
curve here.
Woops. I wanted to plot all
of these points.
I forgot. I've got
to keep going.
I've got 00 -- here we go.
Then we have pi over 6, 06.
That's about right here.
And at pi over 4 it's actually
at 1.
And 1.7. Pi over three is
about 1.7, etcetera.
If you graph these,
you get this shape,
where it gets close
to the asymptotes.
That is 1 cycle
of the tangent curve.
Then it starts repeating.
Now let's look back
at the unit circle again.
When I had all these --
notice when I got to pi
over 3, it was positive 1.7,
then it was undefined.
And then, once I'm over pi
over 2, it's a negative
number again.
So check this out.
Here I am at --
this is like pi
over three somewhere
around here.
It gets to undefined,
then right on the other side
of this asymptotes,
it's negative again.
So you know what,
it does start down here again.
[ Noises ]
It's hard to graph.
Goes like this.
These are the ones,
right here.
And now we have two periods
of the tangent function.
So that's a basic look
at the tangent function,
so we have y equals tangent
of x, the period is pi,
that's how far it goes before
it's repeating,
and the easiest way
to do it is have it go
in between negative pi and pi.
If you had graphed between 0
and pi, it looks a little
funny, although you could
do that.
It's easier
to see a full cycle
if you start at negative pi,
go through 00,
and then it kind of goes
up with these asymptotes.
I didn't talk about the range.
What's the range?
What are the values
for the y coordinates?
Well, it goes forever,
and then it goes up forever,
so the range is all real
numbers, negative infinity
to infinity.
So here's the guides
for graphing tangent function.
For the basic tangent
function, it's pi.
All right, so if you cut
that in half.
Okay, if you start at 00,
you go pi over 2, right?
Half of pi is pi over 2,
so you go pi over 2
to the right
to get 1 asymptote,
pi over 2 to the left
to get the other asymptote.
And then in between there,
it starts down.
It always goes halfway
in between is negative 1,
then 0.
Halfway between on the right
is positive 1.
Woops, sorry.
And then we graph it.
All right,
so we'll do some more tangent
functions on the next video.
This is your math gal,
Julie Harland.
Please visit my website
at yourmathgal.com,
where all of my videos are
organized by topic.