1. Law of Cosines HOMEWORK: Lesson 12.4/1-14

2. 2 Who's Law Is It, Anyway?  Murphy's Law: Anything that can possibly go wrong, will go wrong (at the worst possible moment).  Cole's Law ?? Finely chopped cabbage

3. 3 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 12.5 26° No side opposite from any angle to get the ratio

4. 4 Law of Cosines  Note the pattern A B C a c b

5. We could do the same thing if gamma was obtuse and we could repeat this process for each of the other sides. We end up with the following: LAW OF COSINES Use these to find missing sides Use these to find missing angles

6. 6 Applying the Cosine Law  Now use it to solve the triangle we started with  Label sides and angles Side c first 15 12.5 26° A B C c

7. 7 Applying the Cosine Law  Now calculate the angles use and solve for B 15 12.5 26° A B C c = 6.65

8. 8 Applying the Cosine Law  The remaining angle determined by subtraction 180 – 93.75 – 26 = 60.25 15 12.5 26° A B C c = 6.65

9. Solve a triangle where b = 1, c = 3 and  = 80° Draw a picture. 80   a 1 3 Do we know an angle and side opposite it? No so we must use Law of Cosines. Hint: we will be solving for the side opposite the angle we know. This is SAS a = 2.99

10. Solve a triangle where a = 5, b = 8 and c = 9 Draw a picture.   5 8 9 Do we know an angle and side opposite it? No, so we must use Law of Cosines. Let's use largest side to find largest angle first. This is SSS  84.3

11. 11   5 8 9  84.3

12. 12 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)?  Hint … use the law of cosines! A C

13. 13 105.6 ft 109.05° C A x B 105.6 ft

14. 14 Using the Cosine Law to Find Area  Recall that  We can use the value for h to determine the area b h a A B c C

15. 15 Using the Cosine Law to Find Area  We can find the area knowing two sides and the included angle  Note the pattern b a A B c C

16. Determine the area 16 Try It Out 127° 12m 24m

17. 17 76.3° 42.8° 17.9 Determine the area Missing angle – 180-42.8-76.3 = 60.9° 60.9° Missing side

18. 18 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. 516 412 324

19. 19 516 412 324

20. We'll label side a with the value we found. We now have all of the sides but how can we find an angle? 80   2.99 1 3 Hint: We have an angle and a side opposite it. 80.8  is easy to find since the sum of the angles is a triangle is 180° 19.2