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34 Part I: Technical Mathematics
This law can also be written in terms of its reciprocal:
Part I.D
x y z
sin X sin Y sin Z
( ) = ( ) = ( )
Law of Cosines
Given an oblique triangle such as the one in Figure 4.3, the law of cosines states
2 2 2
x y z 2yzcos
X
= + ( )
= + ( )
2
2xy
( Z)
2 2 2
y x z 2xzcos
Y
2 2
z = x + y
cos .
Three Sides Known
Given the values of three sides, we can find the values of the three angles using
the law of cosines. For example, given the triangle in Figure 4.4 we can find the
values of the three angles:
Y
z
x
X
y
Z
Figure 4.3 Oblique triangle.
Y
z = 5
x = 8
X
y = 10
Z
Figure 4.4
Three sides known.