Unit Circle Reference
Pick or type any angle and see its radians, quadrant, and exact cosine and sine values on an interactive unit circle.
| Degrees | Radians | cos θ | sin θ | (x, y) |
|---|
What does this Unit Circle Reference tool do?
This tool is an interactive unit circle. Choose an angle and it instantly shows the radian value, the quadrant, and the point (cos θ, sin θ) where that angle meets the circle. You can pick one of 16 standard angles or type any angle in degrees, including negative numbers and values above 360. For standard angles the results appear as exact fractions with decimals beside them, and for every other angle you get decimals rounded to four places. A built-in reference table lists all 16 standard angles together, so you can use it as a unit circle chart and a calculator in one place.
How to Use the Unit Circle Reference Tool
- Choose a standard angle. Open the Standard angle dropdown, or click or tap one of the dots on the circle. Each option lists the angle in degrees and in radians.
- Or type your own angle. Enter a number of degrees in the Custom angle box, such as -45, 135, or 750. If the entry is not a valid number, a short message tells you what to type instead.
- Read the circle. A shaded sector and an arc marked θ show the angle, a highlighted radius points to the selected spot, and dashed lines drop to the x-axis and y-axis. The matching quadrant label is highlighted.
- Read the results panel. It lists the angle you entered, its equivalent angle when different, the radians, the quadrant, the point (cos θ, sin θ), cos θ, sin θ, and the x² + y² check.
- Open the Reference table. Tick the Reference table checkbox to see every standard angle in one table. Click or select a row with the keyboard to jump to that angle.
- Adjust the view. Untick Angle labels to hide the labels around the circle if you want a cleaner picture.
- Start over anytime. Click Reset to 0° to return to the starting angle.
Built-In Features
Click, Tap, or Use the Keyboard
Select any of the 16 standard angles by clicking a dot on the circle, picking it from the dropdown, or tabbing to a dot and pressing Enter or Space for quick and easy angle selection.
Any Custom Angle in Degrees
Type a negative value, a decimal, or a number above 360. The tool maps it to the matching position on the circle and shows the equivalent angle.
Exact Values for Standard Angles
Cosine and sine appear as true fractions such as √3/2 or 1/2 for standard angles, with a decimal beside each one for quick comparison and easier understanding of values.
Live Visual Feedback
A shaded sector, an arc marked θ, a highlighted radius, and dashed lines down to the axes update with every angle. The active quadrant label is highlighted too.
Clickable Reference Table
Switch on the table to see degrees, radians, cos θ, sin θ, and the (x, y) point for all 16 standard angles. Click any row to jump to that angle.
Built-In Sanity Check
Each result ends with an x² + y² line that equals 1, which confirms the point really lies on the unit circle.
How to Read the Unit Circle
The unit circle has a radius of 1 and sits at the center of the x and y axes. Angles start at the positive x-axis and turn counterclockwise. Wherever the angle lands on the circle, that point has two coordinates. The x-coordinate is the cosine of the angle and the y-coordinate is the sine. Because the radius is exactly 1, no extra division is needed, and that is what makes the circle such a handy shortcut for trigonometry.
Cosine Is the x-Coordinate
Cosine tells you how far right or left the point sits. It is positive on the right side of the circle, negative on the left, and zero at the very top and bottom.
Sine Is the y-Coordinate
Sine tells you how far up or down the point sits. It is positive in the upper half of the circle, negative in the lower half, and zero on the x-axis.
Signs of Cosine and Sine in Each Quadrant
| Quadrant | Angle Range | cos θ (x) | sin θ (y) |
|---|---|---|---|
| Quadrant I | Between 0° and 90° | Positive | Positive |
| Quadrant II | Between 90° and 180° | Negative | Positive |
| Quadrant III | Between 180° and 270° | Negative | Negative |
| Quadrant IV | Between 270° and 360° | Positive | Negative |
Every point on the circle follows x² + y² = 1. If a cosine and sine pair you wrote by hand does not add up to 1 after squaring, one of the values has a mistake.
Unit Circle Values for 0, 30, 45, 60, and 90 Degrees
These five angles are the foundation of the whole circle. Once you know them, the remaining standard angles are mirror images with different signs.
| Degrees | Radians | cos θ | sin θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | π/6 | √3/2 | 1/2 |
| 45° | π/4 | √2/2 | √2/2 |
| 60° | π/3 | 1/2 | √3/2 |
| 90° | π/2 | 0 | 1 |
A simple memory trick helps here. Sine goes 0, 1/2, √2/2, √3/2, 1 as the angle grows from 0 to 90 degrees, which you can also write as √0/2, √1/2, √2/2, √3/2, √4/2. Cosine uses the same list in reverse order.
