Unit Circle with Reference Angles
Enter an angle to see its position on the unit circle: the coordinates, quadrant, reference angle, and exact sine, cosine, and tangent values, updated live.
The unit circle ties together nearly every idea covered elsewhere on this site, reference angles, quadrants, and exact trig values all show up as a single labeled point here. This page walks through the definition, maps reference angles to coordinates, and lists exact values for all sixteen standard positions around the circle.
What is the unit circle?
The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. For any angle θ measured in standard position, the terminal side crosses the circle at exactly one point, and that point's coordinates are always (cos θ, sin θ).
This works because, in a circle of radius 1, the horizontal distance from the center to any point on the edge is by definition the cosine of the angle, and the vertical distance is the sine. There's no triangle to draw and no ratio to divide. The radius is already 1, so cosine and sine are read directly as coordinates.
The unit circle is what lets trigonometry extend past right triangles. A triangle can only represent angles between 0° and 90°. The unit circle represents every angle, including negative angles and angles greater than 360°, because a point can sit anywhere around the full circle.
Choosing a radius of exactly 1 isn't arbitrary. It's what makes the coordinates equal sine and cosine directly, with no division required. In a circle of radius r, the coordinates of a point at angle θ are (r cos θ, r sin θ); setting r = 1 strips away the multiplication entirely. This is also why the unit circle is the natural home for the Pythagorean identity sin²θ + cos²θ = 1. It's just the equation of the circle itself, x² + y² = 1, rewritten in terms of the angle.
How reference angles map to unit circle coordinates
Every reference angle corresponds to a first-quadrant point on the unit circle, and every other angle with that same reference angle lands at a point with the same coordinate magnitudes. Only the signs change. For example, the reference angle of 210° is 30°. The coordinates at 30° are (√3/2, 1/2). The coordinates at 210° are (−√3/2, −1/2), same numbers, both signs flipped, because 210° is in Quadrant III where both cosine and sine are negative.
This is why memorizing the first-quadrant unit circle values for 30°, 45°, and 60° reveals the entire circle. Once you know the reference angle and the quadrant, you know the exact coordinates everywhere.
Unit circle table: all 16 standard positions
Exact values, not decimals. √2/2, √3/2, and their variants repeat throughout the circle.
| Angle (°) | Angle (rad) | Coordinates (cos, sin) | sin | cos | tan |
|---|---|---|---|---|---|
| 0° | 0 | (1, 0) | 0 | 1 | 0 |
| 30° | π/6 | (√3/2, 1/2) | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | (√2/2, √2/2) | √2/2 | √2/2 | 1 |
| 60° | π/3 | (1/2, √3/2) | √3/2 | 1/2 | √3 |
| 90° | π/2 | (0, 1) | 1 | 0 | - |
| 120° | 2π/3 | (−1/2, √3/2) | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | (−√2/2, √2/2) | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | (−√3/2, 1/2) | 1/2 | −√3/2 | −√3/3 |
| 180° | π | (−1, 0) | 0 | −1 | 0 |
| 210° | 7π/6 | (−√3/2, −1/2) | −1/2 | −√3/2 | √3/3 |
| 225° | 5π/4 | (−√2/2, −√2/2) | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | (−1/2, −√3/2) | −√3/2 | −1/2 | √3 |
| 270° | 3π/2 | (0, −1) | −1 | 0 | - |
| 300° | 5π/3 | (1/2, −√3/2) | −√3/2 | 1/2 | −√3 |
| 315° | 7π/4 | (√2/2, −√2/2) | −√2/2 | √2/2 | −1 |
| 330° | 11π/6 | (√3/2, −1/2) | −1/2 | √3/2 | −√3/3 |
| 360° | 2π | (1, 0) | 0 | 1 | 0 |
How to use the unit circle to find trig values for any angle
- 1
Reduce and find the reference angle
Normalize the angle between 0° and 360°, then apply the quadrant formula to get the reference angle.
- 2
Read the first-quadrant value
Look up the sine and cosine of the reference angle in the table above.
- 3
Apply the sign
Use ASTC to decide whether sine, cosine, and tangent are positive or negative in the original angle's quadrant.
Frequently Asked Questions
What is the unit circle?
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Every point on it has coordinates (cos θ, sin θ), where θ is the angle measured from the positive x-axis. It is the standard tool for defining trig functions at any angle, not just acute ones.
What are the coordinates on the unit circle?
The coordinates of any point on the unit circle are (cos θ, sin θ), where θ is the standard-position angle to that point. For example, at 30° the coordinates are (√3/2, 1/2). The same pattern holds at every angle, including negative angles and angles beyond 360°.
Why is the unit circle important in trigonometry?
The unit circle connects angles directly to sine and cosine values without needing a triangle. Because the radius is 1, the x-coordinate equals cos θ and the y-coordinate equals sin θ automatically, which extends trig functions to any angle, not just angles inside a right triangle.
What is the reference angle for 5π/6 on the unit circle?
5π/6 is in Quadrant II, between π/2 and π. Its reference angle is π − 5π/6 = π/6. The coordinates at 5π/6 are (−√3/2, 1/2), which have the same magnitude as the coordinates at π/6, (√3/2, 1/2), with the x-value sign flipped.
How do you find sin and cos using the unit circle?
Locate the angle on the unit circle, then read the y-coordinate for sine and the x-coordinate for cosine. For angles beyond the first quadrant, find the reference angle first, look up its sine and cosine, then apply the correct sign for the quadrant.