6.1 - Law of Sines

2

An oblique triangle is a triangle that has no right angles.

To solve an oblique triangle, you need to know the measure of at least one side and the measures of any

other two parts of the triangle – two sides, two angles, or one angle and one side.

C

BA

ab

c

3

Four Cases for solving a triangle…

3

Four Cases for solving a triangle…1. Two angles and any side (AAS or ASA)

A

Cc

A

Bc

3

Four Cases for solving a triangle…1. Two angles and any side (AAS or ASA)

2. Two sides and an angle opposite one of them (SSA)

A

Cc

A

Bc

C

c

a

3

Four Cases for solving a triangle…1. Two angles and any side (AAS or ASA)

2. Two sides and an angle opposite one of them (SSA)

3. Three sides (SSS)

A

Cc

A

Bc

a

cb

C

c

a

3

Four Cases for solving a triangle…1. Two angles and any side (AAS or ASA)

2. Two sides and an angle opposite one of them (SSA)

3. Three sides (SSS)

4. Two sides and their included angle (SAS)

A

Cc

A

Bc

a

cb

C

c

a

c

aB

4

Case 1 and Case 2:

4

Case 1 and Case 2: Law of SinesIf ABC is an oblique triangle with sides a, b, and c, then

Acute Triangle

C

BA

bh

c

a

C

BA

bh

c

a

Obtuse Triangle

5

Find the remaining angle and sides of the triangle.

Example 1 (ASA):

5

Find the remaining angle and sides of the triangle.

Example 1 (ASA):

C

BA

b

c

60°

10°

a = 4.5 ft

5

Find the remaining angle and sides of the triangle.

Example 1 (ASA):

The third angle in the triangle is A = 180° – A – B

= 180° – 10° – 60° = 110°

C

BA

b

c

60°

10°

a = 4.5 ft

110°

5

Find the remaining angle and sides of the triangle.

Example 1 (ASA):

The third angle in the triangle is A = 180° – A – B

= 180° – 10° – 60° = 110°

C

BA

b

c

60°

10°

a = 4.5 ft

110°

Use the Law of Sines to find side b and c.

5

Find the remaining angle and sides of the triangle.

Example 1 (ASA):

The third angle in the triangle is A = 180° – A – B

= 180° – 10° – 60° = 110°

C

BA

b

c

60°

10°

a = 4.5 ft

110°

Use the Law of Sines to find side b and c.

4.15 ft

5

Find the remaining angle and sides of the triangle.

Example 1 (ASA):

The third angle in the triangle is A = 180° – A – B

= 180° – 10° – 60° = 110°

C

BA

b

c

60°

10°

a = 4.5 ft

110°

Use the Law of Sines to find side b and c.

4.15 ft

0.83 ft

6

Watch out for SSA!

6

Watch out for SSA!SSA is not a congruency property. SO, if we are given two sides and the NON-included angle in a triangle, there are three possible scenarios…

6

Watch out for SSA!SSA is not a congruency property. SO, if we are given two sides and the NON-included angle in a triangle, there are three possible scenarios…

➡ Option 1: No triangle is formed

6

Watch out for SSA!SSA is not a congruency property. SO, if we are given two sides and the NON-included angle in a triangle, there are three possible scenarios…

➡ Option 1: No triangle is formed➡ Option 2: One triangle is formed

6

Watch out for SSA!SSA is not a congruency property. SO, if we are given two sides and the NON-included angle in a triangle, there are three possible scenarios…

➡ Option 1: No triangle is formed➡ Option 2: One triangle is formed➡ Option 3: Two triangles are formed

6

Watch out for SSA!SSA is not a congruency property. SO, if we are given two sides and the NON-included angle in a triangle, there are three possible scenarios…

➡ Option 1: No triangle is formed➡ Option 2: One triangle is formed➡ Option 3: Two triangles are formed

How is it determined? If …

6

Watch out for SSA!SSA is not a congruency property. SO, if we are given two sides and the NON-included angle in a triangle, there are three possible scenarios…

➡ Option 1: No triangle is formed➡ Option 2: One triangle is formed➡ Option 3: Two triangles are formed

How is it determined? If …opp ≥ adj

0 or 1 triangle

6

Watch out for SSA!SSA is not a congruency property. SO, if we are given two sides and the NON-included angle in a triangle, there are three possible scenarios…

➡ Option 1: No triangle is formed➡ Option 2: One triangle is formed➡ Option 3: Two triangles are formed

How is it determined? If …opp ≥ adj opp ≤ adj

0 or 1 triangle 0 or 2 triangles

7

Use the Law of Sines to solve the triangle.A = 110°, a = 125 inches, b = 100 inches

Example 2 (SSA):

7

Use the Law of Sines to solve the triangle.A = 110°, a = 125 inches, b = 100 inches

Example 2 (SSA):

C

BA

b = 100 in

c

a = 125 in

110°

7

Use the Law of Sines to solve the triangle.A = 110°, a = 125 inches, b = 100 inches

Example 2 (SSA):

C

BA

b = 100 in

c

a = 125 in

110°

There is either 0 or 1 triangle satisfying the given conditions

because opp ≥ adj

7

Use the Law of Sines to solve the triangle.A = 110°, a = 125 inches, b = 100 inches

Example 2 (SSA):

C

BA

b = 100 in

c

a = 125 in

110° 48.74°

There is either 0 or 1 triangle satisfying the given conditions

because opp ≥ adj

7

Use the Law of Sines to solve the triangle.A = 110°, a = 125 inches, b = 100 inches

