Transcendental functions such as exponential, trigonometric and hyperbolic functions (Part - 1) | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


4
Transcendental Functions
So far we have used only algebraic functions as examples when ?nding 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.
4.1 Trigonometri
 Fun
tions When you ?rst encountered the trigonometric functions it was probably in the context of
“triangle trigonometry,” de?ning, 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 de?nition 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.
Page 2


4
Transcendental Functions
So far we have used only algebraic functions as examples when ?nding 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.
4.1 Trigonometri
 Fun
tions When you ?rst encountered the trigonometric functions it was probably in the context of
“triangle trigonometry,” de?ning, 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 de?nition 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.
To de?ne the radian measurement system, we consider the unit circle in the xy-plane:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . .. .. . . . .. ... .. ... ... .... ..... .................................................................................................................................................................................................................................................................................. ...... .... ... ... .. .. .. .. . . .. . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................................................................................................................................................
..................................................................................................................................................................................
x
(cosx,sinx)
y
A
B
(1,0)
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 ?gure, 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 2pr = 2p(1) = 2p, so the radian
measure of the full circular angle (that is, of the 360 degree angle) is 2p.
While an angle with a particular measure can appear anywhere around the circle, we
need a ?xed, conventional location so that we can use the coordinate system to de?ne
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 ?gure, x is the standard location of the angle p/6, that is, the length of the
arc from (1,0) to A is p/6. The angle y in the picture is-p/6, because the distance from
(1,0) to B along the circle is also p/6, but in a clockwise direction.
Now the fundamental trigonometric de?nitions are: the cosine of x and the sine of x
are the ?rst and second coordinates of the point A, as indicated in the ?gure. The angle x
shown can be viewed as an angle of a right triangle, meaning the usual triangle de?nitions
of the sine and cosine also make sense. Since the hypotenuse of the triangle is 1, the “side
opposite over hypotenuse” de?nition 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 de?nitions work only for angles that can “?t” in a right triangle,
namely, angles between 0 and p/2. The coordinate de?nitions, on the other hand, apply
Page 3


4
Transcendental Functions
So far we have used only algebraic functions as examples when ?nding 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.
4.1 Trigonometri
 Fun
tions When you ?rst encountered the trigonometric functions it was probably in the context of
“triangle trigonometry,” de?ning, 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 de?nition 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.
To de?ne the radian measurement system, we consider the unit circle in the xy-plane:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . .. .. . . . .. ... .. ... ... .... ..... .................................................................................................................................................................................................................................................................................. ...... .... ... ... .. .. .. .. . . .. . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................................................................................................................................................
..................................................................................................................................................................................
x
(cosx,sinx)
y
A
B
(1,0)
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 ?gure, 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 2pr = 2p(1) = 2p, so the radian
measure of the full circular angle (that is, of the 360 degree angle) is 2p.
While an angle with a particular measure can appear anywhere around the circle, we
need a ?xed, conventional location so that we can use the coordinate system to de?ne
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 ?gure, x is the standard location of the angle p/6, that is, the length of the
arc from (1,0) to A is p/6. The angle y in the picture is-p/6, because the distance from
(1,0) to B along the circle is also p/6, but in a clockwise direction.
Now the fundamental trigonometric de?nitions are: the cosine of x and the sine of x
are the ?rst and second coordinates of the point A, as indicated in the ?gure. The angle x
shown can be viewed as an angle of a right triangle, meaning the usual triangle de?nitions
of the sine and cosine also make sense. Since the hypotenuse of the triangle is 1, the “side
opposite over hypotenuse” de?nition 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 de?nitions work only for angles that can “?t” in a right triangle,
namely, angles between 0 and p/2. The coordinate de?nitions, on the other hand, apply
to any angles, as indicated in this ?gure:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . .. . .. . . .. ... ... ... ... ...... .................................................................................................................................................................................................................................................................................. ...... .... ... ... .. .. .. . . .. .. . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . .. . .. .. .. .. .. .. ... .... ....... ...................................................
x
A
(cosx,sinx)
The angle x is subtended by the heavy arc in the ?gure, that is, x = 7p/6. Both
coordinates of point A in this ?gure are negative, so the sine and cosine of 7p/6 are both
negative.
