STUDY RESOURCE
Unit Circle Cheat Sheet
All 16 standard angles with exact values, patterns, and quadrant signs — ready to print.
The 1-2-√3 Memory Pattern
You don't need to memorize 16 × 3 = 48 values. You only need to know three reference angles and one sign rule.
Then apply quadrant signs: QI (+,+) · QII (−,+) · QIII (−,−) · QIV (+,−). For the reciprocal functions, use sec = 1/x · csc = 1/y · cot = x/y.
ALL 16 ANGLES AT A GLANCE
Visual unit circle reference
Read degrees around the circle, then use the table below for radians and exact values. Opposite points share the same reference angle and change signs by quadrant.
All 16 Angles — Exact Values
Axes (0°, 90°, 180°, 270°)
| Degrees | Radians | cos θ (x) | sin θ (y) | tan θ | Coordinates |
|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 | (1, 0) |
| 90° | π/2 | 0 | 1 | undefined | (0, 1) |
| 180° | π | −1 | 0 | 0 | (−1, 0) |
| 270° | 3π/2 | 0 | −1 | undefined | (0, −1) |
Quadrant I
| Degrees | Radians | cos θ (x) | sin θ (y) | tan θ | Coordinates |
|---|---|---|---|---|---|
| 30° | π/6 | √3/2 | 1/2 | √3/3 | (√3/2, 1/2) |
| 45° | π/4 | √2/2 | √2/2 | 1 | (√2/2, √2/2) |
| 60° | π/3 | 1/2 | √3/2 | √3 | (1/2, √3/2) |
Quadrant II
| Degrees | Radians | cos θ (x) | sin θ (y) | tan θ | Coordinates |
|---|---|---|---|---|---|
| 120° | 2π/3 | −1/2 | √3/2 | −√3 | (−1/2, √3/2) |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 | (−√2/2, √2/2) |
| 150° | 5π/6 | −√3/2 | 1/2 | −√3/3 | (−√3/2, 1/2) |
Quadrant III
| Degrees | Radians | cos θ (x) | sin θ (y) | tan θ | Coordinates |
|---|---|---|---|---|---|
| 210° | 7π/6 | −√3/2 | −1/2 | √3/3 | (−√3/2, −1/2) |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 | (−√2/2, −√2/2) |
| 240° | 4π/3 | −1/2 | −√3/2 | √3 | (−1/2, −√3/2) |
Quadrant IV
| Degrees | Radians | cos θ (x) | sin θ (y) | tan θ | Coordinates |
|---|---|---|---|---|---|
| 300° | 5π/3 | 1/2 | −√3/2 | −√3 | (1/2, −√3/2) |
| 315° | 7π/4 | √2/2 | −√2/2 | −1 | (√2/2, −√2/2) |
| 330° | 11π/6 | √3/2 | −1/2 | −√3/3 | (√3/2, −1/2) |
How to Use This Cheat Sheet
- Start with QI only. Learn 30°, 45°, 60° values until they're automatic. Everything else is derived from these.
- Apply the sign rule. In QII, x flips negative. In QIII, both flip. In QIV, y flips negative. The absolute values stay the same.
- Remember the axes. At 0°: (1, 0). At 90°: (0, 1). At 180°: (−1, 0). At 270°: (0, −1). Tangent is undefined when x = 0.
- Test yourself. Close this sheet and take the unit circle quiz for random recall, or fill in the unit circle to rebuild all values from memory.
Frequently Asked Questions
What are the 16 standard angles on the unit circle?
The 16 standard angles are 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°. They come from 30°, 45°, and 60° reference angles placed in each of the four quadrants.
How do I memorize unit circle values without brute force?
Use the 1-2-√3 pattern. For 30°-60°-90°: the numerators under the radical are 1, 2, 3 (or their square roots). At 30°: cos=√3/2, sin=1/2. At 45°: cos=sin=√2/2. At 60°: cos=1/2, sin=√3/2. Then apply quadrant sign rules: all six functions are positive in QI; sin and csc in QII; tan and cot in QIII; cos and sec in QIV.
What is the quadrant sign rule (ASTC)?
ASTC stands for All, Sine, Tangent, Cosine. QI: all six functions are positive. QII: sine and cosecant are positive. QIII: tangent and cotangent are positive. QIV: cosine and secant are positive. Every other function is negative within that quadrant.
What is the unit circle cheat sheet used for?
Students use a unit circle cheat sheet to quickly look up exact values for trigonometric functions at standard angles. It is useful when studying for pre-calculus, calculus, ACT, or SAT exams. Memorizing or understanding the patterns on the cheat sheet saves time on tests.