Here are the key points about true bearings: - True bearings are always measured clockwise from North - To find the bearing of point B from point A, start at North and measure the angle clockwise to the line from A to B - The bearing of Y from X is 120° or S60°E - The bearing of X from Y would be N60°W So in summary, true bearings provide a unambiguous way to specify the direction from one point to another by always measuring clockwise from North.

1. Trigonometric Ratios

2. Trigonometric Ratios
 hypotenuse
adjacent
opposite

3. Trigonometric Ratios
opp
sin  
 hypotenuse hyp
adjacent adj
cos  
hyp
opp
opposite tan  
adj

4. Trigonometric Ratios hyp
opp
sin   cosec 
 hypotenuse hyp opp
adjacent adj hyp
cos   sec 
hyp adj
opp adj
opposite tan   cot  
adj opp

5. Trigonometric Ratios hyp
opp
sin   cosec 
 hypotenuse hyp opp
adjacent adj hyp
cos   sec 
hyp adj
opp adj
opposite tan   cot  
adj opp
e.g.  i  sin x  cos 25

6. Trigonometric Ratios hyp
opp
sin   cosec 
 hypotenuse hyp opp
adjacent adj hyp
cos   sec 
hyp adj
opp adj
opposite tan   cot  
adj opp
e.g.  i  sin x  cos 25
x  90  25
x  65

7. Trigonometric Ratios hyp
opp
sin   cosec 
 hypotenuse hyp opp
adjacent adj hyp
cos   sec 
hyp adj
opp adj
opposite tan   cot  
adj opp
e.g.  i  sin x  cos 25  ii  cot  x  20   tan  x  30 
x  90  25
x  65

8. Trigonometric Ratios hyp
opp
sin   cosec 
 hypotenuse hyp opp
adjacent adj hyp
cos   sec 
hyp adj
opp adj
opposite tan   cot  
adj opp
e.g.  i  sin x  cos 25  ii  cot  x  20   tan  x  30 
x  90  25 x  20  x  30  90
x  65 2 x  80
x  40

9.  iii 
a
61
13

10. a
 iii   tan 61
13
a
61
13

11. a
 iii   tan 61
13
a a  13tan 61
61 a  23.5 units (to 1 dp)
13

12. a
 iii   tan 61
13
a a  13tan 61
61 a  23.5 units (to 1 dp)
13
 iv 
32 x
5

13. a
 iii   tan 61
13
a a  13tan 61
61 a  23.5 units (to 1 dp)
13
 iv  5
 sin 32
x
32 x
5

14. a
 iii   tan 61
13
a a  13tan 61
61 a  23.5 units (to 1 dp)
13
 iv  5
 sin 32
x 5
32 x x
sin 32
x  9.4 units (to 1 dp)
5

15. a
 iii   tan 61
13
a a  13tan 61
61 a  23.5 units (to 1 dp)
13
 iv  5
 sin 32
x 5
32 x x
sin 32
x  9.4 units (to 1 dp)
5
v
14

10

16. a
 iii   tan 61
13
a a  13tan 61
61 a  23.5 units (to 1 dp)
13
 iv  5
 sin 32
x 5
32 x x
sin 32
x  9.4 units (to 1 dp)
5
10
v cos  
14
14

10

17. a
 iii   tan 61
13
a a  13tan 61
61 a  23.5 units (to 1 dp)
13
 iv  5
 sin 32
x 5
32 x x
sin 32
x  9.4 units (to 1 dp)
5
10
v cos  
14
14 10
  cos 1
14
   44 25
10