The remaining trigonometric functions can be most easily de?ned in terms of the sine
and cosine, as usual:
tanx =
sinx
cosx
cotx =
cosx
sinx
secx =
1
cosx
cscx =
1
sinx
and they can also be de?ned as the corresponding ratios of coordinates.
Although the trigonometric functions are de?ned in terms of the unit circle, the unit
circle diagram is not what we normally consider the graph of a trigonometric function.
(The unit circle is the graph of, well, the circle.) We can easily get a qualitatively correct
idea of the graphs of the trigonometric functions from the unit circle diagram. Consider
the sine function, y = sinx. As x increases from 0 in the unit circle diagram, the second
coordinate of the point A goes from 0 to a maximum of 1, then back to 0, then to a
minimum of -1, then back to 0, and then it obviously repeats itself. So the graph of
y = sinx must look something like this:
-1
1
p/2 p 3p/2 2p -p/2 -p -3p/2 -2p
..... ...... ...... ....... ........... ................................................................................................................................................................... ......... ...... ...... ..... ...... ...... ...... ...... ......... ..................................................................................................................................................................... .......... ...... ...... ...... ..... ...... ...... ...... ........ ..................................................................................................................................................................... .......... ....... ...... ...... ...... ..... ...... ...... ....... ............. ................................................................................................................................................... ............... ........ ...... ...... ..... .
Page 4


4
Transcendental Functions
So far we have used only algebraic functions as examples when ?nding 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.
4.1 Trigonometri
 Fun
tions When you ?rst encountered the trigonometric functions it was probably in the context of
“triangle trigonometry,” de?ning, 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 de?nition 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.
To de?ne the radian measurement system, we consider the unit circle in the xy-plane:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . .. .. . . . .. ... .. ... ... .... ..... .................................................................................................................................................................................................................................................................................. ...... .... ... ... .. .. .. .. . . .. . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................................................................................................................................................
..................................................................................................................................................................................
x
(cosx,sinx)
y
A
B
(1,0)
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 ?gure, 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 2pr = 2p(1) = 2p, so the radian
measure of the full circular angle (that is, of the 360 degree angle) is 2p.
While an angle with a particular measure can appear anywhere around the circle, we
need a ?xed, conventional location so that we can use the coordinate system to de?ne
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 ?gure, x is the standard location of the angle p/6, that is, the length of the
arc from (1,0) to A is p/6. The angle y in the picture is-p/6, because the distance from
(1,0) to B along the circle is also p/6, but in a clockwise direction.
Now the fundamental trigonometric de?nitions are: the cosine of x and the sine of x
are the ?rst and second coordinates of the point A, as indicated in the ?gure. The angle x
shown can be viewed as an angle of a right triangle, meaning the usual triangle de?nitions
of the sine and cosine also make sense. Since the hypotenuse of the triangle is 1, the “side
opposite over hypotenuse” de?nition 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 de?nitions work only for angles that can “?t” in a right triangle,
namely, angles between 0 and p/2. The coordinate de?nitions, on the other hand, apply
to any angles, as indicated in this ?gure:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . .. . .. . . .. ... ... ... ... ...... .................................................................................................................................................................................................................................................................................. ...... .... ... ... .. .. .. . . .. .. . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . .. . .. .. .. .. .. .. ... .... ....... ...................................................
x
A
(cosx,sinx)
The angle x is subtended by the heavy arc in the ?gure, that is, x = 7p/6. Both
coordinates of point A in this ?gure are negative, so the sine and cosine of 7p/6 are both
negative.
The remaining trigonometric functions can be most easily de?ned in terms of the sine
and cosine, as usual:
tanx =
sinx
cosx
cotx =
cosx
sinx
secx =
1
cosx
cscx =
1
sinx
and they can also be de?ned as the corresponding ratios of coordinates.
