Exercise Set 4.3: Unit Circle Trigonometry

Math 1330, Precalculus The University of Houston Chapter 4: Trigonometric Functions

Sketch each of the following angles in standard position. (Do not use a protractor; just draw a quick sketch of each angle.)

1. (a) 30 (b) 135− (c) 300

2. (a) 120 (b) 60− (c) 210

3. (a) 3π (b) 7

4π (c) 5

6π

−

4. (a) 4π

− (b) 43π (c) 11

6π

5. (a) 90 (b) π− (c) 450

6. (a) 270 (b) 180 (c) 4π−

7. (a) 240− (b) 136π (c) 510−

8. (a) 315− (b) 94π

− (c) 1020

Find three angles, one negative and two positive, that are coterminal with each angle below.

9. (a) 50 (b) 200−

10. (a) 300 (b) 830−

11. (a) 25π (b) 7

2π

−

12. (a) 83π (b) 4

9π

−

Answer the following.

13. List four quadrantal angles in degree measure, where each angle θ satisfies the condition 0 360θ≤ < .

14. List four quadrantal angles in radian measure, where each angle θ satisfies the condition

52 2π πθ≤ < .

15. List four quadrantal angles in radian measure, where each angle θ satisfies the condition 5 134 4π πθ< ≤ .

16. List four quadrantal angles in degree measure, where each angle θ satisfies the condition 800 1160θ< ≤ .

Sketch each of the following angles in standard position and then specify the reference angle or reference number.

17. (a) 240 (b) 30− (c) 60

18. (a) 315 (b) 150 (c) 120−

19. (a) 76π (b) 5

3π (c) 7

4π

−

20. (a) 3π

− (b) 54π (c) 5

6π

−

21. (a) 840 (b) 376π

− (c) 660−

22. (a) 113π (b) 780− (c) 11

4π

−

The exercises below are helpful in creating a comprehensive diagram of the unit circle. Answer the following.

23. Using the following unit circle, draw and then

label the terminal side of all multiples of 2π from

0 to 2π radians. Write all labels in simplest form.

24. Using the following unit circle, draw and then

label the terminal side of all multiples of 4π from

0 to 2π radians. Write all labels in simplest form.

x

y

1 -1

1

-1

0

x

y

1 -1

1

-1

0

Exercise Set 4.3: Unit Circle Trigonometry

Math 1330, Precalculus The University of Houston Chapter 4: Trigonometric Functions

x

y

1

-1

1

-1

25. Using the following unit circle, draw and then

label the terminal side of all multiples of 3π from

0 to 2π radians. Write all labels in simplest form.

26. Using the following unit circle, draw and then

label the terminal side of all multiples of 6π from

0 to 2π radians. Write all labels in simplest form.

27. Use the information from numbers 23-26 to label all the special angles on the unit circle in radians.

28. Label all the special angles on the unit circle in degrees.

Name the quadrant in which the given conditions are satisfied.

29. ( ) ( )sin 0, cos 0θ θ> < 2

30. ( ) ( )sin 0, sec 0θ θ< > 4

31. ( ) ( )cot 0, sec 0θ θ> < 3

32. ( ) ( )csc 0, cot 0θ θ> > 1

33. ( ) ( )tan 0, csc 0θ θ< < 4

34. ( ) ( )csc 0, tan 0θ θ< > 3

Fill in each blank with , , or < > = .

35. ( ) ( )sin 40 _____ sin 140

36. ( ) ( )cos 20 _____ cos 160

37. ( ) ( )tan 310 _____ tan 50

38. ( ) ( )sin 195 _____ sin 15

39. ( ) ( )cos 355 _____ cos 185

40. ( ) ( )tan 110 _____ tan 290

x

y

1

-1

1

-1

x

y

1 -1

1

-1

0

x

y

1 -1

1

-1

0

Exercise Set 4.3: Unit Circle Trigonometry

Math 1330, Precalculus The University of Houston Chapter 4: Trigonometric Functions

60o

30o

10

60o

30o

3

Diagram 1 Diagram 2

Let ( ),P x y denote the point where the terminal side of an angle θ meets the unit circle. Use the given information to evaluate the six trigonometric functions of θ .

41. P is in Quadrant I and 23

y = .

42. P is in Quadrant I and 38

x = .

43. P is in Quadrant IV and 45

x = .

44. P is in Quadrant III and 2425

y = − .

45. P is in Quadrant II and 15

x = − .

46. P is in Quadrant II and 27

y = .

For each quadrantal angle below, give the coordinates of the point where the terminal side of the angle intersects the unit circle. Then give the six trigonometric functions of the angle. If a value is undefined, state “Undefined.”

47. 90

48. 180−

49. 2π−

50. 32π

51. 52π

−

52. 8π

Rewrite each expression in terms of its reference angle, deciding on the appropriate sign (positive or negative). For example,

( ) ( )sin 240 sin 60= − 4tan tan3 3π π⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

( ) ( )sec 315 sec 45− = 7cos cos6 6π π⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

53. (a) ( )cos 300 (b) ( )tan 135

54. (a) ( )sin 45− (b) ( )cot 210

55. (a) ( )sin 140 (b) 2sec3π⎛ ⎞−⎜ ⎟

⎝ ⎠

56. (a) ( )csc 190− (b) 11cos6π⎛ ⎞

⎜ ⎟⎝ ⎠

57. (a) 25csc6π⎛ ⎞

⎜ ⎟⎝ ⎠

(b) ( )cot 460−

58. (a) ( )tan 520 (b) 9sec4π⎛ ⎞−⎜ ⎟

⎝ ⎠

For each angle below, give the coordinates of the point where the terminal side of the angle intersects the unit circle. Then give the six trigonometric functions of the angle.

