1. Re:view Use the Law of Sines to solve: Solve ABC
A=36º, a=10 meters, b=4 meters
A=24.3º, C=54.6º, c=2.68 cm
Solve ABC
32º 5 A B C
6.2 Law of Cosines

2. 6.2: Law of Cosines Objectives:
Use Law of Cosines to solve oblique triangles
Use Law of Cosines to model and solve real-life problems
Use Heron’s Area Formula to find areas of Triangles

3. Must haves for solving Oblique Triangles
2 angles and any side (AAS or ASA)
2 sides and an angle opposite one of them (SSA)
3 sides (SSS)
2 sides and their included angle (SAS)
The first two cases can be solved using the Law of Sines.
The last two cases can be solved using the Law of Cosines
6.2 Law of Cosines

4. Law of Cosines
Given an angle (A, B, C), the side length (a, b, c) is equal to the sum of the squares of the remaining two side lengths minus twice the product of the two side lengths and the cosine of the angle given. #WOW
6.2 Law of Cosines

5. Rules for Law of Cosines
a triangle can only have one obtuse angle.
if the largest angle is acute, the remaining 2 angles must also be acute
6.2 Law of Cosines

6. 3 Sides of a Triangle (SSS)
Use the Law of Cosines to find the angle opposite the longest side.
Use the Law of Sines to find one of the other angles.
Subtract the angles found in step 1 & 2 from 180º
6.2 Law of Cosines

7. Example: SSS Given: ABC Find: All 3 angles of the triangle B
c = 14 ft
b = 19 ft
a = 8 ft
6.2 Law of Cosines

8. 2 Sides and the Included Angle (SAS)
Use the Law of Cosines to find the unknown side
Use the Law of Sines to find one of the missing angles
Find the last angle by subtracting the 2 known angles from 180°
6.2 Law of Cosines

9. Example SAS Given: ABC with A=115º
Find: The remaining 2 angles of the triangle
15 cm
10 cm C B A
6.2 Law of Cosines

10. Re:view
Write the three standard form equations for the Law of Cosines.
Write the formulas for Law of Sines.
What is the area of a triangle?
How else can we re-write the area of a triangle using trig?
6.2 Law of Cosines

11. Heron’s Area Formula
6.2 Law of Cosines

12. Example: Heron’s Formula
Find the area of the triangular region having sides of lengths a=43 meters, b=53 meters, and c=72 meters.
Solve for s.
Apply the side lengths and the length of s to Heron’s formula to solve for the area of the triangle.
6.2 Law of Cosines

13. Example: SSS #2
Given: On a map, Orlando is 178 millimeters due south of Niagra Falls. Denver is 273 millimeters from Orlando, and Denver is 235 millimeters from Niagra Falls.
Find: the bearing of Denver from Orlando and the bearing from Denver to Niagra Falls.
6.2 Law of Cosines

14. Example: SAS #2
Given: A plane flies 810 miles from A to B with a bearing of N75°E. Then it flies 648 miles from B to C with a bearing of N32°E. Draw a diagram to represent the problem and find the straight line distance and bearing from C and A.
6.2 Law of Cosines

15. Homework
Check Blackboard and mrtower.wordpress.com
6.2 Law of Cosines