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36 Part I: Technical Mathematics
Two Sides and an Angle between Them Known
Part I.D
Given the values of two sides and the angle between them, we can find the values
of the remaining side and the other two angles. For example, given the triangle
shown in Figure 4.5, we see that sides x and y are 8 and 10 units, respectively, and
angle Z is 29.69 degrees. We can find the length of side z using the law of cosines:
2 2 2
z x y 2xycos
Z
z
z
z
2 2 2
2
2
= + ( )
= 8 + 10 2( 8)( 10)
cos ( 29. 69)
= 164 160cos ( 29.
69)
= 164 160( 0.
8687)
2
z = 25
2
z = 25
z = 5
We can then apply the law of cosines to find angle X:
2 2 2
x y z 2yzcos
X
= + ( )
2 2 2
8 = 10 + 5 2( 10)( 5) cos( X)
64 125 100cos
64 125 = 125 100cos( X)
125
61
= ( X)
= 100cos( X)
61÷ 100 = 100cos( X)÷ 100
( )
061 . = cos
X
cos 1 ( 061 . )= X
X = 52.
41
Y
z
x = 8
X
y = 10
29.69°
Z
Figure 4.5 Two sides and angle between known.