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Trigonometric Functions
So far we have used only algebraic functions as examples when finding derivatives, that is,
functions that can be built up by the usual algebraic operations of addition, subtraction,
multiplication, division, and raising to constant powers. Both in theory and practice there
are other functions, called transcendental, that are very useful. Most important among
these are the trigonometric functions, the inverse trigonometric functions, exponential
functions, and logarithms. In this chapter we investigate the trigonometric functions.
4.1 Trigonometri Fun tions
When you first encountered the trigonometric functions it was probably in the context of
“triangle trigonometry,” defining, for example, the sine of an angle as the “side opposite
over the hypotenuse.” While this will still be useful in an informal way, we need to use a
more expansive definition of the trigonometric functions. First an important note: while
degree measure of angles is sometimes convenient because it is so familiar, it turns out to
be ill-suited to mathematical calculation, so (almost) everything we do will be in terms of
radian measure of angles.
74 Chapter 4 Trigonometric Functions
To define the radian measurement system, we consider the unit circle in the xy-plane:
x
(cos x, sin x)
y
A
B
An angle, x, at the center of the circle is associated with an arc of the circle which is said
to subtend the angle. In the figure, this arc is the portion of the circle from point (1, 0)
to point A. The length of this arc is the radian measure of the angle x; the fact that the
radian measure is an actual geometric length is largely responsible for the usefulness of
radian measure. The circumference of the unit circle is 2πr = 2π(1) = 2π, so the radian
measure of the full circular angle (that is, of the 360 degree angle) is 2π.
While an angle with a particular measure can appear anywhere around the circle, we
need a fixed, conventional location so that we can use the coordinate system to define
properties of the angle. The standard convention is to place the starting radius for the
angle on the positive x-axis, and to measure positive angles counterclockwise around the
circle. In the figure, x is the standard location of the angle π/6, that is, the length of the
arc from (1, 0) to A is π/6. The angle y in the picture is −π/6, because the distance from
(1, 0) to B along the circle is also π/6, but in a clockwise direction.
Now the fundamental trigonometric definitions are: the cosine of x and the sine of x
are the first and second coordinates of the point A, as indicated in the figure. The angle x
shown can be viewed as an angle of a right triangle, meaning the usual triangle definitions
of the sine and cosine also make sense. Since the hypotenuse of the triangle is 1, the “side
opposite over hypotenuse” definition of the sine is the second coordinate of point A over
1, which is just the second coordinate; in other words, both methods give the same value
for the sine.
The simple triangle definitions work only for angles that can “fit” in a right triangle,
namely, angles between 0 and π/2. The coordinate definitions, on the other hand, apply
76 Chapter 4 Trigonometric Functions
Similarly, as angle x increases from 0 in the unit circle diagram, the first coordinate of
the point A goes from 1 to 0 then to −1, back to 0 and back to 1, so the graph of y = cos x
must look something like this:
Exercises 4.1.
Some useful trigonometric identities are in appendix B.
- Find all values of θ such that sin(θ) = −1; give your answer in radians. ⇒
- Find all values of θ such that cos(2θ) = 1/2; give your answer in radians. ⇒
- Use an angle sum identity to compute cos(π/12). ⇒
- Use an angle sum identity to compute tan(5π/12). ⇒
- Verify the identity cos
(t)/(1 − sin(t)) = 1 + sin(t).
- Verify the identity 2 csc(2θ) = sec(θ) csc(θ).
- Verify the identity sin(3θ) − sin(θ) = 2 cos(2θ) sin(θ).
- Sketch y = 2 sin(x).
- Sketch y = sin(3x).
- Sketch y = sin(−x).
- Find all of the solutions of 2 sin(t) − 1 − sin
(t) = 0 in the interval [0, 2 π]. ⇒
4.2 The Derivative of sin x
What about the derivative of the sine function? The rules for derivatives that we have are
no help, since sin x is not an algebraic function. We need to return to the definition of the
derivative, set up a limit, and try to compute it. Here’s the definition:
d
dx
sin x = lim
∆x→ 0
sin(x + ∆x) − sin x
∆x
Using some trigonometric identities, we can make a little progress on the quotient:
sin(x + ∆x) − sin x
∆x
sin x cos ∆x + sin ∆x cos x − sin x
∆x
= sin x
cos ∆x − 1
∆x
+ cos x
sin ∆x
∆x
4.3 A hard limit 77
This isolates the difficult bits in the two limits
lim
∆x→ 0
cos ∆x − 1
∆x
and lim
∆x→ 0
sin ∆x
∆x
Here we get a little lucky: it turns out that once we know the second limit the first is quite
easy. The second is quite tricky, however. Indeed, it is the hardest limit we will actually
compute, and we devote a section to it.
