BAM!!! Mr. Tarrou. Ok, if you want to see tangent and cotangent graphed with a bunch
of transformations like phase shifts and vertical shifts this is not the video for you. Watch
the next lesson I am going to post. This link is up here above my head. If you however want
to understand why does tangent and cotangent have a period of pi, where sine, cosine, secant,
and cosecant have a period of 2pi, and you want to see the relationship between the unit
circle and the values for tangent, then watch this video. We are going to set up just the
basic tangent and cotangent graphs and then do some more examples in the next video. Here
we have the unit circle. It is set up with all the radian values. If you have been watching
my lessons for graphing trig functions without a calculator, then you will know that we only
care about graphing the easy points. You know, the ones, zeros, and negative one. So let's
look around this unit circle and of course I have got my answers already listed out here.
As we go around the unit circle and we start to work on graphing tangent of theta which
is y over x on the unit circle. Let's look at some of our values and look at how these
values repeat around the unit circle. We have the tangent of zero which is y over x. Of
course that is zero over one which is zero. And now if I just continue to go around the
values on the unit circle like pi over 6, and pi over 4, and so on, you can see that
we have a lot of decimals that we really don't want to graph when we are tying to make these
by hand. So the tangent of pi/6 is y over x so it is going to be one-half over the square
root of three over two and those common denominators are going to cancel out giving you the value
of one over the square root of three. We never leave square roots in the denominator so we
have to rationalize that. The tangent of pi over six is the square root of three over
three. You don't want to try and sketch that value when you are not using a calculator
to help us out. The tangent of pi over four. Well tangent is again y over x, so it is not
like sine, cosine, secant, and cosecant which always use the radius as part of their calculation.
Because of that when you look at pi over four and see that it is y over x you get the value
of one. Anything divided by itself is one. So we only had to forty five degrees or pi
over four radians before we found an easy value to graph of one. If we were talking
about sine or cosine you would have to rotate 90 degrees or pi over two radians to get to
that easier value of zero and one that use to sketch, or graph it. So continuing on to
pi over three. Again tangent is y over x, the y coordinate over the x coordinate on
the unit circle. Again the common denominator of two cancels out and we get another decimal
answer from the square root of three. And then at the tangent of pi over two, again
it is y over x. The y coordinate over the x coordinate, and one divided by zero is undefined.
So a couple of things going on here. We can see that our easy values of zero, one, and
well this is some weird value of undefined. But they are not going to happen at the quadrantal
angles like they do with sine and cosine and thus secant and cosecant as well. So that
is an idea of how they are a little bit different. We will have places where when you are doing,
when you are doing y divided by x or with cotangent which is x over y, that leaves the
possibility of x or y becoming zero. Those will be the quadrantal angles. So there is
a possibility like we see here with the tangent of pi over 2, that it can come out to be undefined.
Our graphs are going to have vertical asymptotes at those points where the x or y's become
zero in the denominator and thus you can't set up the tangent or cotangent function and
you get vertical asymptotes. Ok, around the unit circle I have highlighted the angles
that give us the easy values of one, zero, negative one, or in the case of tangent or
cotangent they can become undefined. Secant and cosecant can have vertical asymptotes
as well. Because secant being r over x and cosecant being y over y, those have the possibility
of having vertical asymptotes as well. They do have vertical asymptotes. So we have these
easy values that we are going to use to sketch the standard tangent and cotangent graphs
in a second. Let's talk about why tangent only has a period of pi instead of 2pi. Think
about the cosine graph. The cosine graph, the cosine of zero is one, the cosine of pi/2
is zero, the cosine of pi is negative one. So we have gone from values of one to negative
one. Then as we continue around our circle, the cosine of 3pi/2 is zero. The cosine of
2pi is back to one. So the values for the cosine, the cosine of zero, the cosine of
pi/2, the cosine of pi, the cosine of 3p/2, the cosine of 2pi. Hopefully my hand motion
is enough. I am assuming if you are watching a lesson about tangent and cotangent, you
have already about sine and cosine graphs. Those values go from one to negative one.
Now you start to climb back up again, but the values are in a different order. So thus,
the period of cosine and sine is a full 2pi. You have to go all the way around the unit
circle before you values start to repeat, and they repeat in the same order. Well look
at tangent here. The tangent of zero, the tangent of pi/4, the tangent of pi/2 where
it is undefined, the tangent of 3p/4 where it is the first time we get negative one,
and the tangent of pi is back to zero again. So our values are zero, one, undefined, negative
one, and we get back to zero. Now watch what happens. The tangent of 5pi/4. The tangent
in quadrant three, that is where we are, has a negative x and a negative y so it is a positive
value and we get the value of one again. Just like we had at pi/4, the other side of the
unit circle or 180 degrees or pi units away. Then go to 3pi/2. We are again doing y divided
by x. X at this value or this quadrantal angle is zero and it is once again undefined. So
at 5pi/4 and at pi/4 you have values of one. At 3pi/2 and pi/2, opposite sides of the circle,
you have undefined again. Then at 7pi/4 because the x is positive and the y is negative we
get a tangent value which is negative one just like it was at 3pi/4. So you see, these
values on the opposite sides of the unit circle are repeating and repeating in the same order.
