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Unit Circle Visualizer – Interactive Sine, Cosine, Tangent Calculator
Move the angle slider or drag on the circle to see live sin, cos, tan values in degrees or radians. Get quadrants, reference angles, exact special values, arc length, and quick trig insights with zero back-end calls.
No uploads. Everything runs locally in your browser.
Status
Example loaded at 45°.
Angle: 45.00° / 0.785 radQuadrant: IReference: 45.00°Special angle: 45°
Unit
Unit circle
Click or drag on the circle area to set the angle. Colors: hypotenuse (cyan), adjacent/cos (purple dashed), opposite/sin (green dashed).
Explanation
- Unit circle point: x = cos(θ), y = sin(θ). Tangent = sin(θ) / cos(θ) where cos(θ) ≠ 0.
- Quadrant rules: I (+,+), II (+/−), III (−,−), IV (−/+); tangent follows sin/cos signs.
- Reference angle is the acute angle to the nearest x-axis: 45.00°.
- Angles wrap: 450° → 90°; −45° → 315°. Radians wrap over 2π similarly.
- Arc length on unit circle equals θ in radians (normalized): 0.785.
Special angle detected: 45°. Exact values shown in the table and cards.
sin(θ)
0.7071
Exact: √2/2
cos(θ)
0.7071
Exact: √2/2
tan(θ)
1
Exact: 1
sec(θ)
1.4142
csc(θ)
1.4142
cot(θ)
1
Insights
Quadrant
I
Sign (sin, cos, tan)
+, +, +
Reference angle
45.00°
Arc length (θ rad)
0.785
Angle type
Special angle
Symmetry
sin(−θ) = −sin(θ); cos(−θ) = cos(θ); tan(−θ) = −tan(θ)
Unit circle table
| Angle (°) | Radians | sin (exact) | cos (exact) | tan (exact) | sin (dec) | cos (dec) | tan (dec) |
|---|---|---|---|---|---|---|---|
| 0° | 0.000 | 0 | 1 | 0 | 0 | 1 | 0 |
| 30° | 0.524 | 1/2 | √3/2 | √3/3 | 0.5 | 0.866 | 0.5774 |
| 45° | 0.785 | √2/2 | √2/2 | 1 | 0.7071 | 0.7071 | 1 |
| 60° | 1.047 | √3/2 | 1/2 | √3 | 0.866 | 0.5 | 1.7321 |
| 90° | 1.571 | 1 | 0 | undefined | 1 | 0 | undefined |
| 120° | 2.094 | √3/2 | -1/2 | -√3 | 0.866 | -0.5 | -1.7321 |
| 135° | 2.356 | √2/2 | -√2/2 | -1 | 0.7071 | -0.7071 | -1 |
| 150° | 2.618 | 1/2 | -√3/2 | -√3/3 | 0.5 | -0.866 | -0.5774 |
| 180° | 3.142 | 0 | -1 | 0 | 0 | -1 | 0 |
| 210° | 3.665 | -1/2 | -√3/2 | √3/3 | -0.5 | -0.866 | 0.5774 |
| 225° | 3.927 | -√2/2 | -√2/2 | 1 | -0.7071 | -0.7071 | 1 |
| 240° | 4.189 | -√3/2 | -1/2 | √3 | -0.866 | -0.5 | 1.7321 |
| 270° | 4.712 | -1 | 0 | undefined | -1 | 0 | undefined |
| 300° | 5.236 | -√3/2 | 1/2 | -√3 | -0.866 | 0.5 | -1.7321 |
| 315° | 5.498 | -√2/2 | √2/2 | -1 | -0.7071 | 0.7071 | -1 |
| 330° | 5.760 | -1/2 | √3/2 | -√3/3 | -0.5 | 0.866 | -0.5774 |
| 360° | 6.283 | 0 | 1 | 0 | 0 | 1 | 0 |
Wave previews
Sine (pink) and cosine (blue) sampled over 0–360°. Marker shows current angle.
Practice mode (lightweight)
Angle: 30°
Try to recall sin, cos, tan before revealing.
What is this tool
This Unit Circle Visualizer is an interactive trigonometry helper that turns the classic unit circle chart into a live calculator. Enter an angle in degrees or radians, drag the slider, or click the circle to move the point; the tool instantly shows sin(θ), cos(θ), tan(θ), quadrant, reference angle, arc length, and exact special values like √2/2 or √3/2. You also get optional sec/csc/cot, symmetry hints, and wave previews that link the circle to sine and cosine graphs, making it a complete trig values calculator for lessons, homework, or quick engineering checks.
Angle normalization handles negative or oversized inputs (e.g., 450° wraps to 90°), while undefined tangent cases at 90° and 270° are clearly labeled to avoid confusion. Special angles are auto-detected and labeled with exact fractions and radicals alongside decimal approximations. All visuals and math run client-side—no logins, no uploads—so you can experiment freely with instant feedback.
Beyond a static unit circle chart, this page adds animation, draggable control, and a mini practice mode. A lightweight quiz prompts common angles, and the unit circle table consolidates degrees, radians, sin, cos, and tan in both exact and decimal forms. The linked wave preview demonstrates how x = cos(θ) and y = sin(θ) trace periodic graphs, reinforcing the geometry-to-graph connection.
Use cases
- Trigonometry learning: explore quadrants, signs, and reference angles by dragging the point and watching live values.
- Exam preparation: rehearse exact special angles (30°/45°/60°/90°) and see decimals to check calculator results.
- Visual understanding of angles: map rotations to coordinates, arc length, and sine/cosine waves in one view.
- STEM quick checks: verify sin/cos/tan for robotics or graphics transforms without opening a heavy CAS tool.
- Teaching demos: project the animated circle to show symmetry, periodicity, and undefined tangent at π/2 and 3π/2.
Pair this with the exponent & power calculator when working with trig-powered exponents, the equation solver for linear follow-ups, and the standard deviation calculator when you need to sanity-check angle datasets in experiments.
Popular ways to use this tool
- Set the angle to 45° or π/4 to see √2/2 values and then animate to watch the point trace the circle.
- Toggle to radians, input −π/3, and observe wrapping plus quadrant sign changes for sin and cos.
- Use the table to memorize special angles, then quiz yourself with the practice mode and reveal answers only after guessing.
- Click on the circle near 210° to demonstrate how tangent stays positive in quadrant III while sin and cos are negative.
- Drag the slider through 0–360° while watching sine/cosine wave markers to connect circular motion with periodic graphs.
Need more math helpers? Jump to the percentage calculator for fast ratios, the fraction calculator for exact fractional forms, or the ratio calculator when breaking down component proportions.
How to use Unit Circle Visualizer
- Choose degrees or radians, then enter an angle or drag the slider/circle point.
- Watch the unit circle update with the point, quadrant, reference angle, and exact values for special angles.
- Review sin, cos, tan (and optional sec/csc/cot), plus wave previews and the unit circle table.
- Copy results, load the example, clear, or start a quick quiz to test your trig recall.
Key features of Unit Circle Visualizer
- Interactive unit circle with draggable angle, slider control, and live point highlighting
- Degrees ↔ radians toggle with normalization for angles beyond 360° / 2π, including negatives
- Exact and decimal values for special angles (√2/2, √3/2, 1/2) plus quadrant and reference angle detection
- Sine, cosine, tangent cards with undefined handling at 90°/270°, optional sec/csc/cot, and arc length
- Unit circle table, quadrant sign rules, symmetry notes, wave previews, and lightweight quiz/practice mode
Unit Circle Visualizer — key benefits
- Completely free with no signup or registration required.
- Runs entirely in your browser — your data is never uploaded.
- Works on desktop, tablet, and mobile devices.
- Instant results with no waiting or processing queues.
- Part of the ToolCircuit suite of free online math tools tools.
FAQ
How do I use this unit circle visualizer?
Pick degrees or radians, type an angle or drag the slider/circle. The point moves instantly, showing sin, cos, tan, quadrant, and reference angle with exact values at special angles.
Does it handle angles beyond 360° or negative angles?
Yes. Angles are normalized (wrap) automatically, so 450° behaves like 90°, and −45° wraps to 315°. Radian inputs wrap around 2π as well.
What happens at undefined tangent angles?
At 90° and 270° (π/2 and 3π/2), tangent is undefined because cos = 0. The tool marks it clearly and avoids division errors.
Can I see exact trig values?
Yes. Common angles like 30°, 45°, 60°, 90°, 120°, 135°, 150°, and their equivalents show exact sin, cos, and tan values alongside decimals.
Is this unit circle calculator free and private?
It’s free, runs entirely in your browser, requires no login, and never uploads your inputs.