18. Exact Ratios
60
60 60

19. Exact Ratios
60
2 2
60 60
2

20. Exact Ratios
30 2
3
60
1

21. Exact Ratios
1
sin 30 
2
30 2 3
3 cos30 

2
60 tan 30 
 1
1 3

22. Exact Ratios
1 3
sin 30  sin 60 

2 2
30 2 3
cos 60 
 1
3 cos30 

2 2
60 tan 30 
 1
tan 60  3
1 3

23. Exact Ratios
1 3
sin 30  sin 60 

2 2
30 2 3
cos 60 
 1
3 cos30 

2 2
60 tan 30 
 1
tan 60  3
1 3
45 2
1
45
1

24. Exact Ratios
1 3
sin 30  sin 60 

2 2
30 2 3
cos 60 
 1
3 cos30 

2 2
60 tan 30 
 1
tan 60  3
1 3
1
sin 45 
2
45 1
2 cos 45 

1 2
45  tan 45  1
1

25. Alternative way of remembering the exact ratios
0 30 45 60 90
sin
cos
tan

26. Alternative way of remembering the exact ratios
0 30 45 60 90
0 1 2 3 4
sin
2 2 2 2 2
cos
tan

27. Alternative way of remembering the exact ratios
0 30 45 60 90
0 1 2 3 4
sin
2 2 2 2 2
4 3 2 1 0
cos
2 2 2 2 2
tan

28. Alternative way of remembering the exact ratios
0 30 45 60 90
0 1 2 3 4
sin
2 2 2 2 2
4 3 2 1 0
cos
2 2 2 2 2
tan
sin

cos

29. Alternative way of remembering the exact ratios
0 30 45 60 90
0 1 2 3 4
sin
2 2 2 2 2
4 3 2 1 0
cos
2 2 2 2 2
tan 0 1 2 3 4
sin
 4 3 2 1 0
cos

30. Measuring Angles

31. Measuring Angles
Angle of Elevation
B

A

32. Measuring Angles
Angle of Elevation
B

A
angle of elevation of B from A

33. Measuring Angles
Angle of Elevation Angle of Depression
B Y


A X
angle of elevation of B from A

34. Measuring Angles
Angle of Elevation Angle of Depression
B Y


A X
angle of elevation of B from A angle of depresion of X from Y

35. Measuring Angles
Angle of Elevation Angle of Depression
B Y


A X
angle of elevation of B from A angle of depresion of X from Y
Note: angle of elevation = angle of depression

36. Measuring Angles
Angle of Elevation Angle of Depression
B Y


A X
angle of elevation of B from A angle of depresion of X from Y
Note: angle of elevation = angle of depression
Compass Bearings N
W E
S

37. Measuring Angles
Angle of Elevation Angle of Depression
B Y


A X
angle of elevation of B from A angle of depresion of X from Y
Note: angle of elevation = angle of depression
Compass Bearings NW N NE
W E
SW S SE

38. Measuring Angles
Angle of Elevation Angle of Depression
B Y


A X
angle of elevation of B from A angle of depresion of X from Y
Note: angle of elevation = angle of depression
Compass Bearings NW NNW N NE
NNE
WNW ENE
W E
WSW ESE
SW SSW S SSE SE

39. True Bearings
Always start NORTH and measure clockwise

40. True Bearings
Always start NORTH and measure clockwise
X
Y

41. True Bearings
Always start NORTH and measure clockwise
X
Y
Bearing of Y from X

42. True Bearings
Always start NORTH and measure clockwise
N
X
30
Y
Bearing of Y from X

43. True Bearings
Always start NORTH and measure clockwise
N
X
30
Y
Bearing of Y from X
120 T
or S60 E

44. True Bearings
Always start NORTH and measure clockwise
N
X
30
Y
Bearing of Y from X Bearing of X from Y
120 T
or S60 E

45. True Bearings
Always start NORTH and measure clockwise
N
X
30 N
60
Y
Bearing of Y from X Bearing of X from Y
120 T
or S60 E

46. True Bearings
Always start NORTH and measure clockwise
N
X
30 N
60
Y
Bearing of Y from X Bearing of X from Y
120 T 300 T
or S60 E or N60 W

47. Exercise 4A; 1 to 3 (pick some), 4acf, 5, 7bd, 8ac, 9bd, 11, 12,
14c, 17, 18, 19b, 21, 23, 24, 25
Exercise 4B; 1b, 2b, 3, 4b, 6, 7, 10