1. The Law of Cosines
What you’ll learn
Use the Law of Cosines to solve oblique triangles.
Solve applied problems using the Law of Cosines.

2. Solving an SAS Triangle
The Law of Sines was good for
ASA - two angles and the included side
AAS - two angles and any side
SSA - two sides and an opposite angle (being aware of possible ambiguity)
Why would the Law of Sines not work for an SAS triangle?
No side opposite from any angle to get the ratio 15
26°
12.5

3. Deriving the Law of Cosines
Split up the triangle into two right triangles by drawing the altitude from C to 𝐴𝐵
Set up ratios from Angle A
𝑆𝑖𝑛 𝐴= ℎ 𝑏 (Solve for h)
𝐶𝑜𝑠 𝐴= 𝑘 𝑏 (Solve for k)
3. Now I use the Pythagorean theorem on the shaded triangle
𝑎 2 = ℎ 2 + (𝑐−𝑘) 2 C
b h a
k c - k
A B c

5. b2 = a2 + c2 -2ac cosB a2 = b2 + c2 -2bc cosA a = 37.9 cm
SAS Applying the Law of Cosines
b2 = a2 + c2 -2ac cosB
= (230)2 + (150)2 - 2(230)(150)cos430
b = m
a2 = b2 + c2 -2bc cosA
= (61)2 + (43)2 - 2(61)(43)cos380
a = cm

6. Solving an SAS Triangle

7. Answer
Find acute
angle B.

8. Solving an SSS Triangle:

9. Answer

10. Answer

11. Example 4: A ship travels 60 miles due east to point B, then adjusts its course northward. After traveling 80 miles in that direction to point C, the ship is 139 miles from its point of departure. Describe the bearing from point B to point C.