4.3 A hard limit
We want to compute this limit:
lim
∆x→ 0
sin ∆x
∆x
Equivalently, to make the notation a bit simpler, we can compute
lim
x→ 0
sin x
x
In the original context we need to keep x and ∆x separate, but here it doesn’t hurt to
rename ∆x to something more convenient.
To do this we need to be quite clever, and to employ some indirect reasoning. The
indirect reasoning is embodied in a theorem, frequently called the squeeze theorem.
THEOREM 4.3.1 Squeeze Theorem Suppose that g(x) ≤ f (x) ≤ h(x) for all x
close to a but not equal to a. If lim
x→a
g(x) = L = lim
x→a
h(x), then lim
x→a
f (x) = L.
This theorem can be proved using the official definition of limit. We won’t prove it
here, but point out that it is easy to understand and believe graphically. The condition
says that f (x) is trapped between g(x) below and h(x) above, and that at x = a, both g
and h approach the same value. This means the situation looks something like figure 4.3.1.
The wiggly curve is x
sin(π/x), the upper and lower curves are x
and −x
. Since the
sine function is always between −1 and 1, −x
≤ x
sin(π/x) ≤ x
, and it is easy to see
that lim
x→ 0
−x
= 0 = lim
x→ 0
x
. It is not so easy to see directly, that is algebraically,
that lim
x→ 0
x
sin(π/x) = 0, because the π/x prevents us from simply plugging in x = 0.
The squeeze theorem makes this “hard limit” as easy as the trivial limits involving x
To do the hard limit that we want, lim
x→ 0
(sin x)/x, we will find two simpler functions
g and h so that g(x) ≤ (sin x)/x ≤ h(x), and so that lim
x→ 0
g(x) = lim
x→ 0
h(x). Not too
surprisingly, this will require some trigonometry and geometry. Referring to figure 4.3.2,
x is the measure of the angle in radians. Since the circle has radius 1, the coordinates of
4.3 A hard limit 79
Finally, the two limits lim
x→ 0
cos x and lim
x→ 0
1 / cos x are easy, because cos(0) = 1. By
the squeeze theorem, lim
x→ 0
(sin x)/x = 1 as well.
Before we can complete the calculation of the derivative of the sine, we need one other
limit:
lim
x→ 0
cos x − 1
x
This limit is just as hard as sin x/x, but closely related to it, so that we don’t have to do
a similar calculation; instead we can do a bit of tricky algebra.
cos x − 1
x
cos x − 1
x
cos x + 1
cos x + 1
cos
x − 1
x(cos x + 1)
− sin
x
x(cos x + 1)
sin x
x
sin x
cos x + 1
To compute the desired limit it is sufficient to compute the limits of the two final fractions,
as x goes to 0. The first of these is the hard limit we’ve just done, namely 1. The second
turns out to be simple, because the denominator presents no problem:
lim
x→ 0
sin x
cos x + 1
sin 0
cos 0 + 1
Thus,
lim
x→ 0
cos x − 1
x
Exercises 4.3.
- Compute lim
x→ 0
sin(5x)
x
⇒ 2. Compute lim
x→ 0
sin(7x)
sin(2x)
⇒
- Compute lim
x→ 0
cot(4x)
csc(3x)
⇒ 4. Compute lim
x→ 0
tan x
x
⇒
- Compute lim
x→π/ 4
sin x − cos x
cos(2x)
⇒
- For all x ≥ 0, 4x − 9 ≤ f (x) ≤ x
− 4 x + 7. Find lim
x→ 4
f (x). ⇒
- For all x, 2x ≤ g(x) ≤ x
− x
x→ 1
g(x). ⇒
- Use the Squeeze Theorem to show that lim
x→ 0
x
cos(2/x) = 0.
80 Chapter 4 Trigonometric Functions
4.4 The Derivative of sin x , ontinued
Now we can complete the calculation of the derivative of the sine:
d
dx
sin x = lim
∆x→ 0
sin(x + ∆x) − sin x
∆x
= lim
∆x→ 0
sin x
cos ∆x − 1
∆x
+ cos x
sin ∆x
∆x
= sin x · 0 + cos x · 1 = cos x.
The derivative of a function measures the slope or steepness of the function; if we
examine the graphs of the sine and cosine side by side, it should be that the latter appears
to accurately describe the slope of the former, and indeed this is true:
sin x
cos x
Notice that where the cosine is zero the sine does appear to have a horizontal tangent
line, and that the sine appears to be steepest where the cosine takes on its extreme values
of 1 and −1.
