Pre-Calculus: Graphing tan, sec, csc, cot

Trig functions can be horrendous!

06/02/2009

~ brittanie

This video lesson is a great introduction to how to graph advanced trig functions like cosecant, secant, tangent, and cotangent. The expertise of Professor Burger and his easy-to-follow instructions are well appreciated.

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The Trigonometric Functions
Graphing Other Trig Functions
Graphing the Tangent, Secant, Cosecant, and Cotangent Functions Page [1 of 3]
Now we have a really good sense of what the sine function looks like and what the cosine function looks like and I
want to now use that information to figure out what do the graphs of the other trig functions look like? In particular, the
tangent and the secant, cosecant, and cotangent, what do they look like?
Well, let’s take a look and start with tangent. I want to actually do this by first just reminding you what the sine and
cosine function look like. So, if this is 2? , this would be ? , this is 1/2? , this is 3/2 ? . And let me graph actually
each of these things separately. First let me graph the sine function. This, by the way, is 1 and this is -1. So what
does sine look like? So, sine starts up, starts down here, goes up, comes down like that. So, I’m just going to do one
period. Very pretty, soothing kind of curve. That’s sine.
Now, let’s graph cosine. So, cosine starts at 1 and then goes down, comes down to here. So, it’s just the thing
shifted by
2
?
radians. That’s cosine. So now I put them together so you can sort of see them in action. You can see
that one’s just a shift of the other. Now, I want to figure out tangent and the reason why I put both of these together is
because I remind you of the identity, which says that tangent is actually the same thing as sine over cosine. So, all I
have to do is take the values here, and create a fraction where in the numerator, I put the red function value, and in
the denominator, I put the purple function value. And that should give me the tangent. So, let’s actually put those
things together and see what happens. So, I’m going to try to keep these exactly lined up. So this is going to be 2? ,
this is going to be ? . This is going to be
2
?
and this is going to be 3
2
?
. Okay, well, let’s see what happens. I now
want to plot what the tangent function looks like. Remember the rules. Tangent is just sine divided by cosine. So
let’s just plot some easy points and see what we get. For example, at zero, what is tangent? Well, I take sine, which
remember, sine is the red one here, so I take that value which is zero and divide it by 1. So that’s zero. So, in fact we
know tangent is going to be zero right here. What about this point, which by the way, is
4
?
, 45°. That’s where sine
and cosine are the same. If I have the same value here as here, what I see is the value of 1, something over itself.
And, in fact, what is that value? It’s the 2 ÷ 2 , on the top, and then on the bottom I have 2 ÷ 2 , so I see
1. So, right here, I see a value of 1. So I put that in.
What about right here at
2
?
? Well, at
2
?
, I take the sine, which is 1 and divide it by the cosine, which is 0. Now wait
a minute. One over zero is undefined. So what’s going on here? This is not defined here. Well, it turns out this is an
asymptote. It turns out that the tangent, of all things, actually has an asymptote, right here, at x =
2
?
. So this curve is
sort of going to head up toward that asymptote. It approaches it but never touches it. What happens here at ? ?
Well, at ? , the sine, you notice, is zero, and the cosine is -1, so
0
?1
is zero. And what happens over here in these
values? Where in these values, look where I am. I’m in this region right here, so I’m taking a sine, which is a positive
value, and dividing it by a negative cosine, so that quotient will be negative. So, in fact these are negative values
here. And it’s asymptotic, so it looks like this.
And then what happens here at three halves? Well, again, notice the cosine is zero. So I have a zero denominator.
This is an asymptote. So the tangent function is really sort of peculiar compared to the sine and cosine, because it
has asymptote at the odd multiples of
2
?
. And what happens? Then what happens here? Well, I’m just going to
repeat this same process because, notice that at 2? , I had 0 ÷ 1, so it’s zero, and in this region, you’ll notice that I’m
negative because I’ve got a negative top sign, but positive cosine, so that quotient is negative. So, in fact, what I see
is this. And this continues. And it continues this way, too. And this is actually a graph of y = tangent x. So the
The Trigonometric Functions
Graphing Other Trig Functions
Graphing the Tangent, Secant, Cosecant, and Cotangent Functions Page [2 of 3]
tangent function, you notice, actually has asymptotes at the odd multiples of
2
?
here, and 3
2
?
, and then again at 5
2
?
and so forth. And you also may notice that the period is not 2? , like it is here. But the period here is just ? . Right?
In ? , I see one period, and I just repeat that. So, in fact, the period of tangent is just ? and you can see it has
asymptotes along these points here, where you’re at odd multiples of
2
?
. And you can see where the zeros are, at
just multiples of ? , right? Zero? , 1? , 2? , and so forth. So that’s the zeros of those. And you can see that’s
exactly what the tangent function looks like. So, there’s the tangent function.
Now, what about like the secant, and so forth?
Well, to see those, I have to take a reciprocal. So, let me actually first begin by sketching in -- so I’ll go ahead and do
secant. If I want to graph secant, what I’d better do is first draw in, what? Sine or cosine? Secant, I want cosine. So
here’s 2? , here’s ? , here’s 1/2? . So, let me first just sketch in lightly the cosine function. This is 1, 1, still 1. That’s
cosine. But I want to graph is the secant. So, y = secant x. And remember that y = secant x is the same thing as y =
