1. Law of Sines and 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.
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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)
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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
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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 – C – B
= 180 – 10 – 60
= 110.
Use the Law of Sines to find side b and c.
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Example: Law of Sines - ASA

6. The Ambiguous Case (SSA) Law of Sines
In earlier examples, you saw that two angles and one side determine a unique triangle.
However, if two sides and one opposite angle are given, then three possible situations can occur: (1) no such triangle exists,
(2) one such triangle exists, or
(3) two distinct triangles satisfy the
conditions.

7. The Ambiguous Case (SSA)

8. Example 3 – Single-Solution Case—SSA
For the triangle in Figure 6.5, a = 22 inches, b = 12 inches, and A = 42 . Find the remaining side and angles.
One solution: a  b
Figure 6.5

9. Now you can determine that
Example 3 – Solution
cont’d
Check for the other “potential angle”
C  180  –  =
( = which is more than 180  so there
is only one triangle.)
Now you can determine that
C  180  – 42  –  = 
Multiply each side by b
Substitute for A, a, and b.
B is acute.

10. Multiply each side by sin C.
Example 3 – Solution
cont’d
Then the remaining side is given by
Law of Sines
Multiply each side by sin C.
Substitute for a, A, and C.
Simplify.

11. 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
C  180 – 110 – 48.74
= 21.26
Since the “given angle is already obtuse, there will only be one triangle.
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Example: Single Solution Case - SSA

12. 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.
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Example: No-Solution Case - SSA

13. 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
C  180 – 58 – 72.2 = 49.8
Check for the other “potential angle”
C  180  – 72.2  = 
(107.8  + 58  =  which is less than 180  so there are two triangles.)
Example continues.
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Example: Two-Solution Case - SSA

14. 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
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Example: Two-Solution Case – SSA continued

15. Definition: Law of Cosines
(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
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Definition: Law of Cosines

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

17. Example: Law of Cosines - SASB A
6.2
75
9.5
Solve the triangle.
Law of Cosines:
Law of Sines:
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Example: Law of Cosines - SAS