An interactive unit circle with exact trig values
The unit circle is a circle of radius 1 centred at the origin. A point on it at angle θ from the positive x-axis has coordinates (cos θ, sin θ), which is why it is the quickest way to learn and remember the trigonometric functions. This interactive unit circle lets you drag that point around — or type an angle in degrees or radians — and instantly see the coordinates, the exact values of sin, cos and tan (and their reciprocals csc, sec and cot), the reference angle, the quadrant and the signs of each function.
Exact values are shown with surds for every multiple of 15°, from √3/2 to (√6 − √2)/4; other angles get decimals. A quiz mode drills the special-angle values, and the printable chart comes filled in, with blanks for practice, or completely blank — on A4 or US Letter, as a PDF, PNG or straight to the printer.
How to use the Unit Circle
- Type an angle — 225, −45, 5pi/4 or π/6 — or drag the point on the circle. Use the arrow keys for one-degree steps (Shift for 15°).
- Read the angle in degrees and radians, the quadrant, the coordinates and the table of sin, cos, tan, csc, sec and cot.
- Choose what dragging snaps to: 15° for the special angles, 5° or 1°.
- Switch to Quiz to practise exact values; the circle jumps to each answered angle.
- Pick a printable chart (filled, practice or blank), the paper size, and print or download the PDF or PNG.
Unit Circle features
Drag, type or use the keyboard
Pointer and touch dragging with snapping, typed angles in degrees or with π, and arrow-key steps.
Exact surds
Values for every multiple of 15°, including tan 15° = 2 − √3, plus csc, sec and cot.
Reference angle and signs
The acute angle to the x-axis and the sign of sin, cos and tan in that quadrant.
Tangent shown geometrically
The green segment on the line x = 1 has length tan θ, next to the blue cos and red sin legs.
Quiz mode
Multiple-choice questions on sin, cos and tan of the 16 standard angles in degrees or radians, with a running score.
Printable charts
Filled, practice and blank unit circle charts with an optional first-quadrant table, as vector PDF or PNG.
Exact values of the special angles
A handy pattern: sin of 0°, 30°, 45°, 60°, 90° is √0/2, √1/2, √2/2, √3/2, √4/2, and cos runs the same list backwards. These come from the 30-60-90 triangle (sides 1, √3, 2) and the 45-45-90 triangle (sides 1, 1, √2).
| θ | 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 | undefined |
Reference angles and the ASTC rule
Every angle has a reference angle: the acute angle between its terminal side and the x-axis. Its trig values have the same size as the reference angle’s; only the sign changes. 225° lies in quadrant III with a reference angle of 45°, so sin 225° = −√2/2, cos 225° = −√2/2 and tan 225° = 1.
The signs follow “All Students Take Calculus”: in quadrant I all functions are positive, in II only sine, in III only tangent and in IV only cosine. Angles beyond 360° or below 0° are coterminal with one between 0° and 360° — add or subtract 360° (2π) until it is.
Degrees, radians and graphs
Radians measure the arc length on the unit circle: a full turn is 2π, so 180° = π and 1 radian ≈ 57.30°. To convert, multiply degrees by π/180. The angle converter handles other units, and the function grapher plots y = sin x or y = tan x so you can watch the unit-circle values trace out the waves.
Unit Circle FAQ
What is the unit circle?
A circle with radius 1 centred at (0, 0). The point at angle θ on the unit circle is (cos θ, sin θ), so its coordinates give the cosine and sine of the angle, and tan θ = sin θ / cos θ.
How do I remember the unit circle?
Learn the first quadrant (0°, 30°, 45°, 60°, 90°) and the √0/2 … √4/2 pattern, then use reference angles and the ASTC sign rule for the other quadrants. The quiz mode helps you practise.
Why is tan 90° undefined?
tan θ = sin θ / cos θ, and cos 90° = 0, so the division is by zero. On the circle, the radius at 90° is parallel to the tangent line x = 1 and never meets it.
How do I find the reference angle?
Quadrant II: 180° − θ. Quadrant III: θ − 180°. Quadrant IV: 360° − θ. The unit circle tool shows it for any angle you enter.
Can I print a blank unit circle?
Yes. Choose “Blank” or “Practice” under Printable chart, then print or download the PDF. The practice version keeps the angles and leaves the coordinates empty.
Does it accept radians?
Yes. Type expressions with π such as 5pi/4 or 2π/3, or switch “Plain numbers are” to radians and type decimals like 2.1.