Unit Circle Calculator

This free unit circle calculator reports cosine, sine, and tangent of an angle θ measured in degrees. Type the default θ = 30° and the point on the rim is (cos 30°, sin 30°) = (√3/2, 1/2) ≈ (0.8660, 0.5000), with tan 30° = 1/√3 ≈ 0.5774. A unit circle is the circle of radius 1 centered at the origin O. From the positive x-axis, θ opens to a ray that meets the circle at P.

No account is required. This widget takes degrees, not radians: 30 in Angle θ (°) is thirty degrees, which the formulas convert by multiplying by π/180 before cosine and sine. How it works beside the form keeps P = (cos θ, sin θ) and tan θ = sin θ / cos θ on separate lines. Cosine is the x-coordinate of P; sine is the y-coordinate. Tangent is that ratio, and it is undefined when cosine is 0.

What is a unit circle?

A unit circle is a circle whose radius is 1. On this page the center is O at (0, 0), so the equation of the rim is x² + y² = 1. Every point P on that rim is at distance 1 from O. The angle θ is the directed opening from the positive x-axis to the ray OP: positive θ goes counterclockwise, negative θ clockwise. Because the radius is 1, the usual adjacent-over-hypotenuse and opposite-over-hypotenuse ratios collapse: the adjacent side of the reference triangle is exactly x, and the opposite side is exactly y.

Sine and cosine as coordinates

Drop a perpendicular from P to the x-axis and you get a right triangle with hypotenuse OP = 1. Then cos θ = adjacent / hypotenuse = x / 1 = x, and sin θ = opposite / hypotenuse = y / 1 = y. That is the whole trick of the unit circle: cosine is the x-coordinate and sine is the y-coordinate of P. At the default 30°, P sits at (√3/2, 1/2). Pythagoras on those coordinates is the identity cos² θ + sin² θ = 1, which is the same statement as x² + y² = 1.

Tangent — sine over cosine

Tangent is sine divided by cosine: tan θ = sin θ / cos θ, or y / x on the unit circle. For θ = 30° that is (1/2) / (√3/2) = 1/√3 ≈ 0.5774, which this calculator writes in the tan output. Division is illegal when the denominator is 0, so tan θ is undefined at 90° and 270° — and at any angle that lands on those after wrapping, such as −90° or 450°. In the script, if |cos θ| is smaller than 10⁻¹² the tan field is Infinity, and the output displays as an em dash rather than a number. The drawing skips a tangent length so the labels for θ, cosine, and sine stay readable.

Degrees versus radians, and how to use this calculator

A full turn is 360° or 2π radians. This calculator never asks for radians in the form: you type degrees. Behind the scenes, θ_rad = θ° × π/180, then cosine and sine run on that radian measure, which is what JavaScript's Math.cos and Math.sin expect. Common conversions: 30° = π/6, 90° = π/2, 180° = π, 360° = 2π. Enter Angle θ (°); the figure and the three outputs update together. A central opening from arc length is the Central Angle Calculator, which reports both radians and degrees. The How to Calculate Circle Area and Circumference tutorial keeps a radius playground if you want the disk, not the trig point.

0°, 90°, 180°, 360°, and negative angles

At 0°, P is (1, 0): cos = 1, sin = 0, tan = 0. At 90°, P is (0, 1): cos = 0, sin = 1, and tan is undefined. At 180°, P is (−1, 0): cos = −1, sin = 0, tan = 0. At 360° the ray has gone all the way around, so you are back at (1, 0) with the same three values as 0°. Angles outside 0°–360° wrap: 390° is the same point as 30°. Negative angles are allowed — there is no minimum on the input — and −30° is the same point as 330°, in the fourth quadrant, with cos (−30°) = √3/2, sin (−30°) = −1/2, tan (−30°) = −1/√3.

Related circle formulas

The shaded disk of radius r, not necessarily 1, is the Area of a Circle Calculator. Expanding (x − h)² + (y − k)² = r² is the Equation of a Circle Calculator; the unit circle is the special case h = 0, k = 0, r = 1. An opening at the center from arc length s and radius r is the Central Angle Calculator. A half-disk is the Semicircle Area Calculator. A tangent length from an external point to a circle about the origin is the Tangent of a Circle Calculator.

Worked example: θ = 30°

Start with the default angle, θ = 30°. Convert to radians only if you are doing it by hand: 30 × π/180 = π/6. Then cos 30° = √3/2 ≈ 0.8660 and sin 30° = 1/2 = 0.5, so P = (√3/2, 1/2) on the unit circle. Tangent is the ratio: tan 30° = (1/2) / (√3/2) = 1/√3 ≈ 0.5774. Check: (√3/2)² + (1/2)² = 3/4 + 1/4 = 1. Those are the three outputs this page prints for 30°.

\[ P=\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right),\quad \tan 30^\circ=1/\sqrt{3} \]

Frequently asked questions

References

• Wikipedia — Unit circle

Assumptions and limitations

These unit-circle results are unofficial. Cosine and sine are JavaScript Math.cos and Math.sin after converting degrees to radians. Tangent is sine / cosine, or Infinity when |cosine| is smaller than 10⁻¹².