Unit Circle Calculator: Exact Sine, Cosine and Tangent
Enter an angle in degrees or radians to get its point on the unit circle, sine, cosine, tangent, reference angle and quadrant. Exact standard angles.
Unit Circle Calculator
Enter an angle in degrees or radians to get its point on the unit circle, its sine, cosine and tangent, its reference angle and its quadrant. The standard angles are answered exactly.
Point on the circle (x, y)
(√3/2, 1/2)
Sine, cosine and tangent
cos θ (the x coordinate)
√3/2
sin θ (the y coordinate)
1/2
tan θ
√3/3
This is one of the sixteen standard angles, so the values are exact rather than rounded. As decimals: cos θ = 0.8660254, sin θ = 0.5, tan θ = 0.57735027.
Where it lands
Quadrant or axis
Quadrant I
Reference angle
30°
Reference angle in radians
π/6rad
The angle
Angle in degrees
30°
Angle in radians
0.52359878rad
Folded into one turn
30°
Folded angle in radians
π/6rad
The unit circle has radius 1 and is centred on the origin. A point on it is (cos θ, sin θ), so the coordinates and the two function values are the same pair of numbers. The tangent is sin θ divided by cos θ, which has no value where cos θ is 0.
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Documentation
Unit Circle Calculator
The unit circle is the circle of radius 1 drawn around the origin of a graph. A unit circle calculator takes one angle and reports where that angle meets the circle: the point (x, y), the sine, cosine and tangent, the reference angle, and the quadrant. This page reads an angle in degrees or in radians and gives exact values such as √3/2 at the sixteen standard angles.
What the calculator does
A visitor types one angle and picks degrees or radians. The page returns:
- the point (x, y) where the angle crosses the circle
- cos θ, sin θ and tan θ
- the reference angle, in degrees and in radians
- the quadrant, or the axis when the angle lands on one
- the angle in both units, and the same angle folded into a single turn
At the sixteen standard angles the answers are printed as exact closed forms, with the decimals shown underneath. Any other angle is shown to eight significant figures. Only the display is rounded.
The unit circle formula
Angles are measured anticlockwise from the positive x-axis. A ray drawn at angle θ crosses the circle at one point, and the two coordinates of that point are the cosine and the sine:
where is the horizontal coordinate, is the vertical coordinate, and is the angle.
The radius is 1, so the coordinates and the two function values are the same pair of numbers under two names. That is the reason the circle is drawn with radius 1 rather than any other size. The tangent is built from the same pair:
Every point on the circle also satisfies Pythagoras' theorem, applied to the right triangle with sides and and hypotenuse 1:
Degrees and radians
The page accepts either unit and reports both. The two are related by half a turn being π radians:
\qquad \theta_{\mathrm{deg}} = \theta_{\mathrm{rad}} \times \frac{180}{\pi}$$ where $\theta_{\mathrm{deg}}$ is the angle in degrees and $\theta_{\mathrm{rad}}$ is the same angle in radians. ### Folding into one turn An angle can be negative and can pass a full turn. The point on the circle repeats every 360°, so the page first folds the angle into the range from 0° up to, but not including, 360°: $$\theta_0 = ((\theta \bmod 360) + 360) \bmod 360$$ where $\theta$ is the angle as typed, in degrees, and $\theta_0$ is the same direction written inside one turn. 750° folds to 30°, and −30° folds to 330°. The fold runs in degrees, where a remainder is exact. ### Reference angle and quadrant The reference angle is the acute angle between the ray and the x-axis. It is never more than 90°: $$\theta_{\mathrm{ref}} = \begin{cases} \theta_0 & \text{if } \theta_0 \le 90 \\ 180 - \theta_0 & \text{if } \theta_0 \le 180 \\ \theta_0 - 180 & \text{if } \theta_0 \le 270 \\ 360 - \theta_0 & \text{otherwise} \end{cases}$$ where $\theta_0$ is the folded angle in degrees. The size of the sine, cosine and tangent depends only on the reference angle. The quadrant supplies the signs. | Quadrant | Folded angle | cos | sin | tan | |---|---|---|---|---| | I | 0° to 90° | + | + | + | | II | 90° to 180° | − | + | − | | III | 180° to 270° | − | − | + | | IV | 270° to 360° | + | − | − | The angles 0°, 90°, 180° and 270° sit on an axis. A quadrant is the open region between two axes, so none of those