Triangles: Triangles: the Ambiguous Casethe Ambiguous Case

By: Rachel Atmadja, 2002

Source: Glencoe Adv. Mathematical Concepts Pg 324 #16

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

32 27

37

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

32 27

37

The ‘magic’ number:

32(sin 37 )= 19.3

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

32 27

37

The ‘magic’ number:

32(sin 37 )= 19.3

19.3 < 27 < 32

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

32 27

37

The ‘magic’ number:

32(sin 37 )= 19.3

19.3 < 27 < 32

Therefore, there are 2 solutions

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

32 27

37

27

Use law of Sines to find A:

Sin 37 Sin A

27 32=

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

32 27

37

27

Sin 37 Sin A

27 32

A = 45.5

=

45.5

Use law of Sines to find A:

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

32 27

37

27

Sin 37 Sin A

27 32

A = 45.5

C = 180 – 37 – 45.5 = 97.5

=

45.5

Now find C:

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

3227

37

27

Sin 37 Sin A Sin 97.5

27 32 c

A = 45.5

C = 97.5

=

45.5

Now use the Law of Sines to find side c

=

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

3227

37

27

Sin 37 Sin A Sin 97.5

27 32 c

A = 45.5

C = 97.5

c = 44.5

=

45.5

Now use the Law of Sines to find side c

=

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

3227

37

27

A = 45.5

C = 97.5

c = 44.5

45.5

Now to find the 2nd set of solution, let’s being by finding A2:

A

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

3227

37

27

A = 45.5 A2 = 134.5

C = 97.5

c = 44.5

45.5

Notice that A and A2 are supplementary angles. To find A2, simply subtract A from 180

A

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

3227

37

27

A = 45.5 A2 = 134.5

C = 97.5 C2 = 8.5

c = 44.5

45.5

Now find C2. You can find C2 simply by subtracting angles B and A2 from 180.

A

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

3227

37

27

A = 45.5 A2 = 134.5

C = 97.5 C2 = 8.5

c = 44.5

45.5

Use the Law of Sines to find side c2:

A

The Ambiguous Case

Find all solutions for the triangle below:

B = 37 a = 32 b = 27

C

BA

3227

37

27

Sin 37 Sin 8.5

27 c2

A = 45.5 A2 = 134.5

C = 97.5 C2 = 8.5

c = 44.5 c2 = 6.6

45.5

Use the Law of Sines to find side c2:

A

=