1. 15. 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. SAS
Use the law of cosines formula to find the side across from the angle you know
Then use the law of sines to find the smaller angle of the two that are left
Then subtract 2 angles you know from 180 to find the last angle

4. Deriving the Law of Cosines
Triangle ABC does not have a right angle so we drop a perpendicular from
Angle B (same as yesterday)
We call this h and now we have a right angle so we can use SOHCAHTOA B c a h r A C b
We also call the section on the bottom left r
Using SOHCAHTOA, sinA=h/c so h=c sinA
cosA=r/c so r=c cosA
Now use the Pythagorean theorem for triangle on the right (see next slide)

5. Deriving the law of cosines
This same process could be used for the other sides to come up with their law of cosines

6. Law of Cosines

7. Example (SAS)– solve the triangleC 15
26°
12.5
A B c

8. Example continued Now calculate the angles
You must do the smallest one first C 15
26°
12.5
A B
c = 6.65

9. SSS
If you are given all 3 sides, you must find the angle across from the largest side first!!!!
After that, you can use the law of sines to find another angle.
Then subtract the two angles you just found from 180 to find the last angle.

10. ExAMPLE (SSS)
Example Solve triangle ABC if a = 9.47 feet, b =15.9 feet, and c = 21.1 feet.

11. Example cont.
Use the law of sines or the law of cosines To show that C  70.1°. Then,

12. Summary of Cases with Suggested Procedures
Case 1:SAA or ASA
Suggested Procedure for Solving
Find the remaining angle using the angle sum formula (A + B + C = 180°).
Find the remaining sides using the law of sines.
Case 2: SSA
Suggested Procedure for Solving
This is the ambiguous case; 0, 1, or 2 triangles.
Find an angle using the law of sines.
Find the remaining angle using the angle sum formula.
Find the remaining side using the law of sines.
If two triangles exist, repeat steps 2 and 3.

13. Summary of Cases with Suggested Procedures
CASE 3: SAS
Suggested Procedure for Solving
Find the third side using the law of cosines.
Find the smaller of the two remaining angles using the law of sines.
Find the remaining angle using the angle sum formula.
CASE 4: SSS
Suggested Procedure for Solving
Find the largest angle using the law of cosines.
Find either remaining angle using the law of sines.
Find the remaining angle using the angle sum formula.

14. FINDING AREA FOR LAW OF COSINES
The law of cosines can be used to derive a formula for the area of a triangle given the lengths of three sides known as Heron’s Formula.
If a triangle has sides of lengths a, b, and c
and if the semiperimeter is
Then the area of the triangle is

15. Using Heron’s Formula to Find an Area
Example The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices?

16. 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