Example 2 (SSA):

C ≈ 180° – 110° – 48.74°

C

BA

b = 100 in

c

a = 125 in

110° 48.74°

21.26°

= 21.26°

There is either 0 or 1 triangle satisfying the given conditions

because opp ≥ adj

7

Use the Law of Sines to solve the triangle.A = 110°, a = 125 inches, b = 100 inches

Example 2 (SSA):

C ≈ 180° – 110° – 48.74°

C

BA

b = 100 in

c

a = 125 in

110° 48.74°

21.26°

48.23 in

= 21.26°

There is either 0 or 1 triangle satisfying the given conditions

because opp ≥ adj

8

Use the Law of Sines to solve the triangle.A = 76°, a = 18 inches, b = 20 inches

Example 3 (SSA):

8

Use the Law of Sines to solve the triangle.A = 76°, a = 18 inches, b = 20 inches

Example 3 (SSA):

There is either 0 or 2 triangles satisfying the given conditions

because opp ≤ adj

8

Use the Law of Sines to solve the triangle.A = 76°, a = 18 inches, b = 20 inches

Example 3 (SSA):

There is no angle whose sine is 1.078.

There is either 0 or 2 triangles satisfying the given conditions

because opp ≤ adjC

AB

b = 20 ina = 18 in

76°

9

Use the Law of Sines to solve the triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA):

9

Use the Law of Sines to solve the triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA):

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

9

Use the Law of Sines to solve the triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA):

There is either 0 or 2 triangles that can be formed (opp ≤ adj)

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

9

Use the Law of Sines to solve the triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA):

72.2°

There is either 0 or 2 triangles that can be formed (opp ≤ adj)

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

9

Use the Law of Sines to solve the triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA):

72.2°

There is either 0 or 2 triangles that can be formed (opp ≤ adj)

49.8°

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

C ≈ 180° – 58° – 72.2° = 49.8°

9

Use the Law of Sines to solve the triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA):

72.2°

10.3 cm

There is either 0 or 2 triangles that can be formed (opp ≤ adj)

49.8°

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

Example continues …

C ≈ 180° – 58° – 72.2° = 49.8°

10

Use the Law of Sines to solve the second triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA) continued:

72.2°

10.3 cm

49.8°

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

a=11.4 cm

10

Use the Law of Sines to solve the second triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA) continued:

C

AB2

b = 12.8 cm

c

a = 11.4 cm58°

72.2°

10.3 cm

49.8°

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

a=11.4 cm

10

Use the Law of Sines to solve the second triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA) continued:

B2 ≈ 180° – 72.2° = 107.8 °

107.8°

C

AB2

b = 12.8 cm

c

a = 11.4 cm58°

72.2°

10.3 cm

49.8°

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

a=11.4 cm

10

Use the Law of Sines to solve the second triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA) continued:

B2 ≈ 180° – 72.2° = 107.8 °

107.8°

C

AB2

b = 12.8 cm

c

a = 11.4 cm58°

14.2°

72.2°

10.3 cm

49.8°

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

C ≈ 180° – 58° – 107.8° = 14.2°

a=11.4 cm

10

Use the Law of Sines to solve the second triangle.A = 58°, a = 11.4 cm, b = 12.8 cm

Example 4 (SSA) continued:

B2 ≈ 180° – 72.2° = 107.8 °

107.8°

C

AB2

b = 12.8 cm

c

a = 11.4 cm58°

14.2°

3.3 cm

72.2°

10.3 cm

49.8°

a = 11.4 cm

C

AB1

b = 12.8 cm

c58°

C ≈ 180° – 58° – 107.8° = 14.2°

a=11.4 cm

11

Area of an Oblique Triangle

11

Area of an Oblique Triangle

11

Area of an Oblique Triangle

Find the area of the triangle.A = 74°, b = 103 inches, c = 58 inches

Example 5:

11

Area of an Oblique Triangle

C

BA

b

c

aFind the area of the triangle.A = 74°, b = 103 inches, c = 58 inches

Example 5:

74°

103 in

58 in

11

Area of an Oblique Triangle

C

BA

b

c

aFind the area of the triangle.A = 74°, b = 103 inches, c = 58 inches

Example 5:

74°

103 in

58 in

11

Area of an Oblique Triangle

C

BA

b

c

aFind the area of the triangle.A = 74°, b = 103 inches, c = 58 inches

Example 5:

74°

103 in

58 in

11

Area of an Oblique Triangle

C

BA

b

c

aFind the area of the triangle.A = 74°, b = 103 inches, c = 58 inches

Example 5:

74°

103 in

58 in

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

Flagpole height: b

14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

Flagpole height: b

14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

Flagpole height: b

14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

Flagpole height: b

16 m14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

Flagpole height: b

16 m14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b

16 m14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b70°

16 m14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b70°

34°

16 m14°

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b70°

34°

16 m14°

B

A

C

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b70°

34°

16 m14°

B

A

C

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b70°

34°

16 m14°

B

A

C

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b70°

34°

16 m14°

B

A

C

12

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°. How tall is the flagpole?

Application:

20°

Flagpole height: b70°

34°

16 m14°

The flagpole is approximately 9.5 meters tall.

B

A

C

Homework:

Assignment #2 p. 436 #7, 8, 13, 19-22, 29-31, 35, 36, 39, 40, 42