4
Transcendental 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.
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 illsuited to mathematical calculation, so (almost) everything we do will be in terms of
radian measure
of angles.
59
60
Chapter 4 Transcendental Functions
To define the radian measurement system, we consider the unit circle in the
xy
plane:
.
.
.
x
(cos
x,
sin
x
)
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 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 2
πr
= 2
π
(1) = 2
π
, so the radian
measure of the full circular angle (that is, of the 360 degree angle) is 2
π
.
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.
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 Fall '11
 GUICHARD
 Trigonometry, Unit Circle, lim, Inverse function, ... ..., Logarithm

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