6.1 Laws of Sines

The Laws of Sine can be used with Oblique triangle

Oblique triangle is a triangle that contains no right angle.

A

B

C

a

b

c

The Laws of Sines

A

B

C

a

b

c

C

c

B

b

A

a

sinsinsin

Using the Law of Sines

Given: How do you find angle B?

aA

B

C

c

b

a

Bm

Cm

Am

6.21

7.16

4.102

b

c

Using the Law of Sines

Given: How do you find side b?

aA

B

C

c

b

a

Bm

Cm

Am

6.21

9.60

7.16

4.102

b

c

Using the Law of Sines

Given: How do you find side b?

aA

B

C

c

b

a

Bm

Cm

Am

6.21

9.60

7.16

4.102

9.60sin4.102sin

6.21 b

b

c

Using the Law of Sines

Given: How do you find side b?

aA

B

C

3.19

7.102sin

9.60sin6.21

9.60sin4.102sin

6.21

b

b

b

c

b

a

Bm

Cm

Am

3.19

6.21

9.60

7.16

4.102

b

c

Using the Law of Sines

Given: How do you find side c?

aA

B

C

c

b

a

Bm

Cm

Am

3.19

6.21

9.60

7.16

4.102

7.16sin4.102sin

6.21 c

b

c

Using the Law of Sines

Given: How do you find side c?

aA

B

C

c

b

a

Bm

Cm

Am

3.19

6.21

9.60

7.16

4.102

36.6

4.102sin

7.16sin6.21

7.16sin4.102sin

6.21

c

c

c

b

c

The Ambiguous Case

Look at this triangle.

If we look at where angle A

Is Acute

A B

C

b a

h

Abh sin

The Ambiguous Case

Look at this triangle.

If we look at

If a = h, then there is one triangle

A B

C

bahAbh sin

The Ambiguous Case

Look at this triangle.

If we look at

If a < h, then there is no triangle

A B

C

ba

hAbh sin

The Ambiguous Case

Look at this triangle.

If we look at

If a > b, then there is one triangle

A B

C

b a

hAbh sin

The Ambiguous Case

Look at this triangle.

If we look at

If h< a <b, then there is two triangles

A B

C

b a

hAbh sin'B

The Ambiguous Case

Do you remember the Hinge Theorem from Geometry.

Given two sides and one angle, two different triangles can be made.

http://mrself.weebly.com/5-5-the-hinge-theorem.html

15 151111

42 42

The Ambiguous Case

Where Angle A is Obtuse.

If a ≤ b, there is no

triangle

A

a

b

The Ambiguous Case

Where Angle A is Obtuse.

If a > b, there is one

triangle

A

a

b

Area of an Oblique triangle

Using two sides and an Angle.

SinBacArea

SinCabArea

SinAbcArea

2

1

2

1

2

1

Find the missing Angles and Sides

Given: 5,8,36 baA

Find the missing Angles and Sides

Given:

5.122

1805.2136

C

C

5,85.122,5.21,36 baCBA

Find the missing Angles and Sides

Given: 5,85.122,5.21,36 baCBA

48.11

36

5.1228

36

8

5.122

c

Sin

Sinc

SinSin

c

HomeworkHomework

Page 416 Page 416

##1, 7, 13, 19,1, 7, 13, 19,

25, 31, 37,25, 31, 37,

43, 4943, 49

Homework

Page 416

# 4, 10, 16, 22,

28, 34, 40,

46, 52