1
cos
. So all I have to do is sort of think about taking each value and looking at its reciprocal. So, actually that’s not
so bad. Because at this point here, that height is 1. So,
1
1
= 1. So, in fact, that point is going to be right here on the
secant.
What about here? Well, there, I’m still at one. The reciprocal of 1 is 1, so that point actually we’ll share in common.
What about this point here? Well, that’s -1 and if you take the reciprocal of -1, we have 1. Now, what happens here?
Well, these are positive values I’m taking the reciprocals of and notice they’re getting smaller and smaller and smaller
and smaller. I’m taking smaller and smaller values, but I’m taking the reciprocal. What happens when you put down 1
over a small number? It starts to actually become pretty large. Like 1 over 1,000, is actually 1,000. So, in fact, while
this is going down, the reciprocal function must be going up. And, in fact, what is it here? Well, here, cosine is zero.
So, I have
1
0
, an asymptote. So, I see actually an asymptote right here, just like with the tangent function, and
similarly, I have one here. And what happens? Well, this function is going to get bigger and bigger and bigger as
these get smaller and smaller and smaller, heading up toward that asymptote. And the same thing happens here
except now they’re negatively valued so, in fact I really should maybe continue these a little bit more. So what you
see is, as you go this way, when you take the reciprocal, you’ll be sort of heading down to negative infinity. Because
those values in reciprocal will get larger, and larger in magnitude with the negative sign. And I have these
asymptotes. So, in fact, this is the graph of a y = secant x. And you’ll notice that the secant function actually has
asymptotes, too. Wherever cosine is zero, we have asymptotes, with them flipping. And you can see it’s sort of
that’s exactly what’s going on here. This is the inverse function. Not a big deal. But exotic. Notice also the period,
by the way, is also what? Well, to get a complete period, here I do need a full thing, so I need, in fact, 2? .
Now what about cosecant? Well then I have to look at the sine and I bet you can figure it out now. So I’ll just do this
one really fast for you. Here’s 2? , here’s ? ,
2
?
, 3
2
?
, here’s 1. First I graph the sine. What I want to show you now
is the graph of y = cosecant of x, which I remind you is 1 over the sine, so I just have to take the reciprocal of each of
these values. So notice wherever it’s zero, we’re going to have an asymptote. So, I’m going to have an asymptote
here, on the y axis, an asymptote here at x = ? , an asymptote here at x = 2? . In fact, I’ll have asymptotes at all the
multiples of ? . And then what does it look like? Well, at this point, I have a height of 1, the reciprocal of 1 is just 1,
so it touches here. But then it has that same exact shape as the secant, but it’s just shifted over, which is not
surprising. Because, in fact, it is just like the sine and cosine relationship. So, I have this wing up here and this wing
up here. And it keeps going of course, then I have more wings up here and so on and so forth. So that is the graph of
the cosecant. I’m not making a big deal about it because it really just follows by looking at this graph and taking each
The Trigonometric Functions
Graphing Other Trig Functions
Graphing the Tangent, Secant, Cosecant, and Cotangent Functions Page [3 of 3]
value and taking a reciprocal, which means one over. So, as you get close to zero, that value is going to get further
and further away from here and it might be negative or positive, depending upon what ends of the axis you’re on.
Anyway, so, now what about cotangent? Let me just do cotangent really, really fast. Cotangent -- and then you’ll
have them all. So, first I’ll graph tangent, just like before. So, here’s 2? , ? , and remember there are asymptotes
here at
2
?
, and here at 3
2
?
. And what does the tangent function do? Well, we just saw that a second ago, it sort of
has this basic flavor. So what’s cotangent going to be? Well, cotangent will have a slightly different look to it
because, remember it equals 1 over tangent. So where are its asymptotes? Its asymptotes will be wherever the
tangent is zero. And what happens at the asymptote here? Well, in some sense this asymptote is because you have
something over zero. So, when you flip something over zero, you have zero over something. Actually, the cotangent
is not only defined but is zero there. It’s sort of like taking something over zero and flipping it, you get zero over
something. So, in fact, what we see is these are going to be the zeros, where this used to be the asymptote, and
where these things have asymptotes, we’re now going to -- where these things have zeros, we’re now going to have
asymptotes. So the asymptote and zero roles are reversed. And what does this look like? Well, if you think about it,
as you get closer and closer to zero here, when you flip them, they’ll just get closer and closer to infinity and as these
get bigger and bigger, when you flip, they’re going to get smaller and smaller. They’re reciprocals, so you get a
picture that looks very much like this. And then here, it looks similarly so. Again, this has period, just ? , and you can
see it has that sort of same flavor but everything is flipped. So, in fact, as you get closer and closer to zero here, the
inverse is getting closer and closer to infinity. As these get closer and closer to infinity, the inverse will get closer and
closer to zero. So, in fact, this is the graph of cotangent.
So, just thinking about what it means to flip things over, you can actually graph all those other functions pretty easily.
And remember that tangent is sine over cosine, you get the tangent function for free.
We’ll take a look at a lot more of these graphing things in detail, up next.