four is inside one, and the page names the axis instead of picking a neighbour. ### The sixteen standard angles Five reference angles carry exact closed forms. | Reference angle | cos | sin | tan | |---|---|---|---| | 0° (0) | 1 | 0 | 0 | | 30° (π/6) | √3/2 | 1/2 | √3/3 | | 45° (π/4) | √2/2 | √2/2 | 1 | | 60° (π/3) | 1/2 | √3/2 | √3 | | 90° (π/2) | 0 | 1 | undefined | Reflecting those rows into the other three quadrants gives sixteen angles: 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315 and 330 degrees. The page generates them by rotating the first-quadrant points a quarter turn at a time, rather than transcribing the table four times, because a transcription is where a wrong sign gets in. 15°, 75° and 105° have closed forms too, but two-term ones such as 2 − √3. They fall outside this family and are answered as decimals. 15° gives cos 0.96592583, sin 0.25881905 and tan 0.26794919. ### Example: 210 degrees 210° is more than 180° and less than 270°, so it lands in quadrant III. Taking 180 away leaves a reference angle of 30°. At a reference angle of 30° the three sizes are √3/2, 1/2 and √3/3. Quadrant III makes both coordinates negative, so the point is (−√3/2, −1/2). The tangent is a negative divided by a negative, so it stays positive at √3/3. In radians the angle is 7π/6 and the reference angle is π/6. As decimals the page shows cos −0.8660254, sin −0.5 and tan 0.57735027. 570° and −150° give the same answers, because both fold to 210°. ### Which constants are exact | Constant | Value | Exact? | Source | |---|---|---|---| | Degrees in one turn | 360 | exact by definition | ISO 80000-3 | | Radians in one turn | 2π | exact by definition | ISO 80000-3; SI Brochure | | cos 30°, sin 60° | √3/2 | exact | NIST DLMF §4.14 | | cos 45°, sin 45° | √2/2 | exact | NIST DLMF §4.14 | | cos 60°, sin 30° | 1/2 | exact | NIST DLMF §4.14 | | tan 90°, tan 270° | no value | exact, a pole | NIST DLMF §4.15 | π is irrational, so it has no exact decimal or binary form and any stored value of it is rounded. The page keeps that rounding out of its answers in two ways. The fold runs in degrees, where 360 divides without remainder. And the standard angles are read from the table of closed forms above, never from a floating-point sine. ### Reading an angle typed in radians A radian entry is already a rounded decimal. π/2 is typed as 1.5707963267948966, and converting that back gives an angle about a ten-trillionth of a degree away from 90 rather than exactly 90. So the page reads any angle within a millionth of a degree of a standard angle as that angle. The tolerance chooses which table is used; it never widens an answer. 30.001° does not become 30°. Angles above a billion degrees are refused. That is a limit of the arithmetic, not of geometry: the gap between the numbers a computer can store grows with their size, and past roughly four and a half billion degrees the gap is wider than the millionth-of-a-degree tolerance, so folding no longer carries enough detail to name an angle. A billion keeps a margin below that point. ### Frequently asked questions **What is the unit circle used for?** It turns sine and cosine into coordinates. Reading one point off the circle gives both values at once, and the picture shows why they repeat every full turn and why their signs change from one quadrant to the next. **What is the exact value of sin 30°?** One half. cos 30° is √3/2, about 0.8660254, and tan 30° is √3/3, about 0.57735027. **Why is tan 90° undefined?** The tangent is the sine divided by the cosine, and cos 90° is 0. Dividing by zero has no answer. The page finds the pole by naming the angle, not by checking whether a computed tangent came back large, because a steep tangent just short of 90° is an ordinary number. **What is a reference angle?** It is the acute angle between the ray and the x-axis, so it always sits between 0° and 90°. 150°, 210° and 330° all have a reference angle of 30°, which is why their sine and cosine share the same three sizes and differ only in sign. **Does a negative angle work?** Yes. A negative angle turns clockwise. −30° points the same way as 330°, so the page returns the point (√3/2, −1/2) and quadrant IV. **Why does the page show the angle twice?** 750° and 30° cross the circle at the same point, but they are not the same rotation. The page echoes the angle as typed, converted into the other unit, and shows the folded angle beside it.