Section 7.2 The Law of Sines
In this section, we discuss solving triangles, and we use the law of sines to solve some triangles.
An
obtuse angle
is an angle between 90
o
and 180
o
.
An
oblique triangle
is a triangle that does not contain a 90
o
angle.
Labeling triangles:
a
,
b
,
c
will be the sides opposite the angles
α
,
β
,
γ
, respectively.
A
,
B
,
C
will be the vertices of angles
α
,
β
,
γ
, respectively.
The Law of Sines
The following relationships hold for any triangle.
•
Law of Sines:
sin
α
a
=
sin
β
b
=
sin
γ
c
.
Solving Triangles:
In the following, S stands for side and A stands for angle, and SAA means
sideangleangle, in that order.
Case 1 :
SAA
or
ASA
(use Law of Sines)
Case 2 :
SSA
(use Law of Sines)
Case 3 :
SAS
(use Law of Cosines)
Case 4 :
SSS
(use Law of Cosines)
The following triangles illustrate the above cases.
We now consider triangles that can be solved with the Law of Sines.
•
Case 1:
SAA
Example.
Solve the triangle if
α
= 50
o
,
β
= 25
o
,
a
= 4.
Solution. Since
α
= 50
o
and
β
= 25
o
,
γ
= 180
o

(25
o
+ 50
o
) = 105
o
.
By the law of sines,
sin
α
a
=
sin
β
b
=
sin
γ
c
. Hence,
sin 50
o
4
=
sin 25
o
b
=
sin 105
o
c
. Thus,
b
=
4 sin 25
o
sin 50
o
= 2
.
2
c
=
4 sin 105
o
sin 50
o
= 5
.
0
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 Fall '11
 Kutter
 Algebra, Angles, Pythagorean Theorem, Sin, triangle, Sines

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