Interactive unit circle with tangent
Follow a point around the circle. Its coordinates are (cos θ, sin θ), and tan θ = sin θ / cos θ. Below the explorer is a complete unit circle chart with exact values.
Make an angle. See the connection.
Quadrant I · both coordinates are positive.
One circle. Three trigonometric functions.
The unit circle has a radius of 1. Starting at the positive x-axis, an angle θ moves counterclockwise around the circle. The point you land on is (cos θ, sin θ). Tangent is the ratio sin θ / cos θ, so it is undefined where cosine is zero.
Use the number field or arrow keys on the slider for precise, keyboard-accessible exploration. Special angles show exact values alongside decimal approximations.
Unit circle chart with sine, cosine & tangent
Every special angle from 0° to 330°, in degrees and radians, with its point (cos θ, sin θ) and tangent. Tangent is undefined at 90° and 270°, where cos θ = 0.
| Degrees | Radians | cos θ | sin θ | tan θ | Quadrant |
|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 | Axis |
| 30° | π/6 | √3/2 | ½ | √3/3 | I |
| 45° | π/4 | √2/2 | √2/2 | 1 | I |
| 60° | π/3 | ½ | √3/2 | √3 | I |
| 90° | π/2 | 0 | 1 | undefined | Axis |
| 120° | 2π/3 | −½ | √3/2 | −√3 | II |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 | II |
| 150° | 5π/6 | −√3/2 | ½ | −√3/3 | II |
| 180° | π | −1 | 0 | 0 | Axis |
| 210° | 7π/6 | −√3/2 | −½ | √3/3 | III |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 | III |
| 240° | 4π/3 | −½ | −√3/2 | √3 | III |
| 270° | 3π/2 | 0 | −1 | undefined | Axis |
| 300° | 5π/3 | ½ | −√3/2 | −√3 | IV |
| 315° | 7π/4 | √2/2 | −√2/2 | −1 | IV |
| 330° | 11π/6 | √3/2 | −½ | −√3/3 | IV |
How to remember the chart
- In quadrant I, the sines of 30°, 45°, and 60° are √1/2 (= ½), √2/2, and √3/2; the cosines run the same list backwards.
- Signs follow the quadrant: “All Students Take Calculus” means all three ratios are positive in I, only sine in II, only tangent in III, and only cosine in IV.
- Each angle in quadrants II–IV reuses a quadrant-I value: 150° has the same reference angle, 30°, as 210° and 330°.
Unit circle questions
Wondering about the why? Start here.
Where is tangent on the unit circle?
Tangent is the ratio of the coordinates: tan θ = sin θ / cos θ = y / x. Geometrically, extend the radius until it meets the vertical line x = 1; the height of that meeting point is tan θ. That is why tangent grows without bound near 90°.
See tangent for every special angleHow does the unit circle show sine and cosine?
A unit circle has radius 1. Measure an angle counterclockwise from the positive x-axis: the point on the circle has coordinates (cos θ, sin θ). Its horizontal coordinate is cosine and its vertical coordinate is sine.
Connect the trigonometric ratiosWhy is tangent undefined at 90° and 270°?
Tangent is sin θ divided by cos θ. At 90° and 270°, cosine is zero, so the ratio would divide by zero. The explorer displays “undefined” at those angles.
Explore sine and cosine graphsWhat is the difference between degrees and radians?
Both measure angles. A full turn is 360° or 2π radians, so 180° equals π radians. Multiply degrees by π/180 to convert to radians; multiply radians by 180/π to convert to degrees.
Convert degrees and radians