Graphics Reference
In-Depth Information
C
a
b
A
B
c
Fig. 4.3.
An arbitrary triangle.
4.5 The Sine Rule
The sine rule relates angles and side lengths for a triangle. Figure 4.3 shows a
triangle labelled such that side
a
is opposite angle
A
,side
b
is opposite angle
B
, etc.
The sine rule states
a
sin
A
=
b
sin
B
c
sin
C
=
4.6 The Cosine Rule
The cosine rule expresses the sin
2
(
β
)+cos
2
(
β
) = 1 relationship for the arbi-
trary triangle shown in Figure 4.3. In fact, there are three versions:
a
2
=
b
2
+
c
2
−
2
bc
cos(
A
)
b
2
=
c
2
+
a
2
−
2
ca
cos(
B
)
c
2
=
a
2
+
b
2
−
2
ab
cos(
C
)
Three further relationships also hold:
a
=
b
cos(
C
)+
c
cos(
B
)
b
=
c
cos(
A
)+
a
cos(
C
)
c
=
a
cos(
B
)+
b
cos(
A
)
4.7 Compound Angles
Two sets of compound trigonometric relationships show how to add and sub-
tract two different angles and multiples of the same angle. The following are
some of the most common relationships:
sin(
A
±
B
) = sin(
A
)cos(
B
)
±
cos(
A
) sin(
B
)
