Unit Circle Chart & Calculator
A complete unit circle chart for precalculus and trigonometry, with all 16 angles in degrees and radians and their (cos, sin) coordinates and tan values. Check any angle in the calculator, print a blank unit circle, or practise with the quiz.
Free unit circle chart download, no sign-in. Drawn in your browser.
Unit circle chart styles
Four looks the AI unit circle generator draws, each on a different chart. The values come from the same exact table as the free chart; only the ink, colour and background change. Click a card to load its prompt.
Unit circle chart examples
Four unit circle chart examples, each a real render of the prompt on its card: the full chart, coordinates with tan values, a cheat sheet with the special right triangles, and a fill-in worksheet. Click a card to load it into the generator above.
The reference chart to keep beside homework.
Use once your course reaches tangent.
A whole trigonometry test on one page.
Fill it in from memory, then check.
What is the unit circle?
The unit circle is the circle of radius 1 centred at the origin, and the point where an angle θ meets it is (cos θ, sin θ). Angles are measured counter-clockwise from the positive x-axis, so every angle lands on one point: its x-coordinate is the cosine, its y-coordinate is the sine, and tan θ = sin θ / cos θ.
About this unit circle chart
- Shows all 16 standard angles in degrees and radians with exact (cos θ, sin θ) coordinates.
- Adds tan θ at every angle on request, with tan 90° and tan 270° marked undefined.
- Highlights any angle with its reference triangle and the cos θ and sin θ legs labelled.
- Calculates the values of any angle in degrees or radians, and finds the angles for a given sine, cosine or tangent value.
- Prints a blank unit circle worksheet with its answer key on page two.
- Downloads as PNG, SVG or PDF from the browser, in colour or black and white.
How to use the unit circle
- 1
Find the angle
Start at the positive x-axis and turn counter-clockwise: 90° (π/2) is the top, 180° (π) the left, 270° (3π/2) the bottom. Negative angles turn clockwise.
- 2
Read the coordinates
The point on the circle is (cos θ, sin θ). At 30° it is (√3/2, 1/2), so cos 30° = √3/2 and sin 30° = 1/2.
- 3
Divide for tangent
tan θ = sin θ / cos θ, so tan 30° = (1/2) ÷ (√3/2) = √3/3. Where cos θ = 0, at 90° and 270°, tan θ is undefined.
Unit circle chart values for all 16 angles
The unit circle is the circle of radius 1 centred at the origin. The point where an angle θ meets it is (cos θ, sin θ), so every row of this unit circle chart is one point: x is the cosine, y is the sine, and tan θ is y divided by x.
| Degrees | Radians | cos θ (x) | sin θ (y) | 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 |
The 16 angles are the multiples of 30° and 45° from 0° to 330°. Every other angle on a unit circle chart reuses these values with the signs of its quadrant.
sec, csc and cot on the unit circle
The other three functions are reciprocals of the first three: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = cos θ/sin θ. In the first quadrant they are:
| Angle | sec θ | csc θ | cot θ |
|---|---|---|---|
| 0° (0) | 1 | undefined | undefined |
| 30° (π/6) | 2√3/3 | 2 | √3 |
| 45° (π/4) | √2 | √2 | 1 |
| 60° (π/3) | 2 | 2√3/3 | √3/3 |
| 90° (π/2) | undefined | 1 | 0 |
Unit circle quadrants and signs
Each quadrant fixes the signs of cos θ and sin θ; the reference angle fixes their size.
| Quadrant | Angles | cos θ (x) | sin θ (y) | tan θ | ASTC |
|---|---|---|---|---|---|
| I | 0° to 90° (0 to π/2) | + | + | + | All positive |
| II | 90° to 180° (π/2 to π) | − | + | − | Sine positive |
| III | 180° to 270° (π to 3π/2) | − | − | + | Tangent positive |
| IV | 270° to 360° (3π/2 to 2π) | + | − | − | Cosine positive |
Angles on an axis belong to no quadrant: at 0°, 90°, 180° and 270° one coordinate is 0 and the other is 1 or −1.
Unit circle with tangent values
Tangent is the one value the coordinates do not show directly. On the unit circle tan θ = sin θ / cos θ = y / x, and it has a picture of its own: extend the radius until it meets the vertical line x = 1, the line that touches the circle at (1, 0). The height where they meet is tan θ.
- tan θ is positive in quadrants I and III, where x and y have the same sign, and negative in II and IV.
- At 90° and 270° the point is (0, ±1), so x = 0 and tan θ is undefined: the radius runs parallel to the line x = 1 and never meets it.
- The first-quadrant values √3/3, 1 and √3 at 30°, 45° and 60° repeat in every quadrant with the quadrant's sign. Switch on tan in the chart above to print them on the unit circle chart.
Unit circle in radians
A radian measures an angle by the arc it cuts on the unit circle: one full turn is an arc of 2π, so 360° = 2π and 180° = π. To convert, multiply degrees by π/180 (150° × π/180 = 5π/6) and radians by 180/π. The 16 radian measures on a unit circle chart fall into four families by denominator:
| Family | Reference angle | Radian measures | Degrees |
|---|---|---|---|
| Denominator 6 | 30° | π/6, 5π/6, 7π/6, 11π/6 | 30°, 150°, 210°, 330° |
| Denominator 4 | 45° | π/4, 3π/4, 5π/4, 7π/4 | 45°, 135°, 225°, 315° |
| Denominator 3 | 60° | π/3, 2π/3, 4π/3, 5π/3 | 60°, 120°, 240°, 300° |
| Quadrantal | 0° or 90° | 0, π/2, π, 3π/2 | 0°, 90°, 180°, 270° |
Read the denominator first: it tells you the reference angle and so the values, and the numerator tells you which quadrant you are in.
