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Law of SinesSolving Oblique Triangles
Prepared by Title V Staff:Daniel Judge, Instructor
Ken Saita, Program Specialist
East Los Angeles College
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Topics
Oblique Triangle DefinitionsThe Law of SinesGeneral Strategies for Using the Law of Sines
ASASAAThe Ambiguous Case SSA
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Trigonometry can help us solve non-right triangles as well. Non-right triangles are know as oblique triangles. There are two categories of oblique triangles—acute and obtuse.
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In an acute triangle, each of the angles is less than 90º.
Acute Triangles
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Obtuse Triangles
In an obtuse triangle, one of the angles is obtuse (between 90º and 180º). Can there be two obtuse angles in a triangle?
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The Law of Sines
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Consider the first category, an acute triangle (, , are acute).
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hNow, sin( ) , so that h a sin( )
ah
But sin( ) , so that h c sin( )c
By transitivity, a sin( ) c sin( )
sin( ) sin( )Which means
c a
Create an altitude, h.
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Let’s create another altitude h’.
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h'sin( ) , so that h' c sin( )
ch'
sin( ) , so that h' b sin( )b
By transitivity, c sin( ) b sin( )
sin( ) sin( )Which means
b c
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Putting these together, we get
sin( ) sin( ) sin( )
a b c
This is known as the Law of Sines.
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The Law of Sines is used when we know any two angles and one side or when we know two sides and an angle opposite one of those sides.
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Fact The law of sines also works for oblique triangles that contain an obtuse angle (angle between 90º and 180º).
is obtuse
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General Strategies for Usingthe Law of Sines
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One side and two angles are known.
ASA or SAA
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ASA
From the model, we need to determine a, b, and using the law of sines.
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First off, 42º + 61º + = 180º so that = 77º. (Knowledge of two angles yields the third!)
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Now by the law of sines, we have the following relationships:
)sin(42 sin(77 ) sin(61 ) sin(77 ) ;
a 12 b 12
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So that
12 sin(42 ) 12 sin(61 )a b
sin(77 ) sin(77 )
12 (0.6691) 12 (0.8746)a b
0.9744 0.9744
a 8.2401 b 10.7709
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SAA
From the model, we need to determine a, b, and using the law of sines.
Note: + 110º + 40º = 180º so that = 30º
ab
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By the law of sines,
)sin(30 sin(40 ) sin(110 ) sin(40 ) ;
a 12 b 12
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Thus,
12 sin(30 ) 12 sin(110 )a b
sin(40 ) sin(40 )
12 (0.5) 12 (0.9397)a b
0.6428 0.5
a 9.3341 b 22.5526
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The Ambiguous Case – SSA
In this case, you may have information that results in one triangle, two triangles, or no triangles.
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SSA – No Solution
Two sides and an angle opposite one of the sides.
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By the law of sines,
sin(57 ) sin( )
15 20
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Thus,20 sin(57 )
sin( )15
20 (0.8387)sin( )
15sin( ) 1.1183 Impossible!
Therefore, there is no value for that exists! No Solution!
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SSA – Two Solutions
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By the law of sines,
sin(32 ) sin( )
30 42
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So that,
42 sin(32 )sin( )
3042 (0.5299)
sin( )30
sin( ) 0.7419
48 or 132
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Case 1 Case 2
48 32 180
100
132 32 180
16
Both triangles are valid! Therefore, we have two solutions.
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Case 1 )sin(100 sin(32 )
c 30
30 sin(100 )c
sin(32 )
30 (0.9848)c
0.5299
c 55.7539
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Case 2 sin(16 ) sin(32 )
c 30
30 sin(16 )c
sin(32 )
30 (0.2756)c
0.5299
c 15.6029
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Finally our two solutions:
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SSA – One Solution
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By the law of sines,
sin(40 ) sin( )
3 2
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2 sin(40 )sin( )
32 (0.6428)
sin( )3
sin( ) 0.4285
25.4 or 154.6
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Note– Only one is legitimate!
40 25.4 180
114.6
40 154.6 180
14.6 Not Possible!
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Thus we have only one triangle.
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By the law of sines,
sin(114.6 ) sin(40 ) ;
b 3
3 sin(114.6 )b
sin(40 )
3 (0.9092)b
0.6428
b 4.2433
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Finally, we have:
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End of Law of Sines
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