3.4 Trigonometric relations 73
To overcome this ambiguity, a principal valuehas been defined for each inverse
function:
(3.18)
They are the values computed, for example, on a pocket calculator or other computer.
EXAMPLE 3.9
Angleπ 26 lies in the first quadrant and is the principal value of the inverse function:
0 Exercises 12, 13
3.4 Trigonometric relations
The sine and cosine rules
The angles and sides of a triangle (Figure 3.14) are related by two rules:
sine rule:
(3.19)
cosine rule:
a
21 = 1 b
21 + 1 c
21 − 12 bc 1 cos 1 A (3.20)
Proofs
For the sine rule the proof is, using Figure 3.15,
sin =,sin AC=
h
c
h
a
sin sin sin ABC
abc
==
cos
−
=
13
26
π
cos cos cos
ππ π
66
13
6
3
2
=−
=
==
xy x
x
=−≤≤
=
−−sin
cos
1122
ππ
(quadrants I and IV)
yyx
xy x
0
22
1≤≤
=−≤≤
−π
ππ
(quadrants I and II)
tan ((quadrants I and IV)
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..Figure 3.15