The unit circle is the single most useful picture in trigonometry. Every angle, measured from the positive x-axis, points to a spot on a circle of radius 1, and the coordinates of that spot are the cosine and sine of the angle. This calculator draws the circle for any angle you enter, marks the 16 standard angles, highlights the reference angle, and gives the exact coordinates when the angle is a special one.
How to use the unit circle calculator
- Type the angle as a number or expression:
120,-45,405,5pi/6. - Choose degrees or radians.
- Read the point P = (cos θ, sin θ) in exact form with decimals, along with tan θ, the quadrant, the reference angle, the coterminal angle and the arc length from (1, 0).
- Check the diagram: the blue ray is the terminal side, the shaded wedge is the reference angle, and the dashed lines drop from P to the axes. The table below lists the coordinates of all 16 standard angles, with yours in bold.
Unit circle formulas
For an angle θ in standard position, the terminal side meets the unit circle at:
The second equation is the Pythagorean identity sin²θ + cos²θ = 1 in disguise. Tangent is the slope of the terminal side, and the arc length on a radius-1 circle equals the angle in radians:
Worked example: θ = −210°
1. Find a coterminal angle. Add 360°: −210° + 360° = 150°. The ray ends in the same place after turning 210° clockwise.
2. Name the quadrant. 150° is in Quadrant II, so x is negative and y is positive.
3. Use the reference angle. 180° − 150° = 30°, and the 30° point is (√3/2, 1/2).
4. Apply the signs. P = (−√3/2, 1/2), or about (−0.866025, 0.5).
5. Read tangent. tan(−210°) = (1/2) ÷ (−√3/2) = −√3/3 ≈ −0.5773502692.
In radians, −210° is −7π/6, and the arc traced from (1, 0) has length 7π/6 ≈ 3.6651914292.
Memorizing the first quadrant
You only need five points; symmetry supplies the other eleven. The sines of 0°, 30°, 45°, 60° and 90° follow a tidy pattern — √0/2, √1/2, √2/2, √3/2, √4/2 — and the cosines run the same list backward:
| θ | Radians | cos θ (x) | sin θ (y) |
|---|---|---|---|
| 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 |
Using symmetry to fill the circle
Each first-quadrant point has three mirror images. Reflecting across the y-axis flips the sign of x; reflecting across the x-axis flips the sign of y. So the reference angle 30° produces four points:
- 30° → (√3/2, 1/2)
- 150° = 180° − 30° → (−√3/2, 1/2)
- 210° = 180° + 30° → (−√3/2, −1/2)
- 330° = 360° − 30° → (√3/2, −1/2)
The same three moves work for 45° and 60°. That is why every quadrant contains the same three “special” points with different signs, and why the reference angle plus the quadrant is enough to find any value.
Reading other functions off the circle
Beyond sine and cosine, the circle shows the reciprocal functions too: sec θ = 1/x, csc θ = 1/y and cot θ = x/y. Points on the axes make some of these undefined; at 180°, for example, y = 0, so csc 180° and cot 180° do not exist. For all six values with steps, use the trigonometric functions calculator. To see how the y-coordinate traces a wave as the angle keeps turning, open trig function graphs.
Frequently asked questions
What is the unit circle?
It is the circle of radius 1 centered at the origin. The terminal side of any angle θ crosses it at the point (cos θ, sin θ), which is why it is used to define sine and cosine for every angle, not just acute ones.
How do I find tan θ from the unit circle?
Divide the y-coordinate by the x-coordinate: tan θ = sin θ ÷ cos θ. At 120°, the point is (−1/2, √3/2), so tan 120° = −√3. Where x = 0, at 90° and 270°, tangent is undefined.
Why is the radius 1?
With radius 1, the hypotenuse of every reference triangle is 1, so the legs equal cos θ and sin θ directly and no division is needed. The arc length from (1, 0) to the point also equals the angle in radians.
Do negative angles work?
Yes. Negative angles turn clockwise from the positive x-axis. −210° ends at the same place as 150°, so both have the point (−√3/2, 1/2). The diagram draws the clockwise arc so you can see the direction.