Although the trigonometric functions are de?ned in terms of the unit circle, the unit
circle diagram is not what we normally consider the graph of a trigonometric function.
(The unit circle is the graph of, well, the circle.) We can easily get a qualitatively correct
idea of the graphs of the trigonometric functions from the unit circle diagram. Consider
the sine function, y = sinx. As x increases from 0 in the unit circle diagram, the second
coordinate of the point A goes from 0 to a maximum of 1, then back to 0, then to a
minimum of -1, then back to 0, and then it obviously repeats itself. So the graph of
y = sinx must look something like this:
-1
1
p/2 p 3p/2 2p -p/2 -p -3p/2 -2p
..... ...... ...... ....... ........... ................................................................................................................................................................... ......... ...... ...... ..... ...... ...... ...... ...... ......... ..................................................................................................................................................................... .......... ...... ...... ...... ..... ...... ...... ...... ........ ..................................................................................................................................................................... .......... ....... ...... ...... ...... ..... ...... ...... ....... ............. ................................................................................................................................................... ............... ........ ...... ...... ..... .
Similarly, as angle x increases from 0 in the unit circle diagram, the ?rst 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 = cosx
must look something like this:
-1
1
p/2 p 3p/2 2p -p/2 -p -3p/2 -2p
.................................................................................. ........ ........ ...... ...... ..... ..... ...... ....... ........ ................................................................................................................................................................. ............. ........ ...... ..... .. .... ..... ...... ....... ....... .......... .................................................................................................................................................................... ........ ...... ....... ..... ...... ..... ...... ....... ......... ..................................................................................................................................................................... ........ ........ ...... ..... ..... ...... ...... ....... ........ ....................................................................................
4.2 The Deriv a tive of sinx
What about the derivative of the sine function? The rules for derivatives that we have are
no help, since sinx is not an algebraic function. We need to return to the de?nition of the
derivative, set up a limit, and try to compute it. Here’s the de?nition:
d
dx
sinx = lim
?x?0
sin(x+?x)-sinx
?x
.
Using some trigonometric identities, we can make a little progress on the quotient:
sin(x+?x)-sinx
?x
=
sinxcos?x+sin?xcosx-sinx
?x
= sinx
cos?x-1
?x
+cosx
sin?x
?x
.
Page 5


4
Transcendental Functions
So far we have used only algebraic functions as examples when ?nding 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.
4.1 Trigonometri
 Fun
tions When you ?rst encountered the trigonometric functions it was probably in the context of
“triangle trigonometry,” de?ning, 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 de?nition 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.
To de?ne the radian measurement system, we consider the unit circle in the xy-plane:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . .. .. . . . .. ... .. ... ... .... ..... .................................................................................................................................................................................................................................................................................. ...... .... ... ... .. .. .. .. . . .. . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................................................................................................................................................................................
..................................................................................................................................................................................
x
(cosx,sinx)
y
A
B
(1,0)
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 ?gure, 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 2pr = 2p(1) = 2p, so the radian
measure of the full circular angle (that is, of the 360 degree angle) is 2p.
While an angle with a particular measure can appear anywhere around the circle, we
need a ?xed, conventional location so that we can use the coordinate system to de?ne
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 ?gure, x is the standard location of the angle p/6, that is, the length of the
arc from (1,0) to A is p/6. The angle y in the picture is-p/6, because the distance from
(1,0) to B along the circle is also p/6, but in a clockwise direction.
Now the fundamental trigonometric de?nitions are: the cosine of x and the sine of x
are the ?rst and second coordinates of the point A, as indicated in the ?gure. The angle x
shown can be viewed as an angle of a right triangle, meaning the usual triangle de?nitions
of the sine and cosine also make sense. Since the hypotenuse of the triangle is 1, the “side
opposite over hypotenuse” de?nition 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 de?nitions work only for angles that can “?t” in a right triangle,
namely, angles between 0 and p/2. The coordinate de?nitions, on the other hand, apply
to any angles, as indicated in this ?gure:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . .. . .. . . .. ... ... ... ... ...... .................................................................................................................................................................................................................................................................................. ...... .... ... ... .. .. .. . . .. .. . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . .. . .. .. .. .. .. .. ... .... ....... ...................................................
x
A
(cosx,sinx)
The angle x is subtended by the heavy arc in the ?gure, that is, x = 7p/6. Both
coordinates of point A in this ?gure are negative, so the sine and cosine of 7p/6 are both
negative.
