Unit Circle Chart
The unit circle is a circle of radius 1 whose points are (cos θ, sin θ) for the angle θ. This chart shows the 16 standard angles in degrees and radians with their exact coordinates, or a blank circle to fill in, drawn to fit your paper size.
Your sheet
Printing: choose Actual size or 100%, not “Fit to page”, so 1 cm prints as 1 cm. How to check.
Where do the values on the unit circle come from?
Point on the circle = (cos θ, sin θ). The exact values come from two triangles: 30-60-90, with sides 1, √3 and 2, giving sin 30° = 1/2 and cos 30° = √3/2; and 45-45-90, with sides 1, 1 and √2, giving sin 45° = cos 45° = √2/2.
Worked example. 150° is 30° short of 180°, so it is the 30° point reflected across the y-axis: cos becomes negative, sin stays positive, and the point is (−√3/2, 1/2). In radians, 150° × π ÷ 180 = 5π/6. The chart’s table lists all sixteen; the same values, one per line, are in data.txt.
Every angle on the chart
| 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 |
How to memorise the unit circle
Learn the first quadrant only: 0°, 30°, 45°, 60°, 90° with cos running 1, √3/2, √2/2, 1/2, 0 and sin running the same list backwards. Then use the quadrant signs for everything else. The radians follow a pattern too: the denominators 6, 4, 3 and 2 repeat around the circle with numerators that count up.
Using the chart in class
Print the filled chart for the wall and the blank one for weekly practice. Our guide to using the unit circle works through reading sine, cosine and tangent, converting units and the special triangles.
Questions about the unit circle
What is the unit circle?
A circle of radius 1 centred at the origin. Any angle θ measured anticlockwise from the positive x-axis meets the circle at the point (cos θ, sin θ), so the circle turns every trigonometric value into a coordinate you can read off.
Why are 30°, 45° and 60° the angles on the chart?
Because their sines and cosines are exact: 1/2, √2/2 and √3/2. They come from the two special right triangles, 30-60-90 and 45-45-90. Every other angle on the chart is one of these three reflected into another quadrant, or a multiple of 90°.
How do I convert degrees to radians?
Multiply by π ÷ 180. So 30° = 30π ÷ 180 = π/6, 45° = π/4, 60° = π/3, 90° = π/2 and 180° = π. Going back, multiply radians by 180 ÷ π. The chart shows both for each angle.
How do I find tan from the chart?
tan θ = sin θ ÷ cos θ, the y-coordinate over the x-coordinate. At 30° that is (1/2) ÷ (√3/2) = √3/3. At 90° and 270° the cosine is 0, so tan is undefined.
Which values are negative in each quadrant?
Quadrant I (0° to 90°): both positive. II (90° to 180°): cos negative, sin positive, so at 135° the point is (-√2/2, √2/2). III: both negative, as at 240°, (-1/2, -√3/2). IV: cos positive, sin negative. The mnemonic is “All Students Take Calculus”: All, Sine, Tangent, Cosine are positive in quadrants I to IV.
How do I use the blank chart?
Untick “Show the values” and print. Fill in each angle in degrees and radians and its coordinates from memory, then check against the filled chart. Students who can fill the blank circle in under five minutes rarely lose marks on trig identities.