Of course, now that we know the derivative of the sine, we can compute derivatives of
more complicated functions involving the sine.
EXAMPLE 4.4.1 Compute the derivative of sin(x
d
dx
sin(x
) = cos(x
) · 2 x = 2x cos(x
EXAMPLE 4.4.2 Compute the derivative of sin
(x
− 5 x).
d
dx
sin
(x
− 5 x) =
d
dx
(sin(x
− 5 x))
= 2(sin(x
− 5 x))
cos(x
− 5 x)(3x
= 2(3x
− 5) cos(x
− 5 x) sin(x
− 5 x).
82 Chapter 4 Trigonometric Functions
- Find all points on the graph of f (x) = 2 sin(x)−sin
(x) at which the tangent line is horizontal.
⇒
- Find an equation for the tangent line to sin
(x) at x = π/3. ⇒
- Find an equation for the tangent line to sec
x at x = π/3. ⇒
- Find an equation for the tangent line to cos
x − sin
(4x) at x = π/6. ⇒
- Find the points on the curve y = x + 2 cos x that have a horizontal tangent line. ⇒
- Let C be a circle of radius r. Let A be an arc on C subtending a central angle θ. Let B be
the chord of C whose endpoints are the endpoints of A. (Hence, B also subtends θ.) Let
s be the length of A and let d be the length of B. Sketch a diagram of the situation and
compute lim
s/d.
4.6 Impli it Differentiation
We have not yet verified the power rule,
d
dx
x
a
= ax
a− 1
, for non-integer a. There is a
close relationship between x
and x
—these functions are inverses of each other, each
“undoing” what the other has done. Not surprisingly, this means there is a relationship
between their derivatives.
Let’s rewrite y = x
as y
= x. We say that this equation defines the function
y = x
implicitly because while it is not an explicit expression y =.. ., it is true that if
x = y
then y is in fact the square root function. Now, for the time being, pretend that
all we know of y is that x = y
; what can we say about derivatives? We can take the
derivative of both sides of the equation:
d
dx
x =
d
dx
y
Then using the chain rule on the right hand side:
1 = 2y
d
dx
y
= 2yy
Then we can solve for y
y
2 y
2 x
x
This is the power rule for x
There is one little difficulty here. To use the chain rule to compute d/dx(y
) = 2yy
we need to know that the function y has a derivative. All we have shown is that if it has a
derivative then that derivative must be x
/2. When using this method we will always
have to assume that the desired derivative exists, but fortunately this is a safe assumption
for most such problems.
4.6 Implicit Differentiation 83
Here’s another interesting feature of this calculation. The equation x = y
defines
more than one function implicitly: y = −
x also makes the equation true. Following
exactly the calculation above we arrive at
y
2 y
2(−x
x
So the single calculation leading to y
= 1/(2y) simultaneously computes the derivatives
of both functions.
We can use the same technique to verify the product rule for any rational power.
Suppose y = x
m/n
. Write instead x
m
= y
n
and take the derivative of each side to get
mx
m− 1
= ny
n− 1
y
. Then
y
mx
m− 1
ny
n− 1
mx
m− 1
n(x
m/n
n− 1
m
n
x
m− 1
x
−m(n−1)/n
m
n
x
m/n− 1
This example involves an inverse function defined implicitly, but other functions can
be defined implicitly, and sometimes a single equation can be used to implicitly define
more than one function. Here’s a familiar example. The equation r
= x
+ y
describes
a circle of radius r. The circle is not a function y = f (x) because for some values of x
there are two corresponding values of y. If we want to work with a function, we can break
the circle into two pieces, the upper and lower semicircles, each of which is a function.
Let’s call these y = U (x) and y = L(x); in fact this is a fairly simple example, and it’s
possible to give explicit expressions for these: U (x) =
r
− x
and L(x) = −
r
− x
But it’s somewhat easier, and quite useful, to view both functions as given implicitly by
r
= x
+ y
: both r
= x
+ U (x)
and r
= x
+ L(x)
are true, and we can think of
r
= x
+ y
as defining both U (x) and L(x).
Now we can take the derivative of both sides as before, remembering that y is not
simply a variable but a function—in this case, y is either U (x) or L(x) but we’re not yet
specifying which one. When we take the derivative we just have to remember to apply the
chain rule where y appears.
d
dx
r
d
dx
(x
+ y
0 = 2x + 2yy
y
− 2 x
2 y
x
y
Now we have an expression for y
, but it contains y as well as x. This means that if we
want to compute y
for some particular value of x we’ll have to know or compute y at that
value of x as well. It is at this point that we will need to know whether y is U (x) or L(x).
