Real life application of angle of elevation and depression.pptx

A lengthy presentation which is a compilation of several areas of trigonometry. Application to real life, angles of elevation and depression.

1. Application of Trigonometry Angles of Elevation and Depression

2. Have you ever seen this instrument?

3. What is it called? It’s a Theodolite !!!

4. Using a Theodolite

5. Surveyors

6. Looking through a Theodolite

7. Looking through a Theodolite

8. Looking through a Theodolite

9. Looking through a Theodolite

10. FACTS Surveyors use two instruments, the transit and the theodolite , to measure angles of elevation and depression. On both instruments , the surveyor sets the horizon line perpendicular to the direction of gravity. Using gravity to find the horizon line ensures accurate measures even on sloping surfaces, industrial and commercial buildings, when planning to set out roads, driveways, retaining walls and site grading.

11. You Need-to-Remember Sin θ = Opposite / Hypotenuse Cos θ = Adjacent / Hypotenuse Tan θ = Opposite / Adjacent To find an angle use inverse Trig Function Trig Fnc -1 (some side / some other side) = angle To Solve Any Trig Word Problem Step 1: Draw a triangle to fit problem Step 2: Label sides from angle’s view Step 3: Identify trig function to use Step 4: Set up equation Step 5: Solve for variable Θ Angle of Elevation or of Depression angle goes here x 33 ° 25 y ° z

12. Example Job Site A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What angle does the ladder make with the ground? x° Step 1: Draw a triangle to fit problem 8 20 Step 2: Label sides from angle’s view adj hyp Step 3: Identify trig function to use S  O / H C  A / H T  O / A Step 4: Set up equation 8 cos x ° = ----- 20 Step 5: Solve for variable cos -1 (8/20) = x x= 66.42 °

13. eye – level/horizontal line of sight

14. eye – level/horizontal line of sight

15. eye – level/horizontal line of sight

16. eye – level/horizontal line of sight

17. eye – level/horizontal line of sight

18. eye – level/horizontal line of sight

19. eye – level/horizontal line of sight

20. eye – level/horizontal line of sight

21. eye – level/horizontal line of sight

22. eye – level/horizontal line of sight

23. eye – level/horizontal line of sight

24. eye – level/Horizontal line of sight

25. eye – level/horizontal line of sight eye – level/horizontal line of sight

26. eye – level/horizontal line of sight eye – level/horizontal line of sight

27. The angles are equal – they are alternate angles eye – level/horizontal line of sight eye – level/horizontal line of sight

28. angle of elevation? angle of depression ? Definition

29. An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram,  1 is the angle of elevation from the tower T to the plane P . An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line.  2 is the angle of depression from the plane to the tower. Definition

30. Definition Angle of Elevation The angle between a horizontal line and the line joining the observer’s eye to some object above the horizontal line is called the angle of elevation.

31. Definition Angle of Depression The angle between a horizontal line and the line joining the observer’s eye to some object below the horizontal line is called the angle of depression.

32. Identifying Angles of Elevation and Depression

33. Example 1: Classifying Angles of Elevation and Depression Classify each angle as an angle of elevation or an angle of depression.  1  3  1 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. 3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.

34. Example 2: Classifying Angles of Elevation and Depression Classify each angle as an angle of elevation or an angle of depression. 2 4 2 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation. 4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

35. Example 2: Classifying Angles of Elevation and Depression Classify each angle as an angle of elevation or an angle of depression.  4  4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.

36. LETS GO!!!!! Use the diagram above to classify each angle as an angle of elevation or angle of depression. 3a.  5 3b.  6  6 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation.  5 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression.

37. CHECK OUT THIS Example A man looks out to sea from a cliff top at a height of 12 metres. He sees a boat that is 150 metres from the cliffs. What is the angle of depression Solution The situation can be represented by the triangle shown in the diagram, where θ is the angle of depression. Using ? ? ? ? to 1 decimal place

38. CHECK OUT THIS Example A ladder is 3.5 metres long. It is placed against a vertical wall 1.75m so that its foot is on horizontal ground is away from the wall. Draw a diagram which represents the information given. Calculate ( i ) the angle of elevation from the ground to the top of the wall. (ii) the angle of depression from the ground to the top of the wall. 5 m 3m

39. Check It Out ! #1 What if…? Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the tower to the plane is 29°. What is the horizontal distance between the plane and the Tower? Simplify the expression.

40. Check It Out ! #1 (cont) What if…? Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the tower to the plane is 29°. What is the horizontal distance between the plane and the Tower? (to the nearest ft) Simplify the expression.

41. Check It Out ! #1 SOLUTION 3500 ft 29° You are given the side opposite A, and x is the side adjacent to A. So write a tangent ratio. Multiply both sides by x and divide by tan 29° . x  6314 ft Simplify the expression.

