THE LAW OF SINES

2.3 I can solve triangles using the Law of Sines

If none of the angles of a triangle is a right angle, the triangle is called

oblique.

All angles are acute Two acute angles, one obtuse angle

To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles.

FOUR CASES CASE 1:

One side and two angles are known (SAA or ASA).

CASE 2:

Two sides and the angle opposite one of them are known (SSA). Ambiguous Case

CASE 3:

Two sides and the included angle are known (SAS).

CASE 4:

Three sides are known (SSS).

S A A

ASA

S A

SAA CASE 1: ASA or SAA Use Law of Sines

A

S A S

CASE 2: SSA - Ambiguous Case Use Law of Sines

S A S

CASE 3: SAS Use Law of Cosine

S S S

CASE 4: SSS Use Law of Cosines

Theorem Law of Sines

b

 10 sin   10 sin 70   9 .

40

c

 10 sin   10 sin 80   9 .

85

a

 12 sin 20  sin 100   4 .

17

b

 12 sin 60  sin 100   10 .

55

The Ambiguous Case: Case 2: SSA

 The known information may result in  One triangle  Two triangles  No triangles

Not possible, so there is only one triangle!

a

 5 sin 132 sin 30 .

5    7 .

37

a

  7 .

37 ,  30  , 

b

 5 ,  17 .

5 ,

c

 3 ,   132 .

5

 1 sin   10 sin 8 45   62 .

1 

or

 2  0 .

88  117 .

9  Two triangles!!

Triangle 1:  1  1  62 .

1   180   45   62 .

1   72 .

9

a

1  8 sin sin 72 .

9  45   10 .

81 

a

1 1  62 .

1  ,  10 .

81 ,  1

b

  8 , 72 .

9  ,

c

 10   45 

Triangle 2:   1 2  117 .

9   180   45   117 .

9   17 .

1  sin

a

2  17 .

1  sin  45 sin

a

2 8 sin 17 .

1   8 45   3 .

33  2

a

2  117 .

9  ,  3 .

33 ,

b

 2  8 ,  17 .

1  ,

c

 10   45 

sin   1 .

28 No triangle with the given measurements!

The Ambiguous Case: Case 2: SSA

 The known information may result in  One triangle  Two triangles  No triangles  The key to determining the possible triangles, if any, lies primarily with the height, h and the fact h = b sin α α b h a

No Triangle

 If a < h = b sin α, then side a is not sufficiently long to form a triangle.

α b a

h = b sin α

 a < h = b sin α

One Right Triangle

 If a = h = b sin α, then side a is just long enough to form a triangle.

α b a

h = b sin α

 a = h = b sin α

Two Triangles

 If a < b and h = b sin α < a , then two distinct triangles can be formed α b a a

h = b sin α

 a < b and h = b sin α < a

One Triangle

 If a ≥ b , then only one triangle can be formed.

 b α a ≥ b  a

h = b sin α

Fortunately we do not have to rely on the illustration to draw a correct conclusion. The Law of Sines will help us.