1. 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? 15
No side opposite from any angle to get the ratio
26°
12.5

3. Deriving the Law of Cosines
Write an equation using Pythagorean theorem for shaded triangle.
b h a
k c - k
A B c

4. Law of Cosines
Similarly
Note the pattern

5. Applying the Cosine Law
Now use it to solve the triangle we started with
Label sides and angles
Side c first C 15
26°
12.5
A B c

6. Applying the Cosine Law
Now calculate the angles
use and solve for B C 15
26°
12.5
A B
c = 6.65

7. Applying the Cosine Law
The remaining angle determined by subtraction
180 – – 26 = 60.25 C 15
26°
12.5
A B
c = 6.65
Experiment with Cosine Law Spreadsheet

8. Wing Span C
The leading edge of each wing of the B-2 Stealth Bomber measures feet in length. The angle between the wing's leading edges is °. What is the wing span (the distance from A to C)?
Hint … use the law of cosines! A

9. Using the Cosine Law to Find Area
Recall that
We can use the value for h to determine the area C
b h a
A B c

10. Using the Cosine Law to Find Area
We can find the area knowing two sides and the included angle
Note the pattern C
b a
A B c

11. Try It Out Determine the area of these triangles 42.8° 127° 76.3° 17.924 12
76.3°

12. Cost of a Lot
An industrial piece of real estate is priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet.
324
412
516