What this calculator does
The unit circle is a circle of radius 1 centred on the origin, and it is the tool trigonometry is built on: for any angle θ measured from the positive x-axis, the point where that angle meets the circle has coordinates (cos θ, sin θ). This calculator takes an angle in either degrees or radians and returns that point, its quadrant, the reference angle, and the sine, cosine and tangent values at that angle.
The habit most people get stuck on is remembering the standard angles: 0°, 30°, 45°, 60°, 90° and their multiples around the circle, and what sin and cos equal at each. Rather than memorising a chart, enter any angle here and the same values come out worked, alongside the quadrant it falls in and how far it sits from the nearest axis.
The formula
Convert the angle to radians if it was entered in degrees, then take cos θ and sin θ directly as the x and y coordinates of the point on the circle. The quadrant follows from the signs of those two values, and the reference angle is the acute angle between the terminal side and the x-axis, found by reflecting the angle back into the first quadrant.
| Term | Meaning |
|---|---|
| θ (theta) | The angle measured anticlockwise from the positive x-axis. |
| Reference angle | The acute angle (0° to 90°) between the terminal side and the nearest part of the x-axis. |
| Quadrant | Which quarter of the plane the point falls in, numbered I to IV anticlockwise from the top right. |
The inputs explained
| Field | What to enter |
|---|---|
| Angle | The angle to evaluate. Any real number is accepted, including negative angles and angles past 360°. |
| Units | Whether the angle above is in degrees or radians. |
When to use it
Building a unit circle table from scratch
Working through each standard angle in turn and recording the coordinates it gives reconstructs the full unit circle table without needing to memorise it beforehand.
Checking a trig identity by hand
Plugging in a specific angle checks whether an identity or a solved equation actually holds at that point, before trusting it more generally.
Reading a unit circle diagram
Given an angle marked on a diagram, this confirms the coordinates, quadrant and sign of each trig function that the diagram is meant to show.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What are the coordinates at the standard unit circle angles?
Coordinates and quadrant at the angles most textbooks list first.
| Angle | Point on the unit circle | Quadrant |
|---|---|---|
| 0° | (1.000, 0) | On an axis |
| 30° | (0.8660, 0.5000) | I |
| 45° | (0.7071, 0.7071) | I |
| 60° | (0.5000, 0.8660) | I |
| 90° | (0, 1.000) | On an axis |
| 120° | (-0.5000, 0.8660) | II |
| 135° | (-0.7071, 0.7071) | II |
| 150° | (-0.8660, 0.5000) | II |
| 180° | (-1.000, 0) | On an axis |
| 270° | (0, -1.000) | On an axis |
| 360° | (1.000, 0) | On an axis |
How does the reference angle change past 90 degrees?
The reference angle folds any angle back into the 0° to 90° range.
| Angle | Quadrant | Reference angle |
|---|---|---|
| 100° | II | 80.00° |
| 150° | II | 30.00° |
| 200° | III | 20.00° |
| 250° | III | 70.00° |
| 300° | IV | 60.00° |
| 350° | IV | 10.00° |
Questions
What is the unit circle used for?
It defines sine, cosine and tangent for any angle, not just the acute angles inside a right-angled triangle. Once an angle can be placed on the unit circle, those three ratios extend naturally to negative angles, angles beyond 90°, and full rotations past 360°.
Why is the radius always 1?
With a radius of exactly 1, the coordinates of the point at angle θ simplify directly to (cos θ, sin θ), with no scaling factor needed. That is the whole reason the construction is useful: it turns the two trig ratios into plain coordinates.
How do I read the unit circle table for negative angles?
A negative angle is measured clockwise instead of anticlockwise. Entering a negative value here returns the same point a positive angle of the same size would reach going the other way round the circle, with sin flipped in sign and cos unchanged.
Why is tan undefined at 90 degrees?
Tangent is sine divided by cosine, and cosine is exactly zero at 90° and 270°. Dividing by zero has no defined result, which is why those two angles are the vertical asymptotes on a graph of tan θ.
For the sine and cosine rule version of trigonometry, used to solve a triangle from its sides and angles, see the trigonometry: solve a triangle calculator. For sin, cos and tan without the coordinate geometry, see the trigonometric functions calculator.