Unit Circle Flashcards: 16 Angles, Radians & Exact Coordinates
Master 16 standard unit-circle positions with 64 two-way cards for degrees, radians, and exact (cos θ, sin θ) coordinates.
About this deck
Unit Circle Flashcards
Master the 16 unique standard positions in 0 ≤ θ < 2π with 64 exact-answer cards. Every card uses the single tag unit-circle.
What the cards practice
- 16 degree-to-radian cards convert each standard degree measure to its exact radian measure.
- 16 radian-to-degree cards retrieve the reverse conversion.
- 16 angle-to-coordinate cards show degrees and radians together and ask for the exact ordered pair (cos θ, sin θ).
- 16 coordinate-to-angle cards ask for the unique radian angle in [0, 2π) and give the equivalent degree measure as brief reference context.
Learning sequence
The fixed sequence has four deterministic passes. The first pass moves progressively from 0° through 330°. The next three passes interleave quadrants, axes, and reference-angle families to avoid long predictable numerical runs. In both coordinate-bearing passes, sign variants from the same absolute-coordinate family recur every four positions and are never adjacent. Direct degree/radian reverse pairs are 24–40 positions apart; direct angle/coordinate reverse pairs are 17–45 positions apart.
Scope and exclusions
The endpoint at 360° = 2π is excluded because it repeats the coordinates of 0; omitting it keeps coordinate-to-angle recall unique in [0, 2π). Reverse single-function prompts such as “which angle has cosine 1/2?” are excluded because sine and cosine values repeat at multiple standard angles. Nonstandard angles are excluded because they require calculation rather than recall of the standard unit-circle positions.
The deck also excludes tangent, cotangent, secant, cosecant, negative and coterminal angles, proofs, generic definitions, calculator approximations, and decimal coordinates. Cards use exact Unicode math in ordinary Markdown instead of renderer-specific syntax or images, so every client and indexed text surface can read them directly.
Related guides
- How to use flashcards for math
- How to use flashcards for AP Calculus AB and BC
- How to use interleaving with flashcards
- How to make better flashcards
Source and license
Authoritative source: OpenStax, Algebra and Trigonometry, first edition, revision AT-2015-002(03/17)-BW, by Jay Abramson, §7.3 Unit Circle, printed pages 604–616 (PDF pages 622–634); original publication year 2015.
© 2017 Rice University. Download for free at https://openstax.org/details/books/algebra-and-trigonometry.
The textbook source is licensed under CC BY 4.0. This deck restates standard mathematical facts as original flashcard prompts and answers; every value was independently computed and verified, and no OpenStax prose, figure, or media was copied. This adapted deck, its metadata, and its original cover are also licensed under CC BY 4.0. Flashcards Open Source App is not affiliated with or endorsed by OpenStax, Study.com, Pearson, or Buffalo State University.
Cards in this deck
Card 1
Question
Convert 0° to radians in [0, 2π).
Answer
0
Card 2
Question
Convert 30° to radians in [0, 2π).
Answer
π/6
Card 3
Question
Convert 45° to radians in [0, 2π).
Answer
π/4
Card 4
Question
Convert 60° to radians in [0, 2π).
Answer
π/3
Card 5
Question
Convert 90° to radians in [0, 2π).
Answer
π/2
Card 6
Question
Convert 120° to radians in [0, 2π).
Answer
2π/3
Card 7
Question
Convert 135° to radians in [0, 2π).
Answer
3π/4
Card 8
Question
Convert 150° to radians in [0, 2π).
Answer
5π/6
Card 9
Question
Convert 180° to radians in [0, 2π).
Answer
π
Card 10
Question
Convert 210° to radians in [0, 2π).
Answer
7π/6
Card 11
Question
Convert 225° to radians in [0, 2π).
Answer
5π/4
Card 12
Question
Convert 240° to radians in [0, 2π).
Answer
4π/3
Card 13
Question
Convert 270° to radians in [0, 2π).
Answer
3π/2
Card 14
Question
Convert 300° to radians in [0, 2π).
Answer
5π/3
Card 15
Question
Convert 315° to radians in [0, 2π).
Answer
7π/4
Card 16
Question
Convert 330° to radians in [0, 2π).
Answer
11π/6
Card 17
Question
For the unit-circle position 30° = π/6, give the exact ordered pair (cos θ, sin θ).
Answer
(√3/2, 1/2)
Card 18
Question
For the unit-circle position 225° = 5π/4, give the exact ordered pair (cos θ, sin θ).
Answer
(−√2/2, −√2/2)
Card 19
Question
For the unit-circle position 120° = 2π/3, give the exact ordered pair (cos θ, sin θ).
Answer
(−1/2, √3/2)
Card 20
Question
For the unit-circle position 270° = 3π/2, give the exact ordered pair (cos θ, sin θ).
Answer
(0, −1)
Card 21
Question
For the unit-circle position 150° = 5π/6, give the exact ordered pair (cos θ, sin θ).
Answer
(−√3/2, 1/2)
Card 22
Question
For the unit-circle position 315° = 7π/4, give the exact ordered pair (cos θ, sin θ).
