Math · Interactive
Unit Circle
The unit circle has a radius of 1, and the point at any angle is simply (cos, sin). Drag the point or the slider below to see sine, cosine, and tangent update live, snap to every common angle, and switch between degrees and radians. Free to use, and exportable into your slides.
Drag the point, or the angle slider, to explore. Toggle degrees and radians, and snap to common angles. Open fullscreen ↗
Use it in your lesson
Drop this into your slides or hand it to students. Free downloads carry a small figviz.io watermark; spend 1 credit to export a clean, high-resolution version.
Remove watermark · high-res · 1 credit each
What is the unit circle?
The unit circle is a circle with a radius of 1 centered at the origin (0, 0) of the coordinate plane. Its equation is x squared plus y squared equals 1. For any angle measured counterclockwise from the positive x-axis, the point where the angle meets the circle has coordinates (cos of the angle, sin of the angle). That single idea links angles to the sine and cosine functions and is why the unit circle is the backbone of trigonometry.
How to read sine, cosine, and tangent
On the unit circle the x-coordinate of the point is the cosine of the angle and the y-coordinate is the sine of the angle. Tangent is sine divided by cosine, so it is the y-coordinate divided by the x-coordinate. Because the radius is 1, sine and cosine always stay between -1 and 1, while tangent is undefined at 90 and 270 degrees where the cosine is 0. Drag the point in the interactive above to watch all three values change in real time.
Degrees, radians, and the common angles
Angles on the unit circle are written in degrees or in radians, where a full turn is 360 degrees or 2 pi radians. The common angles (multiples of 30 and 45 degrees) have exact coordinate values built from 1/2, root 2 over 2, and root 3 over 2. The unit circle chart below lists every common angle with its radian measure and exact coordinates, and the interactive can snap to each one so students connect the picture to the numbers.
How to memorize the unit circle
A reliable trick is to learn the first quadrant (0, 30, 45, 60, 90 degrees) and then reflect it into the other three quadrants, flipping the sign of x in the left half and the sign of y in the bottom half. The sine values in the first quadrant follow the pattern root 0 over 2, root 1 over 2, root 2 over 2, root 3 over 2, root 4 over 2, which simplifies to 0, 1/2, root 2 over 2, root 3 over 2, 1. Practicing with the interactive and a blank printed chart makes the pattern stick.
Unit circle chart (common angles)
| Degrees | Radians | (cos, sin) |
|---|---|---|
| 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) |
| 120° | 2π/3 | (-1/2, √3/2) |
| 135° | 3π/4 | (-√2/2, √2/2) |
| 150° | 5π/6 | (-√3/2, 1/2) |
| 180° | π | (-1, 0) |
| 210° | 7π/6 | (-√3/2, -1/2) |
| 225° | 5π/4 | (-√2/2, -√2/2) |
| 240° | 4π/3 | (-1/2, -√3/2) |
| 270° | 3π/2 | (0, -1) |
| 300° | 5π/3 | (1/2, -√3/2) |
| 315° | 7π/4 | (√2/2, -√2/2) |
| 330° | 11π/6 | (√3/2, -1/2) |
Frequently asked questions
What is the unit circle?
The unit circle is a circle of radius 1 centered at the origin. Its equation is x^2 + y^2 = 1, and the point at an angle has coordinates (cos, sin), which connects angles to the trigonometric functions.
How do you find sine and cosine on the unit circle?
For a given angle, the point where it meets the unit circle has an x-coordinate equal to the cosine of the angle and a y-coordinate equal to the sine of the angle. Tangent is the sine divided by the cosine.
What is a unit circle chart?
A unit circle chart lists the common angles (multiples of 30 and 45 degrees) with their radian measures and exact coordinates (cos, sin). The chart on this page covers every common angle, and you can generate or export your own.
What is the unit circle in radians?
In radians a full turn is 2 pi, so 90 degrees is pi/2, 180 degrees is pi, and 270 degrees is 3pi/2. The interactive lets you toggle between degrees and radians for any angle.
Why is tangent undefined at 90 and 270 degrees?
Tangent equals sine divided by cosine. At 90 and 270 degrees the cosine is 0, and division by zero is undefined, so tangent has no value there and its graph has a vertical asymptote.
How do I use this interactive unit circle?
Drag the point around the circle, or drag the angle slider, to change the angle. The panel shows sine, cosine, tangent, and the coordinates, snapping to common angles, and you can switch between degrees and radians. You can also export it to use in your own slides.