The remaining trigonometric functions can be most easily de?ned in terms of the sine
and cosine, as usual:
tanx =
sinx
cosx
cotx =
cosx
sinx
secx =
1
cosx
cscx =
1
sinx
and they can also be de?ned as the corresponding ratios of coordinates.
Although the trigonometric functions are de?ned in terms of the unit circle, the unit
circle diagram is not what we normally consider the graph of a trigonometric function.
(The unit circle is the graph of, well, the circle.) We can easily get a qualitatively correct
idea of the graphs of the trigonometric functions from the unit circle diagram. Consider
the sine function, y = sinx. As x increases from 0 in the unit circle diagram, the second
coordinate of the point A goes from 0 to a maximum of 1, then back to 0, then to a
minimum of -1, then back to 0, and then it obviously repeats itself. So the graph of
y = sinx must look something like this:
-1
1
p/2 p 3p/2 2p -p/2 -p -3p/2 -2p
..... ...... ...... ....... ........... ................................................................................................................................................................... ......... ...... ...... ..... ...... ...... ...... ...... ......... ..................................................................................................................................................................... .......... ...... ...... ...... ..... ...... ...... ...... ........ ..................................................................................................................................................................... .......... ....... ...... ...... ...... ..... ...... ...... ....... ............. ................................................................................................................................................... ............... ........ ...... ...... ..... .
Similarly, as angle x increases from 0 in the unit circle diagram, the ?rst 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 = cosx
must look something like this:
-1
1
p/2 p 3p/2 2p -p/2 -p -3p/2 -2p
.................................................................................. ........ ........ ...... ...... ..... ..... ...... ....... ........ ................................................................................................................................................................. ............. ........ ...... ..... .. .... ..... ...... ....... ....... .......... .................................................................................................................................................................... ........ ...... ....... ..... ...... ..... ...... ....... ......... ..................................................................................................................................................................... ........ ........ ...... ..... ..... ...... ...... ....... ........ ....................................................................................
4.2 The Deriv a tive of sinx
What about the derivative of the sine function? The rules for derivatives that we have are
no help, since sinx is not an algebraic function. We need to return to the de?nition of the
derivative, set up a limit, and try to compute it. Here’s the de?nition:
d
dx
sinx = lim
?x?0
sin(x+?x)-sinx
?x
.
Using some trigonometric identities, we can make a little progress on the quotient:
sin(x+?x)-sinx
?x
=
sinxcos?x+sin?xcosx-sinx
?x
= sinx
cos?x-1
?x
+cosx
sin?x
?x
.
This isolates the di?cult bits in the two limits
lim
?x?0
cos?x-1
?x
and lim
?x?0
sin?x
?x
.
Here we get a littlelucky: it turns out that once we know the second limit the ?rst 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
sinx
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 o?cial de?nition 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 situationlookssomething like?gure 4.3.1.
The wiggly curve is x
2
sin(p/x), the upper and lower curves are x
2
and -x
2
. Since the
sine function is always between -1 and 1, -x
2
= x
2
sin(p/x)= x
2
, and it is easy to see
that lim
x?0
-x
2
= 0 = lim
x?0
x
2
. It is not so easy to see directly, that is algebraically,
that lim
x?0
x
2
sin(p/x) = 0, because the p/x prevents us from simply plugging in x = 0.
The squeeze theorem makes this “hard limit” as easy as the trivial limits involving x
2
.
To do the hard limit that we want, lim
x?0
(sinx)/x, we will ?nd two simpler functions
g and h so that g(x)= (sinx)/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 ?gure 4.3.2,
x is the measure of the angle in radians. Since the circle has radius 1, the coordinates of