Blank unit circle chart to fill in
A blank unit circle is the standard way to learn the chart: the points and radius lines are drawn, and the labels are left for you. The Blank worksheet tab above prints three common versions, each with its answer key on page two.
- Degrees given: fill in the radians and the coordinates, the usual precalculus test.
- Fully blank: every degree, radian and coordinate box empty, for a memory check.
- Add tan: an extra box under each coordinate pair for the tangent values.
Unit circle quiz and calculator
Two ways to check yourself once the unit circle chart is familiar.
Unit circle quiz
Ten-question rounds in five modes: find a point on an unlabelled circle, convert degrees to radians and back, name the coordinates of an angle, or give sin, cos or tan of it. Wrong answers show the right one on the chart.
Unit circle calculator
Type any angle in degrees or radians, including negative angles and angles past 360°: −45°, 7π/3, 2.5 rad. It returns the quadrant, the reference angle, two coterminal angles and all six trig values, exact wherever the angle is a multiple of 15°.
How to memorize the unit circle
Nobody memorizes a unit circle chart as 48 separate numbers. Learn the first quadrant, then let symmetry and the signs do the rest.
- 1
Learn the 1-2-3 pattern
For 0°, 30°, 45°, 60°, 90°, sin θ runs √0/2, √1/2, √2/2, √3/2, √4/2, which simplifies to 0, 1/2, √2/2, √3/2, 1. cos θ is the same list read backwards.
- 2
Use the hand trick
Hold up your left hand, palm toward you, and give the fingers the angles 0°, 30°, 45°, 60°, 90° from little finger to thumb. Fold the finger of your angle: sin θ = √(fingers on the little-finger side)/2 and cos θ = √(fingers on the thumb side)/2. Fold the ring finger for 30°: one finger on the little-finger side gives sin 30° = 1/2, three on the thumb side give cos 30° = √3/2.
- 3
Reflect with the reference angle
Every angle in quadrants II to IV has the same values as its reference angle, the acute angle to the x-axis: 150°, 210° and 330° all use the 30° values.
- 4
Fix the signs with All Students Take Calculus
All three functions are positive in quadrant I, only Sine in II, only Tangent in III, only Cosine in IV.
What the unit circle is used for
The unit circle extends sine and cosine from the acute angles of a right triangle to every angle, positive, negative or larger than 360°.
Trig values of any angle
Reference angles and quadrant signs give sin, cos and tan of 150°, −45° or 480° from the first-quadrant values.
Graphs of sine and cosine
Following the y-coordinate as θ goes round traces the sine wave, and the x-coordinate the cosine wave, both repeating every 2π.
Solving trig equations
sin θ = 1/2 has two solutions per turn, 30° and 150°: the two points of the circle at height 1/2.
Rotation and waves in physics
Circular motion, alternating current and phasors all describe a point turning on a circle, with cos and sin of the phase as its x and y.
Unit circle chart FAQ
What is the unit circle?+
The unit circle is the circle of radius 1 centred at the origin (0, 0). An angle θ drawn from the positive x-axis meets it at the point (cos θ, sin θ), which is why a unit circle chart lists cosine and sine as coordinates.
Why does the unit circle have a radius of 1?+
In a right triangle sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse. With the hypotenuse equal to 1 the fractions disappear: the adjacent side is cos θ and the opposite side is sin θ, so they become the x and y coordinates of the point.
How do you find the coordinates of an angle on the unit circle?+
Find its reference angle (the acute angle to the x-axis), take the first-quadrant values for that angle, then give them the signs of the quadrant. 210° has reference angle 30° and lies in quadrant III, so its point is (−√3/2, −1/2). The calculator beside the unit circle chart on this page does this for any angle.
How do you find tan on the unit circle?+
Divide the y coordinate by the x coordinate: tan θ = sin θ / cos θ. At 135° the point is (−√2/2, √2/2), so tan 135° = −1. Where x = 0, at 90° and 270°, tan θ is undefined. Switch on tan in the unit circle chart above to print all 16 values.
What are the 16 angles on the unit circle chart?+
0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315° and 330°; in radians 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4 and 11π/6.
How do you convert degrees to radians?+
Multiply by π/180 and simplify: 225° × π/180 = 5π/4. To go back, multiply radians by 180/π: 7π/6 × 180/π = 210°. Every radian label on the unit circle chart above is one of these products, simplified.
How do negative angles work on the unit circle?+
A negative angle is measured clockwise from the positive x-axis. −45° ends where 315° does, so they share the point (√2/2, −√2/2). In general cos(−θ) = cos θ and sin(−θ) = −sin θ.
How do you draw a unit circle?+
Draw the x and y axes and a circle of radius 1 at their crossing. Mark 0°, 90°, 180° and 270° where the circle meets the axes, then 30°, 45° and 60° in the first quadrant, and mirror those three into the other quadrants. Label each point with its angle and its (cos θ, sin θ), or download the finished unit circle chart above as PNG or SVG.
Sources
- 5.2 Unit Circle: Sine and Cosine Functions — Precalculus 2e, OpenStax
- §4.14 Definitions and Periodicity (trigonometric functions) — NIST Digital Library of Mathematical Functions
Keep exploring
Our other diagram tools, and the guides behind them.
Print the unit circle chart you need
Full chart, radians only, first quadrant or a blank worksheet with its key, free in your browser, or as a poster or illustrated chart from the AI generator.
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