Answer
(√2/2, −√2/2)
Card 23
Question
For the unit-circle position 240° = 4π/3, give the exact ordered pair (cos θ, sin θ).
Answer
(−1/2, −√3/2)
Card 24
Question
For the unit-circle position 0° = 0, give the exact ordered pair (cos θ, sin θ).
Answer
(1, 0)
Card 25
Question
For the unit-circle position 210° = 7π/6, give the exact ordered pair (cos θ, sin θ).
Answer
(−√3/2, −1/2)
Card 26
Question
For the unit-circle position 45° = π/4, give the exact ordered pair (cos θ, sin θ).
Answer
(√2/2, √2/2)
Card 27
Question
For the unit-circle position 300° = 5π/3, give the exact ordered pair (cos θ, sin θ).
Answer
(1/2, −√3/2)
Card 28
Question
For the unit-circle position 90° = π/2, give the exact ordered pair (cos θ, sin θ).
Answer
(0, 1)
Card 29
Question
For the unit-circle position 330° = 11π/6, give the exact ordered pair (cos θ, sin θ).
Answer
(√3/2, −1/2)
Card 30
Question
For the unit-circle position 135° = 3π/4, give the exact ordered pair (cos θ, sin θ).
Answer
(−√2/2, √2/2)
Card 31
Question
For the unit-circle position 60° = π/3, give the exact ordered pair (cos θ, sin θ).
Answer
(1/2, √3/2)
Card 32
Question
For the unit-circle position 180° = π, give the exact ordered pair (cos θ, sin θ).
Answer
(−1, 0)
Card 33
Question
Convert π/2 radians to degrees in [0°, 360°).
Answer
90°
Card 34
Question
Convert 7π/6 radians to degrees in [0°, 360°).
Answer
210°
Card 35
Question
Convert π/4 radians to degrees in [0°, 360°).
Answer
45°
Card 36
Question
Convert 2π/3 radians to degrees in [0°, 360°).
Answer
120°
Card 37
Question
Convert 3π/2 radians to degrees in [0°, 360°).
Answer
270°
Card 38
Question
Convert π/6 radians to degrees in [0°, 360°).
Answer
30°
Card 39
Question
Convert 5π/4 radians to degrees in [0°, 360°).
Answer
225°
Card 40
Question
Convert 5π/3 radians to degrees in [0°, 360°).
Answer
300°
Card 41
Question
Convert 0 radians to degrees in [0°, 360°).
Answer
0°
Card 42
Question
Convert 5π/6 radians to degrees in [0°, 360°).
Answer
150°
Card 43
Question
Convert 7π/4 radians to degrees in [0°, 360°).
Answer
315°
Card 44
Question
Convert π/3 radians to degrees in [0°, 360°).
Answer
60°
Card 45
Question
Convert π radians to degrees in [0°, 360°).
Answer
180°
Card 46
Question
Convert 11π/6 radians to degrees in [0°, 360°).
Answer
330°
Card 47
Question
Convert 3π/4 radians to degrees in [0°, 360°).
Answer
135°
Card 48
Question
Convert 4π/3 radians to degrees in [0°, 360°).
Answer
240°
Card 49
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (−1, 0)?
Answer
π
Equivalent degree measure: 180°.
Card 50
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (√3/2, 1/2)?
Answer
π/6
Equivalent degree measure: 30°.
Card 51
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (√2/2, −√2/2)?
Answer
7π/4
Equivalent degree measure: 315°.
Card 52
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (−1/2, √3/2)?
Answer
2π/3
Equivalent degree measure: 120°.
Card 53
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (1, 0)?
Answer
0
Equivalent degree measure: 0°.
Card 54
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (−√3/2, −1/2)?
Answer
7π/6
Equivalent degree measure: 210°.
Card 55
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (−√2/2, √2/2)?
Answer
3π/4
Equivalent degree measure: 135°.
Card 56
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (1/2, −√3/2)?
Answer
5π/3
Equivalent degree measure: 300°.
Card 57
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (0, −1)?
Answer
3π/2
Equivalent degree measure: 270°.
Card 58
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (−√3/2, 1/2)?
Answer
5π/6
Equivalent degree measure: 150°.
Card 59
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (√2/2, √2/2)?
Answer
π/4
Equivalent degree measure: 45°.
Card 60
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (−1/2, −√3/2)?
Answer
4π/3
Equivalent degree measure: 240°.
Card 61
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (0, 1)?
Answer
π/2
Equivalent degree measure: 90°.
Card 62
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (√3/2, −1/2)?
Answer
11π/6
Equivalent degree measure: 330°.
Card 63
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (−√2/2, −√2/2)?
Answer
5π/4
Equivalent degree measure: 225°.
Card 64
Question
In [0, 2π), which unit-circle angle has exact coordinates (cos θ, sin θ) = (1/2, √3/2)?
Answer
π/3
Equivalent degree measure: 60°.
64 cards
Unit Circle Flashcards: 16 Angles, Radians & Exact Coordinates
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