1. 4.1 – Right Triangle Trigonometry

2. Sin Ɵ = opp hyp

3. Sin Ɵ = opp hyp Csc Ɵ = hyp opp

4. Sin Ɵ = opp hyp Cos Ɵ = adj hyp Csc Ɵ = hyp opp

5. Sin Ɵ = opp hyp Cos Ɵ = adj hyp Csc Ɵ = hyp opp Sec Ɵ = hyp adj

6. Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj

7. Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp

8. Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp Csc = 1 sin

9. Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp Csc = 1 sin Sec = 1 cos

10. Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp Csc = 1 sin Sec = 1 cos Cot = 1 tan

11. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Ɵ 8 17 15

12. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp hyp Ɵ 8 17 15

13. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 hyp 17 Ɵ 8 17 15

14. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp hyp 17 opp Ɵ 8 17 15

15. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Ɵ 8 17 15

16. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj hyp Ɵ 8 17 15

17. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15 hyp 17 Ɵ 8 17 15

18. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp hyp 17 adj Ɵ 8 17 15

19. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Ɵ 8 17 15

20. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp adj Ɵ 8 17 15

21. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp = 8 adj 15 Ɵ 8 17 15

22. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp = 8 Cot Ɵ = adj adj 15 opp Ɵ 8 17 15

23. Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp = 8 Cot Ɵ = adj = 15 adj 15 opp 8 Ɵ 8 17 15

24. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ.

25. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Ɵ

26. Ɵ 2

27. Ɵ 2 5

28. Ɵ √21 2 5

29. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp hyp Ɵ √21 2 5

30. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp = √21 hyp 5 Ɵ √21 2 5

31. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp adj Ɵ √21 2 5

32. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

33. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp opp Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

34. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ opp √21 Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

35. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

36. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp hyp 5 adj Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

37. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

38. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 Cot Ɵ = adj adj 2 opp Ɵ √21 2 5

39. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 Cot Ɵ = adj = 2_ adj 2 opp √21 Ɵ √21 2 5

40. Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 Cot Ɵ = adj = 2_ = 2√21 adj 2 opp √21 21 Ɵ √21 2 5

41. Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. 42 18 x

42. Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp 42 18 x

43. Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp Cos 42° = x 18 42 18 x

44. Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp Cos 42° = x 18 18Cos 42° = x 42 18 x

45. Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp Cos 42° = x 18 18Cos 42° = x 13.4 ≈ x 42 18 x

46. Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Ɵ 26 11

47. Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Ɵ 26 11

48. Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Tan Ɵ = 26 11 Ɵ 26 11

49. Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Tan Ɵ = 26 11 Ɵ = tan -1 ( 26 / 11 ) Ɵ 26 11

50. Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Tan Ɵ = 26 11 Ɵ = tan -1 ( 26 / 11 ) Ɵ ≈ 67° Ɵ 26 11

51. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 35° x 10 z Y ZX

52. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj 35° x 10 z Y ZX

53. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj Tan 35° = x 10 35° x 10 z Y ZX

54. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj Tan 35° = x 10 10Tan 35° = x 35° x 10 z Y ZX

55. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj Tan 35° = x 10 10Tan 35° = x x ≈ 7.0 35° x 10 z Y ZX

56. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x 10 10Tan 35° = x x ≈ 7.0 35° x 10 z Y ZX

57. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = x x ≈ 7.0 35° x 10 z Y ZX

58. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ x ≈ 7.0 Cos 35° 35° x 10 z Y ZX

59. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ ≈ 12.2 x ≈ 7.0 Cos 35° 35° x 10 z Y ZX

60. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj35° + Y = 90 ° adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ ≈ 12.2 x ≈ 7.0 Cos 35° 35° x 10 z Y ZX

61. Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj35° + Y = 90 ° adj hypY = 55° Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ ≈ 12.2 x ≈ 7.0 Cos 35° 35° x 10 z Y ZX