Finding Values in the Other Quadrants
For any angle outside the first quadrant, find its reference angle, which is the acute angle between its radius and the x-axis. Take the cosine and sine of that smaller angle, then apply the signs for the quadrant.
| Quadrant | Reference Angle | Example |
|---|---|---|
| Quadrant II | 180° minus θ | 150° has a reference angle of 30° |
| Quadrant III | θ minus 180° | 225° has a reference angle of 45° |
| Quadrant IV | 360° minus θ | 300° has a reference angle of 60° |
Converting Between Degrees and Radians
A full turn is 360 degrees, which equals 2π radians, so half a turn is 180 degrees or π radians. To convert degrees to radians, multiply by π/180. To go back, multiply radians by 180/π. For standard angles the tool does this for you, simplifies the fraction, and also shows a decimal approximation. For example, 210 degrees appears as 7π/6.
| Degrees | Radians | Part of a Full Turn |
|---|---|---|
| 30° | π/6 | One twelfth |
| 45° | π/4 | One eighth |
| 60° | π/3 | One sixth |
| 90° | π/2 | One quarter |
| 180° | π | One half |
| 270° | 3π/2 | Three quarters |
| 360° | 2π | One full turn |
If you also need to convert other everyday measurements, our Unit Converter is a handy companion to this page.
What Happens With Negative Angles and Angles Over 360 Degrees
The circle repeats every 360 degrees, so angles that look very different can land on the same point. A negative angle turns clockwise instead of counterclockwise, and an angle above 360 simply goes around more than once. To find where either one ends up, add or subtract 360 until the result falls between 0 and 360. The tool does this automatically and shows the result in the Equivalent angle row.
| Angle Entered | Equivalent Angle | Where It Lands |
|---|---|---|
| 750° | 30° | Quadrant I |
| 405° | 45° | Quadrant I |
| -45° | 315° | Quadrant IV |
| -90° | 270° | Negative y-axis |
| 540° | 180° | Negative x-axis |
| 1080° | 0° | Positive x-axis |
Angles that share the same landing point are called coterminal angles, and they always have identical cosine and sine values.
Common Reasons to Use a Unit Circle Reference
Checking Homework Answers
Work a problem by hand, then enter the angle to compare your cosine and sine with the exact values on screen.
Revising Before a Test
Click through the 16 standard angles and say each point out loud before the tool reveals it. It is a quick way to test your memory.
Sine and Cosine Graphs
Watch the dashed lines shrink and grow as the angle changes. That is the same height and width that a sine or cosine graph plots.
Placing Points Around a Circle
Programmers and designers use x = cos θ and y = sin θ to place items around a circle. Test the angle here first to see where a point should land.
Useful Tips
Learn the First Quadrant Only
Memorize the values for 0, 30, 45, 60, and 90 degrees. Every other standard angle reuses those numbers with different signs.
Decide the Sign First
Before reading the size of a value, check the quadrant. It tells you at once whether cosine and sine are positive or negative.
Use the Table for Quick Lookups
Keep the Reference table open while you work through problems, then click a row to see where that angle sits on the circle.
Hide Labels for a Cleaner View
If the angle labels feel crowded while you study a single point, untick Angle labels. The dots and the selected angle stay in place.
Frequently Asked Questions
What is the unit circle?
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate grid. Any angle measured from the positive x-axis points to one spot on the circle, and that point’s x-coordinate is the cosine of the angle while its y-coordinate is the sine.
How do I find sin and cos of an angle with this tool?
Pick an angle from the Standard angle dropdown, click a dot on the circle, or type your own value into the Custom angle box. The panel on the right shows cos θ as the x value and sin θ as the y value, along with the full point (cos θ, sin θ).
Can I enter negative angles or angles larger than 360 degrees?
Yes. The custom angle box accepts negative numbers, decimals, and values above 360. The tool also shows an equivalent angle between 0 and 360 degrees, so -45 becomes 315 and 750 becomes 30.
Can I type an angle in radians?
The custom box takes degrees only. The 16 standard angles are labeled in both degrees and radians, so you can read their radian form directly. To use another radian value, multiply it by 180 and divide by π, then type the result in degrees.
Why do some results show square roots and others show decimals?
The 16 standard angles have exact values such as √3/2 and 1/2, so the tool shows those as fractions with a decimal beside them. Any other angle has no simple exact form here, so the tool shows decimals rounded to four places.
Does a large or negative angle still show exact values?
Yes, if its equivalent angle lands on one of the standard angles. For example, 750 degrees has the same position as 30 degrees, so it shows the exact values for 30 degrees.
Does this tool show tangent?
No. It shows cosine and sine, which are the x and y coordinates of the point. Tangent equals sine divided by cosine, so you can work it out from the two values shown. It is undefined when cosine is 0, at 90 and 270 degrees.
What does the Quadrant row say for angles like 90 or 180 degrees?
Those angles sit on an axis rather than inside a quadrant, so the row reads None and names the axis, such as the positive y-axis for 90 degrees or the negative x-axis for 180 degrees.
What is the x² + y² check in the results?
Every point on the unit circle satisfies x² + y² = 1. The tool calculates that sum from the cosine and sine values it shows, so you can confirm the result is a valid point on the circle.
How do I clear my angle and start again?
Click Reset to 0°. It returns the selection to the starting angle of 0 degrees at the point (1, 0).
What does the Reference table show?
Turn on the Reference table checkbox to see all 16 standard angles in one table with degrees, radians, cos θ, sin θ, and the (x, y) point. Clicking a row selects that angle on the circle.