4.6 Implicit Differentiation 85
You might think that the step in which we solve for y
could sometimes be difficult—
after all, we’re using implicit differentiation here because we can’t solve the equation
yx
+ e
y
= x for y, so maybe after taking the derivative we get something that is hard to
solve for y
. In fact, this never happens. All occurrences y
come from applying the chain
rule, and whenever the chain rule is used it deposits a single y
multiplied by some other
expression. So it will always be possible to group the terms containing y
together and
factor out the y
, just as in the previous example. If you ever get anything more difficult
you have made a mistake and should fix it before trying to continue.
It is sometimes the case that a situation leads naturally to an equation that defines a
function implicitly.
EXAMPLE 4.6.3 Consider all the points (x, y) that have the property that the distance
from (x, y) to (x
, y
) plus the distance from (x, y) to (x
, y
) is 2a (a is some constant).
These points form an ellipse, which like a circle is not a function but can viewed as two
functions pasted together. Because we know how to write down the distance between two
points, we can write down an implicit equation for the ellipse:
(x − x
+ (y − y
(x − x
+ (y − y
= 2a.
Then we can use implicit differentiation to find the slope of the ellipse at any point, though
the computation is rather messy.
Exercises 4.6.
In exercises 1–8, find a formula for the derivative y
at the point (x, y):
- y
= 1 + x
⇒
- x
= 7 ⇒
- x
= y
⇒
4 cos x sin y = 1 ⇒
√
x +
√
y = 9 ⇒
tan(x/y) = x + y ⇒
sin(x + y) = xy ⇒
1
x
1
y
= 7 ⇒
- A hyperbola passing through (8, 6) consists of all points whose distance from the origin is a
constant more than its distance from the point (5,2). Find the slope of the tangent line to
the hyperbola at (8, 6). ⇒
- Compute y
for the ellipse of example 4.6.3.
- If y = log
a
x then a
y
= x. Use implicit differentiation to find y
.
- The graph of the equation x
− xy + y
= 9 is an ellipse. Find the lines tangent to this curve
at the two points where it intersects the x-axis. Show that these lines are parallel. ⇒
86 Chapter 4 Trigonometric Functions
- Repeat the previous problem for the points at which the ellipse intersects the y-axis. ⇒
- Find the points on the ellipse from the previous two problems where the slope is horizontal
and where it is vertical. ⇒
- Find an equation for the tangent line to x
= y
+x
at (2,
√
12). (This curve is the kampyle
of Eudoxus.) ⇒
- Find an equation for the tangent line to x
= a
at a point (x
, y
) on the curve,
with x
6 = 0 and y
6 = 0. (This curve is an astroid.) ⇒
- Find an equation for the tangent line to (x
)
= x
− y
at a point (x
, y
) on the curve,
when y
6 = 0. (This curve is a lemniscate.) ⇒
Definition. Two curves are orthogonal if at each point of intersection, the angle between
their tangent lines is π/2. Two families of curves, A and B, are orthogonal trajectories of each
other if given any curve C in A and any curve D in B the curves C and D are orthogonal. For
example, the family of horizontal lines in the plane is orthogonal to the family of vertical lines in
the plane.
- Show that x
− y
= 5 is orthogonal to 4x
= 72. (Hint: You need to find the
intersection points of the two curves and then show that the product of the derivatives at
each intersection point is −1.)
- Show that x
= r
is orthogonal to y = mx. Conclude that the family of circles centered
at the origin is an orthogonal trajectory of the family of lines that pass through the origin.
Note that there is a technical issue when m = 0. The circles fail to be differentiable
when they cross the x-axis. However, the circles are orthogonal to the x-axis. Explain why.
Likewise, the vertical line through the origin requires a separate argument.
- For k 6 = 0 and c 6 = 0 show that y
− x
= k is orthogonal to yx = c. In the case where k and
c are both zero, the curves intersect at the origin. Are the curves y
− x
= 0 and yx = 0
orthogonal to each other?
- Suppose that m 6 = 0. Show that the family of curves {y = mx + b | b ∈ R} is orthogonal to
the family of curves {y = −(x/m) + c | c ∈ R}.
