Trigonometry on the Unit Circle

1. Trig on the Unit Circle
Monday 23rd May 2011

2. y
P(x, y)
1
x

3. y
P(x, y)
1
x

4. y
P(x, y)
1
x
Angles are always taken in an anticlockwise
direction from the positive x axis

5. y
P(x, y)
1
θ x
Angles are always taken in an anticlockwise
direction from the positive x axis

6. y
P(x, y)
1
θ x

7. y
P(x, y)
1
θ x
Consider the coordinates of P

8. y
P(x, y)
1
θ x
x
Consider the coordinates of P

9. y
y P(x, y)
1
θ x
x
Consider the coordinates of P

10. y
P(x, y)
1
θ x

11. y
P(x, y)
1
θ x
This creates a right angled triangle with an angle θ

12. y
P(x, y)
1
θ x
This creates a right angled triangle with an angle θ

13. y
P(x, y)
1
θ x
x
This creates a right angled triangle with an angle θ

14. y
P(x, y)
1 y
θ x
x
This creates a right angled triangle with an angle θ

15. y
P(x, y)
1 y
θ x
x

16. y
P(x, y)
1 y
θ x
x
Using our standard trigonometric ratios

17. y
y
tan θ =
x
P(x, y)
1 y
θ x
x
Using our standard trigonometric ratios

18. y
y
tan θ =
x
P(x, y) y
sin θ = = y
1
1 y
θ x
x
Using our standard trigonometric ratios

19. y
y
tan θ =
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x
Using our standard trigonometric ratios

20. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x

21. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x
In the first quadrant all the trig ratios are positive

22. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x

23. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x
1st Identity:

24. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x
y sin θ
1st Identity: tan θ = =
x cosθ

25. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x

26. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x
2nd Identity (By Pythagoras):

27. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x
2 2
x + y =1
2nd Identity (By Pythagoras):

28. y
1st Quadrant tan θ =
y
x
P(x, y) y
sin θ = = y
1
1 y x
cosθ = = x
θ 1
x
x
2 2
x + y =1
2nd Identity (By Pythagoras): ∴sin 2 θ + cos 2 θ = 1

29. Two important Trig Identities

30. Two important Trig Identities
sin θ
tan θ =
cosθ

31. Two important Trig Identities
sin θ
tan θ =
cosθ
2 2
sin θ + cos θ = 1

32. y
2nd Quadrant 1st Quadrant tan θ =
y
x
P(−x, y) y
sin θ = = y
1 1
y x
cosθ = = x
θ 1
x
x

33. y
2nd Quadrant 1st Quadrant tan θ =
y
x
P(−x, y) y
sin θ = = y
1 1
y x
cosθ = = x
θ 1
x
x
2nd Quadrant (90 - 180 degrees) - θ is always
from the x axis

34. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
P(−x, y) y
sin θ = = y
1 1
y x
cosθ = = x
θ 1
x
x
2nd Quadrant (90 - 180 degrees) - θ is always
from the x axis

35. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
y P(−x, y) y
sin θ = = y sin θ = = y
1 1 1
y x
cosθ = = x
θ 1
x
x
2nd Quadrant (90 - 180 degrees) - θ is always
from the x axis

36. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
y P(−x, y) y
sin θ = = y sin θ = = y
1 1 1
−x y x
cosθ = = −x cosθ = = x
1 θ 1
x
x
2nd Quadrant (90 - 180 degrees) - θ is always
from the x axis

37. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
y 1
P(−x, −y)
3rd Quadrant

38. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
y 1
P(−x, −y)
3rd Quadrant
3rd Quadrant (180 - 270 degrees)

39. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y y y
tan θ = = 1
−x x
P(−x, −y)
3rd Quadrant
3rd Quadrant (180 - 270 degrees)

40. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y y y
tan θ = = 1
−x x
−y
sin θ = = −y P(−x, −y)
1
3rd Quadrant
3rd Quadrant (180 - 270 degrees)

41. y
y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y y y
tan θ = = 1
−x x
−y
sin θ = = −y P(−x, −y)
1
−x 3rd Quadrant
cosθ = = −x
1
3rd Quadrant (180 - 270 degrees)

42. y
−y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y y y
tan θ = = 1
−x x
−y
sin θ = = −y P(x, −y)
1
−x 3rd Quadrant 4th Quadrant
cosθ = = −x
1

43. y
−y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y y y
tan θ = = 1
−x x
−y
sin θ = = −y P(x, −y)
1
−x 3rd Quadrant 4th Quadrant
cosθ = = −x
1
4th Quadrant (270 - 360 degrees)

44. y
−y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y
−y y y tan θ =
tan θ = = 1 x
−x x
−y
sin θ = = −y P(x, −y)
1
−x 3rd Quadrant 4th Quadrant
cosθ = = −x
1
4th Quadrant (270 - 360 degrees)

45. y
−y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y
−y y y tan θ =
tan θ = = 1 x
−x x
−y
−y sin θ = = −y
sin θ = = −y P(x, −y) 1
1
−x 3rd Quadrant 4th Quadrant
cosθ = = −x
1
4th Quadrant (270 - 360 degrees)

46. y
−y 2nd Quadrant 1st Quadrant tan θ =
y
tan θ = x
x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = −x cosθ = = x
1 x 1
θ x
−y
−y y y tan θ =
tan θ = = 1 x
−x x
−y
−y sin θ = = −y
sin θ = = −y P(x, −y) 1
1
−x 3rd Quadrant 4th Quadrant cosθ = x = x
cosθ = = −x 1
1
4th Quadrant (270 - 360 degrees)

47. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = x
cosθ = = −x 1
1
x
−y y −y
tan θ = = tan θ =
−x x x
−y −y
sin θ = = −y sin θ = = −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°

48. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = x
cosθ = = −x 1
1
x
−y y −y
tan θ = = tan θ =
−x x x
−y −y
sin θ = = −y sin θ = = −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°
which ratio is positive in each of the quadrants?

49. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
cosθ =
−x
1
= −x All x
cosθ = = x
1
x
−y y −y
tan θ = = tan θ =
−x x x
−y −y
sin θ = = −y sin θ = = −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°
which ratio is positive in each of the quadrants?

50. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
cosθ =
−x
1
= −x sin All x
cosθ = = x
1
x
−y y −y
tan θ = = tan θ =
−x x x
−y −y
sin θ = = −y sin θ = = −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°
which ratio is positive in each of the quadrants?

51. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
cosθ =
−x
1
= −x sin All x
cosθ = = x
1
x
−y y −y
tan θ = = tan θ =
sin θ =
−x x
−y
= −y
tan sin θ =
−y
x
= −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°
which ratio is positive in each of the quadrants?

52. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
cosθ =
−x
1
= −x sin All x
cosθ = = x
1
x
−y y −y
tan θ = = tan θ =
sin θ =
−x x
−y
= −y
tan cos sin θ =
−y
x
= −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°
which ratio is positive in each of the quadrants?

53. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
−x x
cosθ = = x
cosθ = = −x 1
1
x
−y y −y
tan θ = = tan θ =
−x x x
−y −y
sin θ = = −y sin θ = = −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°

54. 2nd Quadrant y 1st Quadrant
90 - 180° 0 - 90° y
y tan θ =
tan θ = x
−x
y y
sin θ = = y sin θ = = y
1 1
cosθ =
−x
1
= −x S A x
cosθ = = x
1
x
−y y −y
tan θ = = tan θ =
sin θ =
−x x
−y
= −y
T C sin θ =
−y
x
= −y
1 1
−x x
cosθ = = −x cosθ = = x
1 3rd Quadrant 4th Quadrant 1
180 - 270° 270 - 360°
All Stations To Central

55. Example 1
Determine whether sin 243° is positive or negative

56. Example 1
Determine whether sin 243° is positive or negative
Step 1 : Determine which Quadrant the angle is in.

57. Example 1
Determine whether sin 243° is positive or negative
Step 1 : Determine which Quadrant the angle is in.
y
S A
x
T C

58. Example 1
Determine whether sin 243° is positive or negative
Step 1 : Determine which Quadrant the angle is in.
y
S A
x
T C
Remember the angle is always taken anticlockwise
from the positive x - axis

59. Example 1
Determine whether sin 243° is positive or negative
Step 1 : Determine which Quadrant the angle is in.
y
S A
243°
x
T C
Remember the angle is always taken anticlockwise
from the positive x - axis

60. Example 1
Determine whether sin 243° is positive or negative
y
S A
243°
x
T C

61. Example 1
Determine whether sin 243° is positive or negative
So the angle is in the 3rd Quadrant
y
S A
243°
x
T C

62. Example 1
Determine whether sin 243° is positive or negative
So the angle is in the 3rd Quadrant
y
S A
243°
x
T C
Thus sin 243°is negative

63. Example 2
For 0 ≤θ≤360° find all possible values of θ such that
sin θ = 0.6

64. Example 2
For 0 ≤θ≤360° find all possible values of θ such that
sin θ = 0.6
Step 1: Find the corresponding acute angle (i.e in the 1st Q)

65. Example 2
For 0 ≤θ≤360° find all possible values of θ such that
sin θ = 0.6
Step 1: Find the corresponding acute angle (i.e in the 1st Q)
∴θ = 37° (to nearest degree from Calculator)

66. Example 2
For 0 ≤θ≤360° find all possible values of θ such that
sin θ = 0.6
Step 1: Find the corresponding acute angle (i.e in the 1st Q)
∴θ = 37° (to nearest degree from Calculator)
Step 2: Find other quadrants where the ratio is positive

67. Example 2
For 0 ≤θ≤360° find all possible values of θ such that
sin θ = 0.6
Step 1: Find the corresponding acute angle (i.e in the 1st Q)
∴θ = 37° (to nearest degree from Calculator)
Step 2: Find other quadrants where the ratio is positive
y
S A
x
T C

68. Example 2
For 0 ≤θ≤360° find all possible values of θ such that
sin θ = 0.6
Step 1: Find the corresponding acute angle (i.e in the 1st Q)
∴θ = 37° (to nearest degree from Calculator)
Step 2: Find other quadrants where the ratio is positive
y
S A As sin is positive it must be an
angle in the 1st or 2nd Quadrant
x
T C

69. Example 2
For 0 ≤θ≤360° find all possible values of θ such that
sin θ = 0.6
Step 1: Find the corresponding acute angle (i.e in the 1st Q)
∴θ = 37° (to nearest degree from Calculator)
Step 2: Find other quadrants where the ratio is positive
y
S A As sin is positive it must be an
angle in the 1st or 2nd Quadrant
x
T C
Step 3: Find the angle in the other quadrant(s)

70. Example 2
y
S A
37° 37°
x
T C

71. Example 2
Step 3: Find the angle in the other quadrant(s)
y
S A
37° 37°
x
T C

72. Example 2
Step 3: Find the angle in the other quadrant(s)
y
So the two angles are:
37° and
S A
37° 37°
x
T C

73. Example 2
Step 3: Find the angle in the other quadrant(s)
y
So the two angles are:
37° and 180°-37°=143°
S A
37° 37°
x
T C