66 Chapter 3Transcendental functions
(angles are often quoted as multiples of π). The length of an arc of a circle iss 1 = 1 rθ
when θis in radians.
EXAMPLE 3.3Radians and degrees
(1) The angle 40° in units of the radian is
(2) The angle 0.5 rad in degrees is
(3) The length of the arc that subtends an angle ofθ 1 = 12 rad at the centre of a
circle of radiusr 1 = 13 is
s 1 = 1 rθ 1 = 131 × 121 = 16
0 Exercises 3–5
Trigonometric functions for all angles
The geometric definition of the trigonometric functions in terms of ratios of the sides
of a right-angled triangle restricts the functions to values of angles in the range 0° to
90°, or
The definitions can be extended to all values of the angle by consideration of the
coordinates of a point on a circle, as shown in Figure 3.6.
0
2
≤≤θ
π
05 05
360
2
90
.=.× = ≈ .28 6
ºº
º
ππ
40 40
2
360
2
9
ºº 07
º
=× =≈.
ππ
rad
90
2
180 360 2
6
ºººs=, =, = , ins=inº 30
π
ππ
π
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II I
III IV
Figure 3.6