How to
How to read the unit circle
The unit circle is a circle of radius 1 centered at the origin. Each angle θ is measured counterclockwise from the positive x axis, and where it meets the circle, the point's coordinates are exactly (cos θ, sin θ). This unit circle chart marks the 16 special angles with their degrees, radians and exact coordinates. Tap or hover any point to see its sine, cosine and tangent.
- Find the angle. Degrees are on the inner ring, radians just inside the circle. 90° is π/2, 180° is π, and a full turn, 360°, is 2π.
- Read the point. The first number of the pair is cos θ, the second is sin θ. At 60° the point is (1/2, √3/2), so cos 60° = 1/2 and sin 60° = √3/2.
- Work out the tangent. tan θ = sin θ / cos θ. At 60° that is √3. Where cos θ is 0, at 90° and 270°, the tangent is undefined.
- Check the signs. Cosine is negative in Quadrants II and III, on the left; sine is negative in Quadrants III and IV, at the bottom.
- Print it. Download the filled chart for reference, or the blank one to practice filling in from memory.
Exact values of the special angles
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
| 120° | 2π/3 | -1/2 | √3/2 | -√3 |
| 135° | 3π/4 | -√2/2 | √2/2 | -1 |
| 150° | 5π/6 | -√3/2 | 1/2 | -√3/3 |
| 180° | π | -1 | 0 | 0 |
| 210° | 7π/6 | -√3/2 | -1/2 | √3/3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | -1/2 | -√3/2 | √3 |
| 270° | 3π/2 | 0 | -1 | undefined |
| 300° | 5π/3 | 1/2 | -√3/2 | -√3 |
| 315° | 7π/4 | √2/2 | -√2/2 | -1 |
| 330° | 11π/6 | √3/2 | -1/2 | -√3/3 |
Tangents are written with the square root on top, so tan 30° is √3/3; some books write the same value as 1/√3. The chart only needs three numbers, 1/2, √2/2 and √3/2, plus 0 and 1: every special angle outside the first quadrant has a reference angle of 30°, 45° or 60° and takes that angle's values with the signs of its quadrant.
From the circle to the graphs
Unroll the circle and the heights of the points trace the sine wave; the distances across trace the cosine wave. Try y = sin x in the graphing calculator with degrees switched on to see both, or plot any of the points above on the XY graph maker. Printable grids for drawing your own are in the graph paper maker, including polar paper.
Questions people ask
What is a unit circle chart used for?
It is a one-page reference for trigonometry. It shows the special angles in degrees and radians and, for each one, the point where the angle meets a circle of radius 1. That point is (cos θ, sin θ), so the chart gives exact sine and cosine values, and tangent as sine divided by cosine, without a calculator.
How to remember the unit circle chart?
Learn the first quadrant only: at 30, 45 and 60 degrees the cosines are √3/2, √2/2 and 1/2, and the sines are the same three in reverse. Every other special angle reuses those numbers with signs from its quadrant: cosine is negative on the left half, sine is negative on the bottom half.
How do you graph the unit circle?
Draw a circle of radius 1 centered at the origin of an x and y grid, so it crosses the axes at (1, 0), (0, 1), (-1, 0) and (0, -1). Mark the angles counterclockwise from the positive x axis every 30 and 45 degrees, and label each point with its coordinates, cosine first, then sine.
Can you explain the unit circle in an easy way?
Picture a point walking counterclockwise around a circle of radius 1, starting at the right-hand side. After turning through an angle θ, its distance across from the center is cos θ and its height is sin θ. Because the radius is 1, those two distances are the cosine and sine themselves, with no dividing needed.
What grade is the unit circle taught?
In most US schools the unit circle comes in high school, usually in Algebra 2, trigonometry or precalculus, around grades 10 to 12. Right triangle trigonometry comes first, often in geometry. The unit circle then extends sine and cosine to every angle, including angles past 90 degrees and negative angles.