About the Unit Circle Calculator
On the unit circle — a circle of radius 1 centred at the origin — the point reached by turning an angle θ counter-clockwise from the positive x-axis has coordinates (cos θ, sin θ). This unit circle calculator finds that point for any angle you enter, in degrees or radians, and gives sine, cosine and tangent along with their reciprocals cosecant, secant and cotangent.
For the special angles that students memorise — every multiple of 30° and 45° — and also every multiple of 15°, it shows the exact values in radical form, such as sin 150° = 1/2 and cos 150° = −√3/2, plus the angle as a fraction of π. It also gives the reference angle and quadrant, the two ideas that let you reconstruct the whole unit circle from the first quadrant. The table lists all sixteen standard angles as a printable unit circle chart.
Any angle is accepted, including negative angles and angles beyond one full turn; they are reduced to the coterminal angle between 0° and 360° first.
How to use the unit circle calculator
- 1Enter the angle.
- 2Choose degrees or radians.
- 3Read the point (cos θ, sin θ) and the exact sin, cos and tan values.
- 4Use the reference angle and quadrant to see where the signs come from.
- 5Scroll to the chart for all sixteen standard angles at once.
Formula and method
A point on the unit circle at angle θ forms a right triangle with hypotenuse 1, so its horizontal leg is cos θ and its vertical leg is sin θ; tangent is their ratio, sin θ ÷ cos θ, and is undefined where cos θ = 0 (90° and 270°). The reciprocals are csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = cos θ/sin θ.
Exact values come from the 30-60-90 and 45-45-90 triangles: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2 (and the half-angle values for 15° and 75°). For any other quadrant, find the reference angle — the acute angle to the x-axis — use its first-quadrant values, and attach the sign that x and y have in that quadrant.
- θ
- Angle measured counter-clockwise from the positive x-axis
- x, y
- Coordinates of the point on the circle
- θ_ref
- Reference angle (acute angle to the x-axis)
Worked examples
θ = 150°
150° is in Quadrant II with reference angle 30°. sin 30° = 1/2 and cos 30° = √3/2; in Quadrant II x is negative, so the point is (−√3/2, 1/2) and tan 150° = −√3/3 ≈ −0.5774. In radians it is 5π/6.
θ = 90° (tangent undefined)
At 90° the point is straight up at (0, 1). Because cos 90° = 0, tan 90° = 1/0 and sec 90° are undefined.
θ = 7π/4 radians
7π/4 rad = 315°, in Quadrant IV with a 45° reference angle. The point is (√2/2, −√2/2) and tan = −1.
θ = −60° (negative angle)
Turning 60° clockwise lands at the same place as 300°. The reference angle is 60°, so sin = −√3/2 ≈ −0.8660, cos = 1/2 and csc = −2/√3 ≈ −1.1547.
Frequently asked questions
What is the unit circle?+
A circle of radius 1 centred at the origin. The point at angle θ has coordinates (cos θ, sin θ), which is why it is used to define sine and cosine for every angle, not just acute ones.
How do you remember the unit circle values?+
Learn the first quadrant — sines of 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2 — then use reference angles and the signs in each quadrant (All Students Take Calculus) for the rest.
Why is tan 90° undefined?+
tan θ = sin θ ÷ cos θ, and cos 90° = 0. Dividing by zero is undefined, so tan 90° and tan 270° have no value; the tangent graph has vertical asymptotes there.
What is a reference angle?+
The acute angle between the terminal side of θ and the x-axis. 150°, 210° and 330° all have a 30° reference angle, so their sines and cosines share the magnitudes of 30° and differ only in sign.
How do I convert degrees to radians on the unit circle?+
Multiply by π/180. So 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π and 360° = 2π.