Unit Circle Calculator
Enter any angle in degrees or radians and read sin, cos and tan straight off the unit circle. Common angles like 30°, 45° and 60° are returned as exact radical values; other angles come back as decimals — with a labelled diagram showing the point (cos θ, sin θ) and its projections.
Enter an angle and choose whether it's in degrees or radians.
What is the unit circle?
The unit circle is the circle of radius 1 centred at the origin of the coordinate plane. Every point on it can be written as (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. That single picture defines sine and cosine for every real angle — positive, negative or bigger than a full turn.
How the unit circle produces sin, cos, tan — piece by piece
Each card explains one thing this tool reads from the angle you pick — how the circle coordinates give cosine and sine, how tangent comes out as their ratio, and why certain reference angles have exact radical values instead of decimals.
Coordinates on the circle are (cos θ, sin θ)
Walk counter-clockwise from the positive x-axis by angle θ. The x-coordinate of where you land is cos θ and the y-coordinate is sin θ. This is how the tool reads any angle.
Tangent as the ratio sin / cos
Once the tool has sine and cosine, tangent falls out as their ratio — equivalently, the slope of the radius drawn from the origin to the point.
Why 30°/45°/60° come out as neat radicals
The reference table entries are exact because two special right triangles (45-45-90 and 30-60-90) have simple radical side ratios. Reflecting those into every quadrant produces the 16 classic angles. Any other angle gives a decimal.
Reference table of common angles
| 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 | undefined |
| 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 | undefined |
| 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 |
Every entry in this table is exact — no rounding. Angles beyond 360° or below 0° reduce to one of these by adding or subtracting full turns.
Features of this calculator
- Switch between degrees and radians with a single toggle — no manual conversion needed
- Returns exact values (½, √2/2, √3/2, √3, …) for the 16 common unit-circle angles
- Falls back to accurate decimals for arbitrary angles, including angles > 360° or negatives
- Labelled SVG unit circle draws the terminal side, the point (cos θ, sin θ) and the sine/cosine projections
- Correctly flags tan as undefined at 90°, 270° and every 180° offset
- Full reference table of the classic 16 angles for quick lookup or memorisation
- Also returns sec, csc and cot θ as exact radicals or decimals, with undefined flagged correctly
- Shows the reference angle and the quadrant formula used to derive it
- Quadrant indicator with the ASTC sign chart for sin, cos and tan
- Coterminal angles (θ + 360° and θ − 360°) in both degrees and radians
- Built-in inverse trig tool: arcsin, arccos and arctan with domain checking
Frequently asked questions
+Why is tan θ undefined at 90° and 270°?
Because tan θ = sin θ / cos θ, and cos 90° = cos 270° = 0. Dividing by zero is undefined. On the unit circle, the radius at 90° points straight up — its slope is infinite, so tan blows up.
+What about angles bigger than 360° or negative angles?
Add or subtract multiples of 360° (or 2π) to bring the angle into the range [0°, 360°). Sine, cosine and tangent are periodic, so 400° gives the same values as 40°, and −30° gives the same values as 330°. The calculator does this normalisation automatically.
+How do I convert between degrees and radians?
radians = degrees × π / 180
degrees = radians × 180 / π
+How does the unit circle extend trig beyond right triangles?
Right-triangle definitions of sin and cos only work for angles between 0° and 90°. The unit circle uses coordinates instead — cos is the x-coordinate of the point on the circle, sin is the y-coordinate — so every real angle gets a well-defined value, including obtuse and negative ones.
+What are sec, csc and cot θ?
They're reciprocals: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ. This calculator lists all three alongside sin, cos and tan, and flags them as undefined whenever the function they invert is 0.
+What is the reference angle and the ASTC rule?
The reference angle is the acute angle your terminal side makes with the x-axis — 180° − θ in Quadrant II, θ − 180° in Quadrant III, and 360° − θ in Quadrant IV. ASTC ("All Students Take Calculus") is a memory aid for which functions are positive in each quadrant: All in Q1, Sin in Q2, Tan in Q3, Cos in Q4.
+What are coterminal angles?
Coterminal angles share the same terminal side because they differ by a full turn — θ + 360° and θ − 360° (or θ ± 2π in radians) always land on the same point of the unit circle and give identical sin, cos and tan values.
+How do I find an angle from a trig ratio?
Use the inverse functions: arcsin, arccos and arctan. The tool above accepts any value and returns the angle in both degrees and radians, and rejects inputs outside [−1, 1] for arcsin/arccos since those functions have a restricted domain.