Trigonometry

1. Trigonometry

2. Describe what a bearing is.

3. Describe what a bearing is. • A bearing is a measurement of an angle from North in a clockwise direction.

4. Do you know how to write a vector?

5. Do you know how to write a vector? • A vector is writtenlike this

6. An example of a vector. 3 -4

7. An example of a vector. 3 -4 Vector

8. Problems

9. A taut guy wire to the top of a transmission mast is anchored in the same horizontal plane as the foot of the mast. The wire is 50 m long and makes an angle of 62 degrees with the horizontal. How far is the lower end of the wire from the foot of the mast?

10. Draw a diagram • A taut guy wire to the top of a transmission mast is anchored in the same horizontal plane as the foot of the mast. The wire is 50 m long and makes an angle of 62 degrees with the horizontal. How far is the lower end of the wire from the foot of the mast? 50 m 62 x

11. Solve using cosine 50 m 62 x

12. An aeroplane is flying at 300 km/hr. How far (a) north (b) East of its starting-point is the aeroplane after one hour if the direction of flight is (i) North; (ii)) N 40 degrees E; (iii) N 60 degrees E.

13. (i) • An aeroplane is flying at 300 km/hr. How far (a) north (b) East of its starting-point is the aeroplane after one hour if the direction of flight is (i) North; (ii)) N 40 degrees E; (iii) N 60 degrees E. (a) 300 km (b) 0 km

14. (a) (ii) • An aeroplane is flying at 300 km/hr. How far (a) north (b) East of its starting-point is the aeroplane after one hour if the direction of flight is (i) North; (ii)) N 40 degrees E; (iii) N 60 degrees E. 40 N 300

15. (a) (ii) • An aeroplane is flying at 300 km/hr. How far (a) north (b) East of its starting-point is the aeroplane after one hour if the direction of flight is (i) North; (ii)) N 40 degrees E; (iii) N 60 degrees E. 40 N 300

16. (a) (iii) • An aeroplane is flying at 300 km/hr. How far (a) north (b) East of its starting-point is the aeroplane after one hour if the direction of flight is (i) North; (ii)) N 40 degrees E; (iii) N 60 degrees E. 60 N 300

17. (b) (ii) • An aeroplane is flying at 300 km/hr. How far (a) north (b) East of its starting-point is the aeroplane after one hour if the direction of flight is (i) North; (ii)) N 40 degrees E; (iii) N 60 degrees E. E 40 300

18. (b) (iii) • An aeroplane is flying at 300 km/hr. How far (a) north (b) East of its starting-point is the aeroplane after one hour if the direction of flight is (i) North; (ii)) N 40 degrees E; (iii) N 60 degrees E. E 60 300

19. From a point 42 m above water-level at low tide the angle of depression of a buoy in the water was 57 degrees. At high tide the angle of depression was 55 degrees. Find the horizontal distance of the buoy from the viewer and the rise of the tide.

20. Draw a diagram • From a point 42 m above water-level at low tide the angle of depression of a buoy in the water was 57 degrees. At high tide the angle of depression was 55 degrees. Find the horizontal distance of the buoy from the viewer and the rise of the tide. 57 42 x

21. Draw a diagram • From a point 42 m above water-level at low tide the angle of depression of a buoy in the water was 57 degrees. At high tide the angle of depression was 55 degrees. Find the horizontal distance of the buoy from the viewer and the rise of the tide. 57 33 42 x

22. From a point 42 m above water-level at low tide the angle of depression of a buoy in the water was 57 degrees. At high tide the angle of depression was 55 degrees. Find the horizontal distance of the buoy from the viewer and the rise of the tide. 57 33 42 x

23. From a point 42 m above water-level at low tide the angle of depression of a buoy in the water was 57 degrees. At high tide the angle of depression was 55 degrees. Find the horizontal distance of the buoy from the viewer and the rise of the tide. 27.3 55 x 42

24. Fred wishes to estimate the height of a building. He steps out a distance of 60 m from the foot of the building and finds the angle of elevation of the top of the building is 38 degrees. Find the height of the building if his eyes are at a height of 1.7 m.

25. Draw a diagram • Fred wishes to estimate the height of a building. He steps out a distance of 60 m from the foot of the building and finds the angle of elevation of the top of the building is 38 degrees. Find the height of the building if his eyes are at a height of 1.7 m. h 38 60 1.7

26. Draw a diagram h 38 60 1.7

27. A step ladder with both sets of legs 3.5 m long and hinged at the top is tied with a rope to prevent the feet of the ladder being more than 1.7 m apart. What is the angle between the two parts when the feet are fully apart?

28. Draw a diagram • A step ladder with both sets of legs 3.5 m long and hinged at the top is tied with a rope to prevent the feet of the ladder being more than 1.7 m apart. What is the angle between the two parts when the feet are fully apart? 3.5 3.5 1.7

29. Draw a diagram x 3.5 3.5 3.5 0.85 1.7

30. x 3.5 3.5 3.5 0.85 1.7

31. From a point A on a straight and level road the angle of elevation of the top of a tower at the end of the road is 30 degrees. After walking along the road to B the angle of elevation of the top of the tower is 50 degrees. How long is AB if the tower is 45 m high?

32. Draw a diagram • From a point A on a straight and level road the angle of elevation of the top of a tower at the end of the road is 30 degrees. After walking along the road to B the angle of elevation of the top of the tower is 50 degrees. How long is AB if the tower is 45 m high? 45 30 50 x

33. Draw a diagram 45 30 50 z x y

34. x = 40.1 m 45 30 50 z x y

35. B Angle BAC is 32 degrees and AB and CB are 30 and 20 m resp. Find angle BCD A E C D

36. B Angle BAC is 32 degrees and AB and CB are 30 and 20 m resp. Find angle BCD 30 h 20 32 A E C D

37. B 30 h 20 32 A E C D

38. B 30 h 20 32 A E C D

39. From the top of a building 120 m above the ground the angles of depression of the top and bottom of another building are 40 and 70 degrees respectively. Find the distance apart of the buildings and the height of the lower one.

40. Draw a diagram • From the top of a building 120 m above the ground the angles of depression of the top and bottom of another building are 40 and 70 degrees respectively. Find the distance apart of the buildings and the height of the lower one. 70 40 120 h x

41. Find x 70 20 120 x

42. Find y 40 50 y 43.7 120 h 43.7

43. Two small pulleys are placed 8 cm apart in a horizontal line and an inextensible string of length 16 cm is placed over the pulleys. Equal masses hang symmetrically at each end of the string and the middle point is pulled down vertically until it is in line with the masses. How far does each mass rise?

44. Draw a diagram • Two small pulleys are placed 8 cm apart in a horizontal line and an inextensible string of length 16 cm is placed over the pulleys. Equal masses hang symmetrically at each end of the string and the middle point is pulled down vertically until it is in line with the masses. How far does each mass rise? 8 cm x y

45. Originally the masses are hanging down 4 cm. • Two small pulleys are placed 8 cm apart in a horizontal line and an inextensible string of length 16 cm is placed over the pulleys. Equal masses hang symmetrically at each end of the string and the middle point is pulled down vertically until it is in line with the masses. How far does each mass rise? 8 cm x y

46. Originally the masses are hanging down 4 cm. 8 cm x y

47. Originally the masses are hanging down 4 cm. 4 x y

48. Solve the equations by substitution 4 x y

49. 4 x y Height that it rises is 1 cm

50. In a right-angled triangle, one of the sides including the right angle is 7 cm longer than the other. If the perimeter is 40 cm, find the lengths of the three sides.