Unit Circle Reference
Explore the unit circle online for free with our interactive unit circle reference. Click any of the 16 standard angles to instantly see exact sin, cos, and tan values in fraction and radical form, plus decimal approximations, radian equivalents, and all six trig functions. No signup required — all calculations run locally in your browser.
Click any point on the unit circle to see exact sin, cos, and tan values. All 16 standard angles are shown with exact values in fraction and radical form. All calculations run locally in your browser.
| Degrees | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undef |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/3 |
| 180° | π | 0 | -1 | 0 |
| 210° | 7π/6 | -1/2 | -√3/2 | √3/3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 |
| 270° | 3π/2 | -1 | 0 | undef |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 |
| 330° | 11π/6 | -1/2 | √3/2 | -√3/3 |
Why Use Our Unit Circle Reference?
Instant Unit Circle Reference
Click any of the 16 standard angles on the unit circle and instantly see exact sin, cos, and tan values in fraction and radical form. Our unit circle reference processes everything in your browser with zero wait time.
Secure Unit Circle Reference Online
All unit circle calculations happen locally in your browser. Your inputs never leave your device, ensuring 100% privacy every time you use our unit circle reference online.
Unit Circle Reference Online — No Installation
Use our unit circle reference directly in any browser with no downloads, plugins, or software required. Access exact trig values for all standard angles from any device, instantly.
Complete Trig Values — All 6 Functions
Our unit circle reference shows sin, cos, tan, csc, sec, and cot for every standard angle. Includes exact radical form, decimal approximations, radian equivalents, and a full quick-reference table.
Common Use Cases for Unit Circle Reference
Trigonometry Homework & Exams
Students use our unit circle reference to quickly verify sin, cos, and tan values for standard angles during homework and exam prep. Memorize the unit circle faster by seeing exact values alongside the visual diagram.
Calculus & Differential Equations
Calculus students rely on the unit circle reference when evaluating limits, derivatives, and integrals involving trigonometric functions. Exact radical values like √3/2 and √2/2 are essential for clean analytical solutions.
Physics & Engineering Problems
Engineers and physicists use the unit circle reference to resolve vectors, analyze wave functions, and compute phase angles. Exact trig values eliminate rounding errors in force diagrams and circuit analysis.
Computer Graphics & Game Development
Developers use the unit circle reference to implement rotation matrices, sprite animations, and camera transformations. Standard angles like 45°, 90°, and 135° appear constantly in 2D and 3D rendering pipelines.
Teaching & Classroom Demonstrations
Teachers use our interactive unit circle reference to demonstrate how sin and cos relate to coordinates on the circle. The visual quadrant shading and click-to-reveal format makes abstract concepts immediately concrete.
Signal Processing & Fourier Analysis
Signal processing engineers reference the unit circle when working with discrete Fourier transforms, phasors, and complex exponentials. Radian values at standard angles are critical for frequency domain analysis.
Understanding the Unit Circle
What is the Unit Circle?
The unit circle is a circle with radius 1 centered at the origin of a coordinate plane. Every point on the unit circle has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. The unit circle is the foundation of trigonometry because it defines sin, cos, and tan for all angles — not just those in right triangles. Our unit circle reference displays all 16 standard angles (0° through 360° in 30° and 45° increments) with their exact values in fraction and radical form.
How Our Unit Circle Reference Works
- 1Click Any Angle Point: Click any of the 16 labeled points on the interactive unit circle diagram. The selected angle is highlighted and its exact trig values appear instantly in the info panel.
- 2View Exact & Decimal Values: The unit circle reference shows sin θ, cos θ, and tan θ in exact form (e.g. √3/2, 1/2) alongside decimal approximations. Reciprocal functions csc, sec, and cot are also displayed.
- 3Look Up Any Custom Angle: Use the custom angle lookup to enter any degree value. If it matches a standard angle, the circle highlights it. For non-standard angles, decimal approximations are computed instantly in your browser.
Key Values on the Unit Circle
- 0° / 360° (0 / 2π): sin = 0, cos = 1, tan = 0. The point (1, 0) on the positive x-axis. Starting and ending position of a full rotation.
- 30° (π/6) and 150° (5π/6): sin = 1/2, cos = ±√3/2, tan = ±√3/3. These angles appear frequently in 30-60-90 triangle problems and physics force decomposition.
- 45° (π/4) and 135° (3π/4): sin = cos = ±√2/2, tan = ±1. The 45° angle produces equal sin and cos values, making it essential for isosceles right triangles and diagonal vectors.
- 90° (π/2) and 270° (3π/2): sin = ±1, cos = 0, tan = undefined. These are the axis intercepts where the tangent function is undefined due to division by zero.
Important Notes About the Unit Circle
The unit circle reference covers angles from 0° to 360°. Angles beyond 360° are coterminal — they share the same sin, cos, and tan values as their equivalent angle within 0°–360°. Negative angles are measured clockwise; for example, −30° is coterminal with 330°. The tangent function is undefined at 90° and 270° because cos θ = 0 at those points, making tan θ = sin θ / cos θ a division by zero.
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Frequently Asked Questions About Unit Circle Reference
A unit circle reference is a diagram showing a circle with radius 1 centered at the origin, with all standard angles labeled alongside their exact sin, cos, and tan values. Our unit circle reference is interactive — click any angle to instantly see its exact trig values in fraction and radical form, plus decimal approximations.
The 16 standard angles on the unit circle are 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°. These come from the 30-60-90 and 45-45-90 special right triangles repeated across all four quadrants.
These values come from the 30-60-90 special right triangle with hypotenuse 1. In that triangle, the side opposite 30° has length 1/2 (so sin 30° = 1/2) and the side adjacent to 30° has length √3/2 (so cos 30° = √3/2). The unit circle extends these ratios to all angles.
Tangent is defined as sin θ / cos θ. At 90° and 270°, cos θ = 0, which makes the division undefined. On the unit circle, these are the top and bottom points (0, 1) and (0, −1) where the x-coordinate is zero.
Angles greater than 360° are coterminal with angles in the 0°–360° range. Subtract 360° repeatedly until the angle falls within that range, then look up the values. For example, 390° − 360° = 30°, so sin 390° = sin 30° = 1/2. Use the custom angle lookup in our unit circle reference to compute values for any angle.
Degrees and radians are two ways to measure the same angles. A full circle is 360° or 2π radians. To convert degrees to radians, multiply by π/180. The standard angles in radians are π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), and so on. Our unit circle reference shows both degree and radian labels.
A common memory trick for sin values in Q1 is the sequence 0, 1/2, √2/2, √3/2, 1 for 0°, 30°, 45°, 60°, 90°. Cos values in Q1 are the reverse: 1, √3/2, √2/2, 1/2, 0. In other quadrants, apply the appropriate sign based on the ASTC rule (All Students Take Calculus): All positive in Q1, Sin positive in Q2, Tan positive in Q3, Cos positive in Q4.
Absolutely. All calculations in our unit circle reference happen locally in your browser. No data is ever sent to any server. Your inputs remain completely private every time you use our unit circle reference online.
Yes! Our unit circle reference is 100% free with no signup, no usage limits, and no premium features. Use it as often as you need — completely free, forever.