1. Digital Lesson
Law of Sines

2. Definition: Oblique Triangles
An oblique triangle is a triangle that has no right angles. C B A a b c
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.
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Definition: Oblique Triangles

3. Solving Oblique Triangles
The following cases are considered when solving oblique triangles.
Two angles and any side (AAS or ASA) A C c A B c
2. Two sides and an angle opposite one of them (SSA) C c a
3. Three sides (SSS) a c b c a B
4. Two sides and their included angle (SAS)
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Solving Oblique Triangles

4. Definition: Law of Sines
The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.)
Law of Sines
If ABC is an oblique triangle with sides a, b, and c, then C B A b h c a C B A b h c a
Acute Triangle
Obtuse Triangle
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Definition: Law of Sines

5. Example: Law of Sines - ASA
Example (ASA):
Find the remaining angle and sides of the triangle. C B A b c
60
10
a = 4.5 ft
The third angle in the triangle is A = 180 – A – B
= 180 – 10 – 60
= 110.
4.15 ft
110
0.83 ft
Use the Law of Sines to find side b and c.
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Example: Law of Sines - ASA

6. Example: Single Solution Case - SSA
Example (SSA):
Use the Law of Sines to solve the triangle.
A = 110, a = 125 inches, b = 100 inches C B A
b = 100 in c
a = 125 in
110
21.26
48.74
48.23 in
C  180 – 110 – 48.74
= 21.26
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Example: Single Solution Case - SSA

7. Example: No-Solution Case - SSA
Example (SSA):
Use the Law of Sines to solve the triangle.
A = 76, a = 18 inches, b = 20 inches C A B
b = 20 in
a = 18 in
76
There is no angle whose sine is
There is no triangle satisfying the given conditions.
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Example: No-Solution Case - SSA

8. Example: Two-Solution Case - SSA
Example (SSA):
a = 11.4 cm C A B1
b = 12.8 cm c
58
Use the Law of Sines to solve the triangle.
A = 58, a = 11.4 cm, b = 12.8 cm
49.8
72.2
10.3 cm
C  180 – 58 – 72.2 = 49.8
Two different triangles can be formed.
Example continues.
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Example: Two-Solution Case - SSA

9. Example: Two-Solution Case – SSA continued
Example (SSA) continued:
72.2
10.3 cm
49.8
a = 11.4 cm C A B1
b = 12.8 cm c
58
Use the Law of Sines to solve the second triangle.
A = 58, a = 11.4 cm, b = 12.8 cm
B2  180 – 72.2 = 
C  180 – 58 – 107.8 = 14.2 C A B2
b = 12.8 cm c
a = 11.4 cm
58
14.2
107.8
3.3 cm
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Example: Two-Solution Case – SSA continued

10. Area of an Oblique TriangleC B A b c a
Find the area of the triangle.
A = 74, b = 103 inches, c = 58 inches
Example:
103 in
74
58 in
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Area of an Oblique Triangle

11. The flagpole is approximately 9.5 meters tall.
Application:
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.
20 A
70
Flagpole height: b
34 B
16 m C
14
The flagpole is approximately 9.5 meters tall.
Copyright © by Brooks/Cole, Cengage Learning. All rights reserved.
Application