42. Check It Out! #2 What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? ( to the nearest foot) By the Alternate Interior Angles Theorem,  F = 3°. Write a tangent ratio. Multiply both sides by x and divide by tan 3°. x  1717 ft Simplify the expression. 3°

43. Check It Out! #3 You sight a rock climber on a cliff at a 32 o angle of elevation. The horizontal ground distance to the cliff is 1000 ft. Find the line of sight distance to the rock climber. 1000 ft x

44. Check It Out! #4 An airplane pilots sights a life raft at a 26 o angle of depression. The airplane’s altitude is 3 km. What is the airplane’s surface distance d from the raft? 3 km d

45. Check It Out! #2 Meteorology One method that meteorologists could use to find the height of a layer of clouds above the ground is to shine a bright spotlight directly up onto the cloud layer and measure the angle of elevation from a known distance away.

46. Find the height of the cloud layer in the diagram to the nearest 10 m. Check It Out! #2 (cont) Meteorology

47. Example 6: Finding Distance by Using Angle of Depression An ice climber stands at the edge of a crevasse that is 115 ft wide. The angle of depression from the edge where she stands to the bottom of the opposite side is 52º. How deep is the crevasse at this point?

48. Example 6 Continued Draw a sketch to represent the given information. Let C represent the ice climber and let B represent the bottom of the opposite side of the crevasse. Let y be the depth of the crevasse.

49. Example 6 Continued By the Alternate Interior Angles Theorem, m  B = 52°. Write a tangent ratio. y = 115 tan 52° Multiply both sides by 115 . y  147 ft Simplify the expression.

50. Example 8: Shipping Application An observer in a lighthouse is 69 ft above the water. He sights two boats in the water directly in front of him. The angle of depression to the nearest boat is 48º. The angle of depression to the other boat is 22º. What is the distance between the two boats? Round to the nearest foot.

51. Example 8 Application Step 1 Draw a sketch. Let L represent the observer in the lighthouse and let A and B represent the two boats. Let x be the distance between the two boats.

52. Example 8 Continued Step 2 Find y . By the Alternate Interior Angles Theorem, m  CAL = 58°. . In ∆ ALC, So

53. Step 3 Find z . By the Alternate Interior Angles Theorem, m  CBL = 22°. Example 8 Continued In ∆ BLC, So

54. Step 4 Find x . So the two boats are about 109 ft apart. Example 8 Continued x = z – y x  170.8 – 62.1  109 ft

55. Check It Out! Example 9 A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot.

56. Step 1 Draw a sketch. Let P represent the pilot and let A and B represent the two airports. Let x be the distance between the two airports. Check It Out! Example 9 Continued 78° 19° 78° 19° 12,000 ft

57. Step 2 Find y . By the Alternate Interior Angles Theorem, m  CAP = 78°. Check It Out! Example 9 Continued In ∆ APC, So

58. Step 3 Find z . By the Alternate Interior Angles Theorem, m  CBP = 19°. Check It Out! Example 9 Continued In ∆ BPC, So

59. Step 4 Find x . So the two airports are about 32,300 ft apart. Check It Out! Example 9 Continued x = z – y x  34,851 – 2551  32,300 ft

60. Homework Pg 558 #1-9 Pg 579 extra practice #1-12

61. Lesson Quiz: Part I Classify each angle as an angle of elevation or angle of depression. 1.  6 2.  9 angle of depression angle of elevation

62. Lesson Quiz: Part II 3. A plane is flying at an altitude of 14,500 ft. The angle of depression from the plane to a control tower is 15°. What is the horizontal distance from the plane to the tower? Round to the nearest foot. 4. A woman is standing 12 ft from a sculpture. The angle of elevation from her eye to the top of the sculpture is 30°, and the angle of depression to its base is 22°. How tall is the sculpture to the nearest foot? 54,115 ft 12 ft

63. Problem 1 Julian is at the base of the building and he wishes to know its height. He walks along to a point 90 ft from the base of the building, and from that point he measures the angle of elevation of the top of the building to be 50 °. What is the height of the building? Round off answer to the nearest whole number.

64. Solution Let h = height of the building The height of the building is 107ft.

65. Problem 2 If a kite is 150 ft high and when 800 ft of string is out, what is the measure of the angle does the kite make with the ground?

66. Solution Let - angle that the kite make with the ground The angle that the kite make with the ground is 11°.

67. Problem 3 A 37 ft flag pole casts a 21 ft shadow. What is the angle of elevation of the sun? Round off your answer to the nearest whole number and let theta be the angle of elevation of the sun.

68. Solution Let - angle of elevation of the sun The angle of elevation of the sun is 60°.

69. Problem 4 From the top of a 115 ft lighthouse, the angle of depression of a boat on the sea is 10°15’. Find the distance of the boat from the base of the lighthouse. Let x be the distance of the boat from the boat of the lighthouse

70. Problem 5 Aris stands 105 ft away from the base of a tree. He measures the angle of elevation to the top of the tree to be 72°. How tall is the tree? Let h be the height of the tree.