6.1 Law of Sines:The Ambiguous Case

ASS/SSA

What have we learned so far?

a c a > c

Obtuse

a c

Acute

1 triangle

What about a < c ?0 triangle 1 triangle

A

a

B

c

Cb

In our triangle, we are given measurements for an acute angle A and sides a and c.

A

a

B

c

Cb

What happens, now, when a < c ?

A

a

B

c

Cb

For purposes you’ll see shortly, we must find the length of the altitude, called h.

h

A

a

B

c

Cb

How can we find h ?

h

sinh

Ac

sinh c A

A

a

B

c

C

No triangle can be formedb

h

What happens when a < h ?

a

A

B

c

Cb

h

What happens when a = h ?

One right triangle is formed

a

A

B

c

Cb

h

What happens when a > h ?

Some of the lengths are too long, and some are too short. Two, however, are just right.

a

A

B

c

Cb

h

So when a > h ...

a

A

B

c

Cb

h

One triangle is formed

…and when a > h ...

B

A second obtuse triangle is also formed

a

A

B

c

Cb

h

In conclusion, given ASS/SSA, where a < c...

a < h a = h a > h

0 triangle 1 triangle 2 triangles

So here are ALL cases for the Law of Sines Ambiguous Case:

a c a > c

Obtuse

a c

Acute

1 triangle0 triangle 1 triangle

a < c

a < h a = h a > h

0 triangle 1 triangle 2 triangles