Solutions: Unit Circle

Solutions to Try Its

1. cos(t)=−√22,sin(t)=√22cos(t)=−√22,sin(t)=√22

2. cos(π)=−1cos(π)=−1, sin(π)=0sin(π)=0

3. sin(t)=−725sin(t)=−725

4. approximately 0.866025403

5. π3π3

6. a. cos(315∘)=√22,sin(315∘)=−√22cos(315∘)=√22,sin(315∘)=−√22
b. cos(−π6)=√32,sin(−π6)=−12cos(−π6)=√32,sin(−π6)=−12

7. (12,−√32)(12,−√32)

Solutions to Odd-Numbered Exercises

1. The unit circle is a circle of radius 1 centered at the origin.

3. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, tt, formed by the terminal side of the angle tt and the horizontal axis.

5. The sine values are equal.

7. I

9. IV

11. √32√32

13. 1212

15. √22√22

17. 0

19. −1

21. √32√32

23. 60∘60∘

25. 80∘80∘

27. 45∘45∘

29. π3π3

31. π3π3

33. π8π8

35. 60∘60∘, Quadrant IV, sin(300∘)=−√32,cos(300∘)=12sin(300∘)=−√32,cos(300∘)=12

37. 45∘45∘, Quadrant II, sin(135∘)=√22sin(135∘)=√22, cos(135∘)=−√22cos(135∘)=−√22

39. 60∘60∘, Quadrant II, sin(120∘)=√32sin(120∘)=√32, cos(120∘)=−12cos(120∘)=−12

41. 30∘30∘, Quadrant II, sin(150∘)=12sin(150∘)=12, cos(150∘)=−√32cos(150∘)=−√32

43. π6π6, Quadrant III, sin(7π6)=−12sin(7π6)=−12, cos(7π6)=−√32cos(7π6)=−√32

45. π4π4, Quadrant II, sin(3π4)=√22sin(3π4)=√22, cos(4π3)=−√22cos(4π3)=−√22

47. π3π3, Quadrant II, sin(2π3)=√32sin(2π3)=√32, cos(2π3)=−12cos(2π3)=−12

49. π4π4, Quadrant IV, sin(7π4)=−√22sin(7π4)=−√22, cos(7π4)=√22cos(7π4)=√22

51. √779√779

53. −√154−√154

55. (−10,10√3)(−10,10√3)

57. (−2.778,15.757)(−2.778,15.757)

59. [−1,1][−1,1]

61. sint=12,cost=−√32sint=12,cost=−√32

63. sint=−√22,cost=−√22sint=−√22,cost=−√22

65. sint=√32,cost=−12sint=√32,cost=−12

67. sint=−√22,cost=√22

69. sint=0,cost=−1

71. sint=−0.596,cost=0.803

73. sint=12,cost=√32

75. sint=−12,cost=√32

77. sint=0.761,cost=−0.649

79. sint=1,cost=0

81. −0.1736

83. 0.9511

85. −0.7071

87. −0.1392

89. −0.7660

91. √24

93. −√64

95. √24

97. √24

99. 0

101. (0,−1)

103. 37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds