Unit Circle

About this tool

This unit circle chart shows the 16 standard angles from 0° to 330° with their radian measures and exact (cos θ, sin θ) coordinates, such as (√3/2, 1/2) at 30°. Select any point, or type any angle in degrees or radians, to get exact and decimal values of sin, cos, tan, sec, csc and cot, the quadrant, the reference angle and, for angles outside 0° to 360°, the coterminal angle. The circle has radius 1, so the point at angle θ is (cos θ, sin θ); tan θ = sin θ ÷ cos θ, and sec, csc and cot are the reciprocals of cos, sin and tan. Exact values come from the reference angle (0°, 30°, 45°, 60° or 90°), with the sign set by the quadrant. Each label set can be switched off, down to a blank practice chart, and the chart prints on one page or downloads as a PNG. Exact radical forms are given for the 16 standard angles only; other angles, such as 15° or 2.5 radians, show values rounded to 6 decimals. Everything runs in your browser.

Why use it

• Exact values such as √3/2 and 2√3/3 for all six functions, with decimals alongside.

• Handles any angle: negative angles, angles above 360°, decimal radians and π forms like 7π/4.

• The selected angle is drawn as a right triangle, with cos θ as the horizontal side and sin θ as the vertical side.

• Prints on one page and downloads as a 1536-pixel-wide PNG, with or without labels.

• Runs in your browser, with no sign-up.

How to use

1. Read the chart: each of the 16 points is labeled with its angle in degrees, its angle in radians and its exact coordinates (cos θ, sin θ).
2. Click or tap a point, or Tab to it and press Enter, to see sin, cos, tan, sec, csc and cot as exact and decimal values. The left and right arrow keys step around the circle.
3. For any other angle, type it in the Any angle box, such as -45, 765 or 5π/6, and choose Degrees or Radians for plain numbers.
4. Read the quadrant and the reference angle and, for angles outside 0° to 360°, the coterminal angle.
5. Use the Degrees, Radians and Coordinates buttons to show or hide each set of labels, then click Print chart or Download PNG.

When it helps

• Memorizing the unit circle before a trigonometry or precalculus test.
• Checking an exact value such as sin 225° or sec 5π/6 during homework.
• Finding the reference angle and coterminal angle of a negative or very large angle.
• Printing a labeled unit circle as a handout, or a blank one as a practice sheet.

Examples

Tips

• Switch off Degrees, Radians and Coordinates to get a blank circle with only the 16 points, then print it as a quiz.
• Type π forms directly, such as 3pi/4 or -π/6. They are always read as radians, whichever unit is selected.
• Reference angle by quadrant: 180° − θ in Quadrant II, θ − 180° in Quadrant III and 360° − θ in Quadrant IV.
• √3/3 and 1/√3 are the same number; this tool writes tan 30° and cot 60° as √3/3, with the root in the numerator.
• Angles other than the 16 standard ones, such as 15°, show decimal values rounded to 6 places.

Frequently Asked Questions

Glossary

Unit circle

A circle of radius 1 centered at (0, 0); the point at angle θ is (cos θ, sin θ).

Radian

The angle that cuts off an arc as long as the radius; a full turn is 2π radians, or 360°.

Reference angle

The smallest angle between an angle's terminal side and the x-axis, from 0° to 90°.

Coterminal angles

Angles that differ by whole turns of 360° (2π) and so share the same terminal side.

Quadrantal angle

An angle such as 0°, 90°, 180° or 270° whose terminal side lies on an axis instead of inside a quadrant.