59. 45

60. 60

61. 210

62. 135

63. 53π

64. 116π

An alternate method of finding trigonometric functions of 30o, 45o, or 60o is shown below.

65.

(a) Find the missing side measures in each of the diagrams above.

(b) Use right triangle trigonometric ratios to

find the following, using Diagram 1:

( )sin 30 _____= ( )sin 60 _____=

( )cos 30 _____= ( )cos 60 _____=

( )tan 30 _____= ( )tan 60 _____=

(c) Repeat part (b), using Diagram 2. (d) Use the unit circle to find the trigonometric

ratios listed in part (b). (e) Examine the answers in parts (b) through

(d). What do you notice?

Exercise Set 4.3: Unit Circle Trigonometry

Math 1330, Precalculus The University of Houston Chapter 4: Trigonometric Functions

66.

(a) Find the missing side measures in each of the diagrams above.

(b) Use right triangle trigonometric ratios to

find the following, using Diagram 1:

( )sin 45 _____= ( )csc 45 _____=

( )cos 45 _____= ( )sec 60 _____=

( )tan 45 _____= ( )cot 60 _____=

(c) Repeat part (b), using Diagram 2. (d) Use the unit circle to find the trigonometric

ratios listed in part (b). (e) Examine the answers in parts (b) through

(d). What do you notice? The following two diagrams can be used to quickly evaluate the trigonometric functions of any angle having a reference angle of 30o, 45o, or 60o. Use right trigonometric ratios along with the concept of reference angles, to evaluate the following. (Remember that when converting degrees to radians,

30 , 45 , 606 4 3π π π

= = = .)

67. (a) ( )sin 240 (b) ( )tan 135

68. (a) ( )cos 330 (b) ( )csc 225−

69. (a) cos4π⎛ ⎞−⎜ ⎟

⎝ ⎠ (b) 11sec

6π⎛ ⎞

⎜ ⎟⎝ ⎠

70. (a) 2sin3π⎛ ⎞

⎜ ⎟⎝ ⎠

(b) 7cot6π⎛ ⎞−⎜ ⎟

⎝ ⎠

Use either the unit circle or the right triangle method from numbers 65-70 to evaluate the following. (Note: The right triangle method can not be used for quadrantal angles.) If a value is undefined, state “Undefined.”

71. (a) ( )tan 30 (b) ( )sin 135−

72. (a) ( )cos 180 (b) ( )csc 60−

73. (a) ( )csc 150− (b) ( )sin 270

74. (a) ( )sec 225 (b) ( )tan 240−

75. (a) ( )cot 450− (b) ( )cos 495

76. (a) ( )sin 210− (b) ( )cot 420−

77. (a) ( )csc π (b) 2cos3π⎛ ⎞

⎜ ⎟⎝ ⎠

78. (a) sin3π⎛ ⎞−⎜ ⎟

⎝ ⎠ (b) 5cot

4π⎛ ⎞

⎜ ⎟⎝ ⎠

79. (a) sec6π⎛ ⎞−⎜ ⎟

⎝ ⎠ (b) 3tan

4π⎛ ⎞

⎜ ⎟⎝ ⎠

80. (a) 11csc4π⎛ ⎞

⎜ ⎟⎝ ⎠

(b) 3sec2π⎛ ⎞−⎜ ⎟

⎝ ⎠

81. (a) 10cot3π⎛ ⎞−⎜ ⎟

⎝ ⎠ (b) 7sec

4π⎛ ⎞

⎜ ⎟⎝ ⎠

82. (a) ( )tan 5π− (b) 11cos2π⎛ ⎞−⎜ ⎟

⎝ ⎠

Use a calculator to evaluate the following to the nearest ten-thousandth. Make sure that your calculator is in the appropriate mode (degrees or radians). Note: Be careful when evaluating the reciprocal trigonometric functions.

For example, when evaluating ( )csc θ on your calculator, use the

identity ( ) ( )1

sincsc

θθ = . Do NOT use the calculator key labeled

( )1sin θ− ; this represents the inverse sine function, which will be

discussed in Section 5.4.

83. (a) ( )sin 37 (b) ( )tan 218−

84. (a) ( )tan 350 (b) ( )cos 84−

85. (a) ( )csc 191 (b) ( )cot 21

60o

30o

2 1

3 1

1 45o

45o

2

Diagram 1 Diagram 2

45o

45o

8

7

45o

45o

Exercise Set 4.3: Unit Circle Trigonometry

Math 1330, Precalculus The University of Houston Chapter 4: Trigonometric Functions

86. (a) ( )cot 310 (b) ( )sec 73

87. (a) cos5π⎛ ⎞⎜ ⎟⎝ ⎠

(b) 11csc7π⎛ ⎞−⎜ ⎟

⎝ ⎠

88. (a) 10sin9π⎛ ⎞−⎜ ⎟

⎝ ⎠ (b) ( )cot 4.7π

89. (a) ( )tan 4.5− (b) ( )sec 3

90. (a) ( )csc 0.457− (b) ( )tan 9.4