1. Chapter 5 Analytic Trigonometry Pre-Calculus OHHS Mr. J. Focht

2. 5.5 Law of Sines Investigating the Law of Sines –Solving Triangles (AAS, ASA) –The Ambiguous Case (SSA) –Applications 5-5

3. Solving Triangles If all triangles were right triangles, we could end the chapter now. 5-5

4. Law of Sines Simple trig and the Pythagorean Theorem won’t work on this triangle. a A b B c C 5-5

5. Solve the triangle. Find the missing pieces. A B C 22 112º 29º ? ? ? We are given 2 angles and a non-included side. AAS 5-5

6. Law of Sines Solve the triangle. A B C a 22 c 112º 29º Set up a proportion with 1 missing entry. Solve for a. a = 42.142.1 5-5

7. Law of Sines Solve for C C = 180º - 29º - 112º = 39º A B C 42.1 22 c 112º 29º 39º Use Law of Sines = 28.6 28.6 5-5

8. Now You Try Solve this triangle 5-5

9. Law of Sines - ASA 20  132  46AB C a b C = 180  - 20  - 132  = 28  28  When possible, find the largest side first. 72.8 5-5

10. Law of Sines - ASA 20  132  46AB C b 28  72.8 33.5 5-5

11. Now You Try 22 5-5

12. Law of Sines - SSA Suppose you were given 2 sides and an angle (non-included) to make a triangle. Angle A must be across from side a. Could you do it? Maybe – it depends on the measurements. Certain measurements would allow us to make 1 triangle, or 2 triangles, or no triangles. b a A 5-5

13. How Many Solutions? 63º A a = 18 b = 25 Find the height h=b sin A h h=25 sin 63º = 22.3 a is too short to fit. No Answer! The height is the shortest side to make the triangle. 22.3 5-5

14. 105º A b = 55 How Many Solutions? a = 73 Since a > b, we can form 1, and only 1, triangle. 5-5

15. 72.2º A b = 22 a = 21 Find All Solutions First find the height. h= b sinA Since h < a, a will fit. At least 1 triangle can be formed. But this is not the only way to draw it. Hint – don’t leave out the lefties h = 20.95 5-5

16. 72.2º A b = 22 a = 21 Find All Solutions Use your left hand to draw it. h = 20.95 a = 21 Bad news – You have 2 sets of solutions to solve 72.2º A b = 22 a = 21 h = 20.95 5-5

17. 72.2º A b = 22 a = 21 First Solution Using the Law of Sines C B c 5-5

18. 72.2º A b = 22 a = 21 c Second Solution B C Using the Law of Sines But we can see B is obtuse. B = 180º - 85.9º = 94.1º 5-5

19. Now You Try How many solutions? Do not solve. 1) A = 36 , a = 2, b =7 2) C = 36 , a =17, c = 16 3) C = 30 , a =18, c = 9 5-5

20. Now You Try Find all solutions. A = 64 , a =16, b =17 5-5

21. Story Problem A baseball fan is sitting directly behind home plate in the last row of the upper deck of Comiskey Park in Chicago. The angle of depression to home plate is 29º 54’, and the angle of depression to the pitcher’s mound is 24º 12’. In major league baseball, the distance between home plate and the pitcher’s mound is 60.5 feet. How far is the fan from home plate? What is our first step? 5-5

22. Draw a Picture First find θ θ=29º54’ - 24º12’ = 5º 42’ 5-5

23. Use Law of Sines d=249.7020342 d ≈ 249.7 5-5

24. Now You Try Surveying a Canyon Two markers A and B on the same side of a canyon rim are 56 ft apart. A third marker C, located across the rim, is positioned so that  BAC = 72  and  ABC = 53 . (a) Find the distance between C and A. (b) Find the distance between the two canyon rims. (Assume they are parallel.) 51.9 ft 5-5

25. Last Word What about SSS and SAS ? They can’t be solved using the Law of Sines. We will need to learn something else. The Law of Cosines 5-5

26. Home Work P. 484, #2, 6, 10, 14, 16, 18, 20, 22, 24, 25, 26, 30, 40, 47- 52 5-5