3.2 Trigonometric functions 65
EXAMPLE 3.2For the triangle in Figure 3.3, 3
21 + 14
21 = 15
2and
0 Exercises 2
Units of angle
The ordinary unit of angle is the degree. Figure 3.4 shows the right angle (90°), the
angle on a line (180°) and the angle round a point (360°).
The unit of angle that is always used in mathematical and
scientific applications is the radian (SI symbol rad), defined in
terms of the properties of the circle. Figure 3.5 shows a circle
of radius r, and an arc of length ssubtending the angle θat the
centre. The length of the arc is proportional to the size of the
angle; for example, doubling the angle θdoubles the length s.
It follows that the ratio of sto the circumference of the circle
(2πr)is equal to the ratio of θto the complete angle around the
centre (360°):
(3.7)
so that
(3.8)
The unit of angle, the radian, is defined by
(3.9)
(rad is the SI unit of angle; sometimes a superscript cis used, so that 1
c1 = 11 rad).
In practice, the symbol for the unit is omitted, and an angle given without unit is
assumed to be in radians; for example
1
360
2
rad=≈57 18′
º
º
π
θ=×
s
r
360
2
º
π
s
2 πr 360
=
θ
º
sin cos
22222224
5
3
5
43
5
25
2
θθ+=
=
=
55
= 1
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.90
◦180
◦360
◦•••
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..r
r
s
θ
Figure 3.5
Figure 3.4