B. Triangles and Angles

1. B. Triangles and Angles Pre-Calculus 20 P20.4 Expand and demonstrate understanding of the primary trigonometric ratios including the use of reference angles (0° ≤ θ ≤ 360°) and the determination of exact values for trigonometric ratios. P20.5 Demonstrate understanding of the cosine law and sine law, including the ambiguous case.

2. Key Terms:

4. 1. Standard Position of Angles • P20.4 • Expand and demonstrate understanding of the primary trigonometric ratios including the use of reference angles (0° ≤ θ ≤ 360°) and the determination of exact values for trigonometric ratios.

5. 1. Standard Position of Angles • An Angle is formed by 2 rays with a common end point. • In a triangle angles are formed when on ray rotates. • The starting point is called the initial arm and the end point is called the terminal arm • When the angle rotation is counterclockwise then the angle are positive

7. Investigate p.75

8. Angles in Standard Position have their vertex at the origin and the initial arm on the positive x-axis

10. For each angle in standard position there is an acute angle called a Reference Angle • Reference angles are formed between the terminal arm and the x-axis • They are always positive and acute

11. The Trig ratios for the angles in Standard Position and its reference angle are the same (except the sign may change) • The right triangle that contains the Reference Angle and one leg on the x-axis is called the Reference Triangle

14. Special Right Triangles: • We have already looked at the 30-60-90 triangle.

15. There is also a 45-45-90 triangle.

16. Example 1

17. Example 2

18. Example 3

19. Example 4

20. Key Ideas:

21. Practice • Ex. 2.1 (p.83) #1-14 #6-20

22. 2. Trig Ratios and Angles • P20.4 • Expand and demonstrate understanding of the primary trigonometric ratios including the use of reference angles (0° ≤ θ ≤ 360°) and the determination of exact values for trigonometric ratios.

23. 2. Trig Ratios and Angles • Suppose θ is any angle in Standard Position, and P(x,y) is a point on the terminal arm at a distance r from the origin.

24. We can use the triangle to determine the 3 primary trig ratios in terms of x, y and r.

25. These will hold true for all four quadrants except the sign will change (+ or -) depending on the quad your are in.

26. This is referred to as the CAST Rule which indicates which trig ratios are + in which quadrants • All quadrants have 2 quadrants that they are positive in.

28. Example 1

29. Example 2 • Determine the exact value of sin240

30. Example 3

31. Example 4

33. Example 5

34. Example 6

35. Key Ideas

36. Practice • Ex. 2.2 (p.96) #1-16, 18, 19, 21 #1-9 odds in each, 10-25 odds

37. 3. The Law of Sines • P20.5 • Demonstrate understanding of the cosine law and sine law, including the ambiguous case.

38. Investigate the Law of Sinesp.100

39. The Law of Sines

40. Example 1

41. What is the distance between Friends Cabin and shore?

42. Example 2

43. The Ambiguous Case • When solving a triangle, we must analyze the info to determine the number solutions that are possible • If we are given ASA, then there is one unique triangle that exists • However, if we are given SSA, the Ambiguous Case may occur

44. In the Ambiguous Case there are 4 possible solutions • 1 oblique triangle • 1 right triangle • No Triangles • 2 Triangles

45. Lets take a look at the 4 possible answers given ∆ABC, ∠A, side b and side a

46. Example 3

47. Example 4

48. Key Ideas

50. Practice • Ex. 2.3 (p.108) #1-14 #4-9 odds in each, 11-13 odds