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? 2 15 12.5 26° No side opposite from any angle to get the ratio

3. LAW OF COSINES 3

4. EXAMPLE (SAS)– SOLVE THE TRIANGLE 4 15 12.5 26° A B C c

5. EXAMPLE CONTINUED Now calculate the angles use and solve for B 5 15 12.5 26° A B C c = 6.65

6. EXAMPLE CONTINUED The remaining angle determined by subtraction 180 – 93.75 – 26 = 60.25 6 15 12.5 26° A B C c = 6.65

7. WING SPAN The leading edge of each wing of the B-2 Stealth Bomber measures 105.6 feet in length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)? 7 A C

8. EXAMPLE (SSS) – SOLVE THE TRIANGLE Example Solve triangle ABC if a = 9.47 feet, b =15.9 feet, and c = 21.1 feet. It is a good idea to first find the angle opposite the the longest side- side b in this case.

9. EXAMPLE CONT. Verify with either the law of sines or the law of cosines that B  45.1°. Then,

10. SUMMARY OF CASES WITH SUGGESTED PROCEDURES Case 1:SAA or ASASuggested Procedure for Solving 1.Find the remaining angle using the angle sum formula (A + B + C = 180°). 2.Find the remaining sides using the law of sines. Case 2: SSASuggested Procedure for Solving This is the ambiguous case; 0, 1, or 2 triangles. 1.Find an angle using the law of sines. 2.Find the remaining angle using the angle sum formula. 3.Find the remaining side using the law of sines. If two triangles exist, repeat steps 2 and 3.

11. SUMMARY OF CASES WITH SUGGESTED PROCEDURES CASE 3: SASSuggested Procedure for Solving 1.Find the third side using the law of cosines. 2.Find the smaller of the two remaining angles using the law of sines. 3.Find the remaining angle using the angle sum formula. CASE 4: SSSSuggested Procedure for Solving 1.Find the largest angle using the law of cosines. 2.Find either remaining angle using the law of sines. 3.Find the remaining angle using the angle sum formula.

12. 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

13. 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?

14. 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. 14 516 412 324