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Unit Circle
Interactive unit circle for trigonometry class. Set degrees from 0 to 360, read radians, sine, cosine, and tangent, and print a chart.
Any degree measure; negatives and turns past 360 wrap onto the circle. The slider moves in whole degrees from 0 to 360.
0° through 360°
Examples
Standard angles
Radius 1. Angles run counterclockwise from the positive x-axis. The dashed leg drops to the x-axis.
Unit circle. Point at 30 degrees. Coordinates (0.8660, 0.5000). Sine 0.5000, cosine 0.8660, tangent 0.5774.
Formulas
(x, y) = (cos θ, sin θ)
radians = degrees × π / 180
tan θ = sin θ / cos θ
With your angle
radians = 30° × π / 180 = π/6 ≈ 0.5236
(cos 30°, sin 30°) = (√3/2, 1/2) ≈ (0.8660, 0.5000)
sin 30° = 1/2 ≈ 0.5000
cos 30° = √3/2 ≈ 0.8660
tan 30° = (1/2) / (√3/2) = √3/3 ≈ 0.5774
Coordinates (cos θ, sin θ)
(0.8660, 0.5000)
sin 30°
1/2
0.5000
cos 30°
√3/2
0.8660
tan 30°
√3/3
0.5774
Quadrant I. Sine, cosine, and tangent are positive. Reference angle 30°. Opposite |sin θ| = 1/2, adjacent |cos θ| = √3/2, hypotenuse = 1.
Radians: π/6 ≈ 0.5236
First-quadrant chart
| Degrees | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
The other quadrants use these absolute values with the sign for that quadrant. Print chart includes this table and the labeled circle.
This unit circle is for trigonometry class, when you need the point where an angle meets the circle of radius 1. Move the slider or type a degree measure from 0 to 360, and the diagram draws the terminal side, the reference triangle in the correct quadrant, and the coordinates (cos θ, sin θ). Standard angles such as 30°, 45°, and 60°, in every quadrant, also show exact sine, cosine, and tangent, including the sign. Tangent is marked undefined at 90° and 270°, and a print button makes a unit circle chart with all 16 standard angles in degrees, radians, and exact coordinates, plus a first-quadrant table.
Reading the point on the circle
The unit circle is centered at the origin and has radius 1. An angle is measured counterclockwise from the positive x-axis, and the ray meets the circle at (cos θ, sin θ). Use the slider, type any degree measure (negatives and turns past 360 wrap onto the circle, so −30° matches 330°), or choose a standard angle. The diagram moves the point, draws the reference triangle in that quadrant, and lists the radian measure together with sine, cosine, and tangent.
Exact values, signs, and the chart
Angles of 0°, 30°, 45°, 60°, and 90°, together with the related angles around the circle, have exact sine, cosine, and tangent built from 1/2, √2/2, and √3/2. In quadrant I every value is positive. In quadrant II the cosine and tangent pick up a minus sign. In quadrant III the sine and cosine are negative and the tangent is positive. In quadrant IV the sine and tangent are negative and the cosine is positive. The right triangle drawn to the x-axis is the reference triangle, and its acute angle with that axis is the reference angle.
The table lists the first-quadrant angles in exact form. Print chart opens a page with that table and a circle labeled in degrees and radians, ready for a notebook or a quiz review. At 90° and 270°, cosine is 0, so tangent, which is sine divided by cosine, is undefined.
Frequently Asked Questions
What is the unit circle?
The unit circle is the circle of radius 1 centered at the origin. For an angle measured counterclockwise from the positive x-axis, the point on the circle is (cos θ, sin θ). Sine is the y-coordinate, cosine is the x-coordinate, and tangent is sine divided by cosine.
How do you turn degrees into radians?
Multiply the degree measure by π and divide by 180. The tool writes that step with your number filled in. For 30°, 30 × π / 180 simplifies to π/6, about 0.5236 radians.
What are the exact values at 30°, 45°, and 60°?
At 30° (π/6), sine is 1/2, cosine is √3/2, and tangent is √3/3. At 45° (π/4), sine and cosine are √2/2 and tangent is 1. At 60° (π/3), sine is √3/2, cosine is 1/2, and tangent is √3. Around the circle those absolute values stay the same, and the sign follows the quadrant.
Why is tangent undefined at 90° and 270°?
Tangent equals sine divided by cosine, and cosine is the x-coordinate. At 90° the point is (0, 1), and at 270° the point is (0, −1), so cosine is 0. Dividing by 0 has no result, and the tool says tangent is undefined.
How do the signs change in each quadrant?
In quadrant I, sine, cosine, and tangent are positive. In quadrant II, sine is positive while cosine and tangent are negative; in quadrant III, tangent is positive while sine and cosine are negative; in quadrant IV, cosine is positive while sine and tangent are negative. A negative angle wraps onto the same point as a positive one: type −30° and the tool shows the 330° point.