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.