So you only have to go half way around the circle or pi radians before your tangent values
once again begin to repeat and repeat in the same order unlike cosine that starts at one
goes to negative one at pi and you start to climb up again. You see those values repeat,
but they are repeating in the opposite order. You have not finished your first period yet.
So sine and cosine, secant and cosecant have a period of 2pi, where as tangent and cotangent
have a period of only pi. Let's set up cotangent for a second. The cotangent of theta is x
over y. So all these yellow values are my tangent values. Cotangent is the reciprocal
of tangent. Well when you flip zero, you get an undefined value. When you flip one, you
get one. So the cotangent of zero is undefined, that will be a vertical asymptote. The cotangent
of pi/4 is one. The cotangent of pi/2. If you take x over y, zero divided by one is
zero. At 3pi/4 you get negative one again, because the reciprocal of negative one is
negative one. If you take zero, the cotangent of pi, if you take zero and you flip it you
get an undefined value. Or, the cotangent of pi is x over y, again you cannot divide
by zero so that is going to be undefined. There is your first period of cotangent. These
values repeat around the bottom of the unit circle. Let's look at a couple of graphs,
shall we:) Ok. nanananan... Camera is a little bit crooked but. Let's get going anyway. y
equals the tangent of x. We are going to set up the parent function. No transformations.
We have a fake amplitude of one. Our leading coefficient is one. I still like to point
out this idea of fake amplitude, not because tangent has amplitude. When a graph goes up
and down forever, there is not height to take half of. I like my kids to see that when they
go down their t-table by hand, if their values are not separated by units of one, the fake
amplitude as I call it, that hopefully they can see their mistake on the t table. Period
is pi over b, not 2pi over b like it is with sine, cosine, secant, and cosecant, but pi
over b. That is pi over one which is pi. There is no parenthesis, no addition or subtraction
of a constant at the end, so the phase shift and the vertical shift are both zero. We are
still going to count by a quarter of the period. So if you have been watching my videos about
graphing sine, cosine, secant, and cosecant, I am going to say the same things. Start our
t table at the phase shift and count by a quarter of the period. Here are the values
that we had just on the unit circle a minute ago. Alrighty then!!! So here we have got
a graph set up with the values that I have in my t table. Well if you connect these dots
you get sort of a weird picture. When we graphed sine and cosine we got a nice flowing graph.
This has a break right dead center of the period. It is zero, one, and then boom it
is undefined. Back to negative one and zero. So you might the first time you graph tangent
want to extend your t table until you get used to pattern. Now I know that five points
will set up one period of a trig function when you start at the phase shift and you
count by a quarter of the period. So even this looks funny, I do know that it is one
period. So let's just start repeating the pattern. Zero, one, asymptote, so now I am
at zero so my next tick mark over again made off my values from the t table must be the
value of one. Then at 3pi/2 I have another vertical asymptote because it was zero, one,
asymptote, zero, one, asymptote. So this is, actually this is my second period of tangent,
because the first period of tangent is generally looked at as the one that is centered around
the origin. So one, zero, negative one, so one, zero, negative one. And what is next?
Vertical asymptote. Good! This is right here in the center is what you would see in your
textbook when it shows you the first parent graph, or the first period of tangent. You
might notice some things about this. If you take this graph and you rotate 180 degrees
it will line up with itself, so this is an odd function. It has symmetry about the origin.
It goes up and down forever so the range is all real numbers or negative infinity to positive
infinity. The domain is going to have this, it is almost all real numbers except you have
got the asymptotes where tangent is undefined. So the domain is going to be all real numbers
except for where the tangent is function is undefined. That is going to change based on
the period of the graph and the phase shift. But, you can just find one place where there
is a vertical asymptote, and then say it is all real numbers except whatever that x value
is, in this case pi/2, and then plus or minus n times whatever your period is. In this case
it will be pi/2 plus or minus n*pi because this is the parent function and my period
is pi units. Your asymptotes are going to be separated by whatever the period is for
your tangent or cotangent graph. Ok, I have two minutes left. Let me set up the basic
cotangent function. I will get with you in just a second. Let's see here. What if this
were the cotangent of x? Well the cotangent of zero as we saw on the unit circle is undefined,
then it goes to one, then zero, then it becomes negative one. Then at pi, the cotangent of
pi is undefined. Well, let's see what that would look like. At zero cotangent is going
to have an asymptote because it is undefined. At pi/4 it is at one. If you take one and
you flip it you still get one. At pi/2 it is going to be zero. At 3pi/4 it is negative
one. At pi it is undefined. So, this is your first period for cotangent. Sort of like your
parent function, your parent graph for y equals the cotangent of x. I am out of time. The
next video we will be actually doing some examples that have some transformations in
them. I hope I have helped you out. Go do your homework.