4.7 Limits revisited
We have defined and used the concept of limit, primarily in our development of the deriva-
tive. Recall that lim
x→a
f (x) = L is true if, in a precise sense, f (x) gets closer and closer to
L as x gets closer and closer to a. While some limits are easy to see, others take some
ingenuity; in particular, the limits that define derivatives are always difficult on their face,
since in
lim
∆x→ 0
f (x + ∆x) − f (x)
∆x
both the numerator and denominator approach zero. Typically this difficulty can be re-
solved when f is a “nice” function and we are trying to compute a derivative. Occasionally
such limits are interesting for other reasons, and the limit of a fraction in which both nu-
merator and denominator approach zero can be difficult to analyze. Now that we have
88 Chapter 4 Trigonometric Functions
As x approaches π, both the numerator and denominator approach zero, so it is not
obvious what, if anything, the quotient approaches. We can often compute such limits by
application of the following theorem.
THEOREM 4.7.5 L’Hˆopital’s Rule For “sufficiently nice” functions f (x) and
g(x), if lim
x→a
f (x) = 0 = lim
x→a
g(x) or both lim
x→a
f (x) = ±∞ and lim
x→a
g(x) = ±∞, and if
lim
x→a
f
(x)
g
(x)
exists, then lim
x→a
f (x)
g(x)
= lim
x→a
f
(x)
g
(x)
. This remains true if “x → a” is replaced by
“x → ∞” or “x → −∞”.
This theorem is somewhat difficult to prove, in part because it incorporates so many
different possibilities, so we will not prove it here. We also will not need to worry about
the precise definition of “sufficiently nice”, as the functions we encounter will be suitable.
EXAMPLE 4.7.6 Compute lim
x→π
x
sin x
in two ways.
First we use L’Hˆopital’s Rule: Since the numerator and denominator both approach
zero,
lim
x→π
x
sin x
= lim
x→π
2 x
cos x
provided the latter exists. But in fact this is an easy limit, since the denominator now
approaches −1, so
lim
x→π
x
sin x
We don’t really need L’Hˆopital’s Rule to do this limit. Rewrite it as
lim
x→π
(x + π)
x − π
sin x
and note that
lim
x→π
x − π
sin x
= lim
x→π
x − π
− sin(x − π)
= lim
x→ 0
x
sin x
since x − π approaches zero as x approaches π. Now
lim
x→π
(x + π)
x − π
sin x
= lim
x→π
(x + π) lim
x→ 0
x
sin x
as before.
EXAMPLE 4.7.7 Compute lim
x→∞
2 x
− 3 x + 7
x
+ 47x + 1
in two ways.
4.7 Limits revisited 89
As x goes to infinity both the numerator and denominator go to infinity, so we may
apply L’Hˆopital’s Rule:
lim
x→∞
2 x
− 3 x + 7
x
+ 47x + 1
= lim
x→∞
4 x − 3
2 x + 47
In the second quotient, it is still the case that the numerator and denominator both go to
infinity, so we are allowed to use L’Hˆopital’s Rule again:
lim
x→∞
4 x − 3
2 x + 47
= lim
x→∞
So the original limit is 2 as well.
Again, we don’t really need L’Hˆopital’s Rule, and in fact a more elementary approach
is easier—we divide the numerator and denominator by x
lim
x→∞
2 x
− 3 x + 7
x
+ 47x + 1
= lim
x→∞
2 x
− 3 x + 7
x
+ 47x + 1
x
x
= lim
x→∞
x
x
x
x
Now as x approaches infinity, all the quotients with some power of x in the denominator
approach zero, leaving 2 in the numerator and 1 in the denominator, so the limit again is
EXAMPLE 4.7.8 Compute lim
x→ 0
sec x − 1
sin x
Both the numerator and denominator approach zero, so applying L’Hˆopital’s Rule:
lim
x→ 0
sec x − 1
sin x
= lim
x→ 0
sec x tan x
cos x
Exercises 4.7.
Compute the limits.
- lim
x→∞
√
x
√
x
− x ⇒ 2. lim
x→ 0
√
9 + x − 3
x
⇒
- lim
t→ 1
(1/t) − 1
t
− 2 t + 1
⇒ 4. lim
x→ 2
2 −
√
x + 2
4 − x
⇒
- lim
t→∞
t + 5 − 2 /t − 1 /t
3 t + 12 − 1 /t
⇒ 6. lim
y→∞
√
y + 1 +
√
y − 1
y
⇒
- lim
x→ 1
√
x − 1
√
x − 1
⇒ 8. lim
x→ 0
(1 − x)
− 1
x
⇒
- lim
t→ 0
(
t +
1
t
)
((4 − t)
− 8) ⇒ 10. lim
t→ 0
(
1
t
1
√
t
)
(
√
t + 1 − 1) ⇒
