1. Law of Sines & Law of Cosines
Sections 6-1/2
Law of Sines & Law of Cosines

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.
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)
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
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 Houghton Mifflin Company, Inc. 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
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.
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 Houghton Mifflin Company, Inc. 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 Houghton Mifflin Company, Inc. 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
Area of an Oblique Triangle

11. 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)
Solving Oblique Triangles

12. Definition: Law of Cosines
The last two cases (SSS and SAS) can be solved using the Law of Cosines.
(The first two cases can be solved using the Law of Sines.)
Law of Cosines
Standard Form
Alternative Form
Definition: Law of Cosines

13. Example: Law of Cosines - SSS
Find the three angles of the triangle. C B A 8 6 12
117.3
26.4
36.3
Find the angle opposite the longest side first.
Law of Sines:
Example: Law of Cosines - SSS

14. Example: Law of Cosines - SASB A
6.2
75
9.5
Solve the triangle.
67.8
9.9
Law of Cosines:
37.2
Law of Sines:
Example: Law of Cosines - SAS

15. Homework
WS 13-1