Unit Circle Trigonometry

Here are 100 questions about the unit circle in trigonometry, organized roughly by difficulty and topic type:

Basic Definitions & Understanding (1–20)

1. What is the radius of the unit circle?
2. What are the coordinates of the center of the unit circle?
3. Why is it called the “unit” circle?
4. In which quadrants is the x-coordinate positive?
5. In which quadrants is the y-coordinate negative?
6. What is the equation of the unit circle?
7. How is the point (1, 0) related to the unit circle?
8. What angle (in degrees) corresponds to the point (0, 1)?
9. What angle (in radians) corresponds to the point (-1, 0)?
10. What is the reference angle when the terminal side is in quadrant II?
11. How many degrees are in one full rotation around the unit circle?
12. How many radians are in one full rotation?
13. What is the relationship between degrees and radians (conversion formula)?
14. On the unit circle, what does the x-coordinate represent?
15. On the unit circle, what does the y-coordinate represent?
16. What is the Pythagorean identity that comes directly from the unit circle?
17. If cos θ = x, what does sin θ equal (in terms of x)?
18. Why are sine and cosine called circular functions?
19. What point on the unit circle corresponds to θ = 0°?
20. What point corresponds to θ = 2π radians?

Quadrantal Angles & Special Angles (21–40)

21. Give the exact coordinates of θ = 30°.
22. Give the exact coordinates of θ = 45°.
23. Give the exact coordinates of θ = 60°.
24. Give the exact coordinates of θ = 90°.
25. Give the exact coordinates of θ = 120°.
26. Give the exact coordinates of θ = 135°.
27. Give the exact coordinates of θ = 150°.
28. Give the exact coordinates of θ = 180°.
29. Give the exact coordinates of θ = 210°.
30. Give the exact coordinates of θ = 225°.
31. Give the exact coordinates of θ = 240°.
32. Give the exact coordinates of θ = 270°.
33. Give the exact coordinates of θ = 300°.
34. Give the exact coordinates of θ = 315°.
35. Give the exact coordinates of θ = 330°.
36. Give the exact coordinates of θ = 360° or 0°.
37. What is cos 0°? sin 0°?
38. What is sin 90°? cos 90°?
39. What is sin 180°? cos 180°?
40. What is cos 270°? sin 270°?

Signs & Quadrants (41–55)

41. In which quadrants is sine positive?
42. In which quadrants is cosine positive?
43. In which quadrants is tangent positive?
44. In which quadrant is both sine and cosine negative?
45. In which quadrant is sine positive but cosine negative?
46. What is the only quadrant where both secant and cosecant are negative?
47. If sin θ < 0 and cos θ > 0, which quadrant is θ in?
48. If tan θ < 0 and cos θ < 0, which quadrant is θ in?
49. What is the sign of cot θ in quadrant III?
50. Where is sec θ undefined?
51. Where is csc θ undefined?
52. Where is tan θ undefined?
53. Where is cot θ undefined?
54. In which two quadrants is sin θ = –cos θ?
55. In which two quadrants is cos θ = –sin θ?

Reference Angles (56–70)

56. What is the reference angle of 210°?
57. What is the reference angle of 315°?
58. What is the reference angle of 7π/6?
59. What is the reference angle of 5π/3?
60. What is the reference angle of –π/6?
61. What is the reference angle of 480°?
62. What is the reference angle of 11π/6?
63. How do you find the reference angle when θ is in quadrant II?
64. How do you find the reference angle when θ is in quadrant III?
65. How do you find the reference angle when θ is in quadrant IV?
66. What is the reference angle of 750°?
67. What angle between 0° and 90° has the same sine as 150°?
68. What angle between 0° and 90° has the same cosine as 300°?
69. What angle between 0° and π/2 has the same tangent as 225°?
70. Why do we use reference angles to find trig values?

Identities & Relationships (71–85)

71. Write sin²θ + cos²θ = 1 using unit circle reasoning.
72. If cos θ = 3/5 and θ is in quadrant I, find sin θ using the unit circle idea.
73. If sin θ = –√7/4 and θ is in quadrant III, find cos θ.
74. Explain why cos(θ + 2π) = cos θ using the unit circle.
75. Explain why sin(θ + π) = –sin θ using the unit circle.
76. What happens to cos θ when you add π to the angle?
77. What is the period of the cosine function according to the unit circle?
78. Why is tan(θ + π) = tan θ?
79. On the unit circle, why does sin(–θ) = –sin θ?
80. Why is cosine an even function?
81. Why is sine an odd function?
82. What point on the unit circle corresponds to θ = –π/3?
83. If θ = 5π/3, what positive coterminal angle < 2π has the same trig values?
84. What is the smallest positive coterminal angle of –7π/4?
85. Explain why the unit circle repeats every 2π radians.

Mixed / Slightly Harder / Application-style (86–100)

86. If a point on the unit circle has x = –√2/2, what are the two possible y-values?
87. A point on the unit circle has y = √3/2. What are the two possible x-values?
88. At what angles is |sin θ| = |cos θ| on the unit circle?
89. At what angles is sin θ = cos θ on the unit circle?
90. At what angles is sin θ = –cos θ?
91. Find all angles θ where cos θ = –1/2 in one full rotation.
92. Find all angles θ where sin θ = √2/2 in [0, 2π).
93. How many solutions does sin θ = 0.8 have in one full rotation?
94. How many solutions does cos θ = –0.3 have in [0, 4π]?
95. If the terminal point is (–√3/2, –1/2), what is θ in radians (0 ≤ θ < 2π)?
96. Convert 300° to radians and give the corresponding point.
97. A central angle of 2.4 radians is marked on the unit circle. Estimate sin(2.4) and cos(2.4).
98. What is the y-coordinate when x = 0.8 on the unit circle (positive branch)?
99. If tan θ = –√3 and cos θ > 0, find sin θ and cos θ.
100. Explain how the unit circle makes it possible to define trigonometric functions for any real number (not just 0°–90°).