Unit Circle

Unit circle chart: degrees, radians and exact values

This trig circle chart lists the standard angles in one counterclockwise turn. Read coordinates in the order (cos θ, sin θ), not (sin θ, cos θ). The circle starts at (1, 0) on the positive x-axis. A full turn, 360° or 2π radians, returns to the same point as 0° and is shown separately below the table.

360° = 2π: cos θ = 1, sin θ = 0, tan θ = 0. It is the same circle position as 0°, with one additional complete rotation.

How to read the point on the circle

The radius is always 1. A line from the origin to the selected point forms the terminal side of the angle. Its horizontal projection gives the x-coordinate, which is cosine; its vertical projection gives the y-coordinate, which is sine. The two coordinates satisfy x² + y² = 1, also written cos² θ + sin² θ = 1.

For example, 30° reaches (√3/2, 1/2). This means cos 30° = √3/2 and sin 30° = 1/2. Tangent compares the vertical and horizontal coordinates: tan θ = sin θ / cos θ. At 30°, simplifying that ratio gives √3/3. At 90°, the point is (0, 1), so cosine is zero and tangent is undefined—not zero.

Degrees, radians and equivalent angles

Degrees divide a turn into 360 parts. Radians measure an angle using the length of its arc relative to the radius, so a unit circle has 2π radians in one turn. To convert degrees to radians, multiply by π/180. Thus 45° becomes π/4 and 150° becomes 5π/6. To convert radians to degrees, multiply by 180/π.

Check the unit selector before entering a number: 30 radians is not 30°. Radian input accepts a finite decimal or a simple pi multiple such as 2pi, pi/6 or 3π/4. It does not evaluate functions, sums or general code. A zero divisor is invalid. The unit selector converts a valid result to its equivalent representation within one turn.

Angles can continue past a complete turn or run clockwise. Adding or subtracting 360° (2π radians) gives a coterminal angle with the same sine and cosine. For −30°, rotate clockwise from the positive x-axis; the same point is labeled 330° when viewed counterclockwise. The original rotation and the one-turn diagram describe the same endpoint.

Quadrants and reference angles

A reference angle is the acute angle between the terminal side and the x-axis. It helps connect an angle in another quadrant to a familiar first-quadrant triangle. At 150°, the reference angle is 30°. The point is left of the origin and above it, so cosine is negative and sine is positive: (−√3/2, 1/2). Tangent is negative because those signs differ.

In quadrant I both coordinates are positive; in II only sine is positive; in III sine and cosine are negative, making tangent positive; in IV only cosine is positive. Points on an axis are quadrant boundaries. This calculator labels them “On an axis” and does not assign them an acute reference angle.

Exact values and numerical limits

Special-angle fractions and radicals come from 30–60–90 and 45–45–90 triangles and the four axis points. Other angles normally require a numerical calculation. The tool labels those outputs as approximate rather than inventing an exact fraction from rounded digits. Very small values may use scientific notation, and an input near an axis must not be mistaken for the axis itself. A tiny negative angle keeps its signed representation if expressing it just below a full turn would round away the difference.

Use the common-angle buttons when you need a standard exact value; use the input or keyboard-controlled diagram to explore other angles. “Copy values” copies plain text only after the browser permits clipboard access. If copying is blocked, select the visible values manually. The static chart remains readable when JavaScript is unavailable.