Quick answer
On the unit circle a point at angle θ is (cos θ, sin θ). At 60° the point is (0.5, 0.8660) in quadrant I.
Enter an angle and choose degrees or radians. The calculator returns the coordinates and quadrant.
Related tools: Circle Calculator: Radius, diameter, circumference and area of a circle from any one of them. Trigonometry Calculator: All six trigonometric functions of an angle. SOHCAHTOA Calculator: Find the other two sides of a right triangle from one side and an acute angle.
How to use the Unit Circle Calculator
- Enter angle.
- Choose unit: Degrees and Radians.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Angle | A number | — |
| Unit | Choose one: Degrees; Radians | Starts at Degrees |
Formula and method
Formula
(x, y) = (cos θ, sin θ)
IEEE-754 double precision; shown to at most 6 decimal places.
Point on the circle of radius 1 at angle θ measured counter-clockwise from the positive x-axis.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Angle = 60; Unit = Degrees.
| Result | Value |
|---|---|
| Point (cos θ, sin θ) | (0.5, 0.866025) |
| Quadrant | I |
Example 2
Inputs: Angle = 180; Unit = Degrees.
| Result | Value |
|---|---|
| Point (cos θ, sin θ) | (-1, 0) |
| Quadrant | On an axis |
Example 3
Inputs: Angle = 225; Unit = Degrees.
| Result | Value |
|---|---|
| Point (cos θ, sin θ) | (-0.707107, -0.707107) |
| Quadrant | III |
The first line is the headline result (point (cos θ, sin θ)). Below it the unit Circle Calculator lists quadrant, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Reading the circle
The x-coordinate is the cosine and the y-coordinate the sine. At 0° the point is (1, 0), at 90° it is (0, 1), at 180° (−1, 0) and at 270° (0, −1).
Signs by quadrant
Quadrant I: both positive. II: sine positive. III: both negative. IV: cosine positive. Any angle size is accepted: 420° gives the same point as 60°.
How the result changes with angle
Holding every other input at the first worked example, this table changes only angle. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Angle | Point (cos θ, sin θ) | Quadrant |
|---|---|---|
| 30 | (0.866025, 0.5) | I |
| 60 | (0.5, 0.866025) | I |
| 120 | (-0.5, 0.866025) | II |
| 240 | (-0.5, -0.866025) | III |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Angle: left empty shows "Enter angle." beside that field and no result.
Choosing between the Unit Circle Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Circle Calculator](/math/circle-calculator/) | i know the and value | Area |
| [Trigonometry Calculator](/math/trigonometry-calculator/) | angle and unit | sin |
| [SOHCAHTOA Calculator](/math/sohcahtoa-calculator/) | acute angle, known side length and known side is the | Opposite |
| [Reference Angle Calculator](/math/reference-angle/) | angle | Reference angle |
| [Inverse Sine Calculator](/math/inverse-sine/) | value | arcsin(0.5) |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- Angles of any size
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for angle to confirm this.
Frequently asked questions
What is the point at 45°?
At 45° the point is (cos 45°, sin 45°) = (0.7071, 0.7071).
Why is the radius 1?
So sine and cosine equal the coordinates.
What is the point at 120°?
At 120° the point is (−0.5, 0.8660), in quadrant II.
What does the Unit Circle Calculator calculate?
Coordinates on the unit circle for an angle. It returns Point (cos θ, sin θ) and Quadrant.
What formula does the Unit Circle Calculator use?
(x, y) = (cos θ, sin θ). Point on the circle of radius 1 at angle θ measured counter-clockwise from the positive x-axis.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Angles of any size.
What does the “Unit” option change?
It sets how the unit Circle is worked out. The choices are Degrees and Radians.
What is an example of a unit Circle result?
With angle 60 and unit Degrees, the point (cos θ, sin θ) is (0.5, 0.866025).
Related calculators
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
- Trigonometry Calculator — All six trigonometric functions of an angle.
- SOHCAHTOA Calculator — Find the other two sides of a right triangle from one side and an acute angle.
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
- Inverse Sine Calculator — arcsin (inverse trigonometric function) with the result in degrees and radians.
- Coterminal Angle Calculator — Angles sharing the same terminal side.
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.