(text reference: Sections 6.2 and 6.4
custom text pgs. 56 - 76)
V. Olds 2012
Unit 3 33
3 Trigonometric Functions
In this chapter, our main focus is on doing Calculus with Trigonometric Functions, which is in text
section 6.4. The Calculus itself is quite straightforward. There will be a small number of new speciﬁc
derivatives to remember – two would suﬃce, but it will probably be easier to memorize a few others,
as well. But there aren’t any new “diﬀerentiation rules”, just the same old rules applied to new
kinds of functions. So there’s really not much that’s new here in the way of Calculus. However, it
is certainly worthwhile doing some review of what Trigonometric Functions are and how they work.
As well, the units in which we measure angles in Calculus may be new to you. All the review you
really need us covered in section 6.2 (and these notes), but you may also ﬁnd it useful to look at text
section 6.1, and you should certainly do the assigned homework exercises from section 6.2, which
reviews the things you need to know about trig functions, before working on the Trig Derivatives
covered in section 6.4.
Perhaps the most important thing for you to get out of the review we do is something that sounds
so basic it shouldn’t be a problem. Trig functions are functions. Well, of course they are! Obviously
they are! It says so right there in the name “trig functions”. However, forgetting that, or not un-
derstanding it in the ﬁrst place, seems to be the biggest problem that students have in this section
of the material. Because with functions come composite functions. That is, if we apply a function
to anything more complicated than x, or t, or whatever the variable may happen to be called, then
what we have is a composite function. And when you do Calculus with a composite function, then
you need the Chain Rule. With trig functions (just like with exponential and logarithmic functions),
there’s always lots of Chain Rule needed. Keeping in mind that these trig functions are functions!
will help you to recognize when those rules are needed.
Perhaps a good place to start is reviewing what we mean by a function, to help you see that
these “new” functions we’re going to be working with are functions. Because in previous courses
when you’ve done trigonometry, you weren’t really thinking of these functions as functions. So let’s
start by recalling what a function is.
Deﬁnition 3.1. Afunction fis a rule that assigns to each element of the domain exactly one
value from the range.
That is, for each value xin the domain of f,fassociates the unique value f(x) with the value
x. And of course the variable might not be called x. It could be tor yor θor λor ... whatever. For
no particular reason, we tend to use tas the variable a lot of the time with trig functions.
Now, some trig functions. You probably recall that there are a number of them. (Six, in fact.)
But there are two main trig functions, and then all the others are deﬁned in terms of those two. So
we start with them.
Deﬁnition 3.2. For any angle t,thas a unique sine value, denoted sin t, and also a unique cosine
value, denoted cos t.
That is, there is a sine function, denoted f(t) = sin t, whose domain is the set of all angles,
and which associates a particular value with the angle t. (Aha! Maybe that’s the problem. Maybe
the reason that some students have diﬃculty realizing that this is a function is because we don’t
necessarily use the brackets. We normally write functions as f(t) or g(t), but then we write the sine
function as sin t, not sin(t). If that’s going to cause you a problem, just imagine that sin is always
followed by invisible brackets.) Getting back to what we were saying, this sine function is a rule
that associates with every number (angle) tin its domain a unique value, which we call sin t. So
that’s a function. Similarly, there is another function, the cosine function, denoted g(t) = cos t,
34 Unit 3
which has the same domain and associates with each number in its domain, i.e. with each angle, a
unique value or number, called cos t. So this is another number associated with the same angle.
Notice that just saying “sin” or “cos” by itself has no mathematical meaning, except as the name
of a function. It’s like just saying “f”. We can talk about the function f, but we can’t do math
with it. We do math with function values, either unspeciﬁed values, like f(x) or f(t), or speciﬁc
function values, like f(1) or f(3). Likewise, we can talk about the function sin or the function cos
(i.e. the sine function or the cosine function), but we can’t do any math with them. We can only
do math with function values, like sin tand cos t. This is just like with logarithmic functions. Just
saying “log” or “ln” doesn’t mean anything by itself, except to name a function. It’s only when we
put a number with it (whether an unspeciﬁed number, i.e. a variable, or a speciﬁc number) that we
have something meaningful that we can do math with. Like logbxor ln 5. So the function name sin
or cos must always be followed by an angle, whether a speciﬁc angle, or an unspeciﬁed angle such
as t. Or maybe t2or 2π−3t.
In doing trigonometry in High School, you probably mostly, or maybe even exclusively, measured
angles in degrees. Everybody knows that there are 360 degrees in a circle, right? Sure! And a right
angle is 90◦. However, as alluded to earlier, in Calculus we generally use a diﬀerent unit of mea-
surement for angles. (Why? Who knows! That’s just the way it is.) Instead of measuring angles in
degrees, we measure them in radians. What’s a radian? Well here’s a deﬁnition.
Deﬁnition 3.3. Consider a segment of a circle. Let rbe the radius of the circle and sbe the arc
length of the segment. Then the angle between the radii producing the segment is tradians, where
For instance, for a full circle, the arc length (i.e. the perimeter of the circle) is s= 2πr. So we
see that there are t=s
r= 2πradians in a full circle.
Similarly, for a right angle, what we have is a circle segment which is one quarter of a full circle.
So the arc length is s=1
4(2πr) = π
2rand we see that a right angle is s
Notice: You don’t need to remember, or even understand, anything about arc length. The only
thing you need to remember here is:
There are 2πradians in a circle, so 2πradians = 360◦.
And that means that when we measure angles in radians, there is almost always a πin the
measurement. That is, an angle measured in radians is generally some multiple of π. So we have
angles like 2π,π(that’s a semi-circle, i.e. the angle is a straight line), π
2(we already saw that that’s
a right angle), or maybe 5π
6. Which means that we express speciﬁc trig function values as things
like sin π
6and cos 3π
As you know, we don’t use calculators in this course. In your previous study of trigonometry,
you may have relied on your calculator to tell you what the sine or cosine value of a particular
angle was. Can’t do that here. So there are some trig function values that you’ll have to know, i.e.
memorize. Just for a few basic angles. The angles whose trig function values you need to know
are the multiples of π
4. (Note: π
6= 30◦and π
4= 45◦. These are the same angles you most
likely encountered a lot when you did trig before.)