Unit Circle Calculator
Drag the point around the circle and watch the angle, the six trig functions, and the sine and cosine waves update live — with exact values at every special angle.
Drag the point around the circle, use the slider, or pick a common angle.
- Radians
- π/4 0.7854 rad
- Gradians
- 50 gon
- Quadrant
- Quadrant I
Trig functions
| Exact | Decimal | |
|---|---|---|
| sin | √2/2 | 0.7071 |
| cos | √2/2 | 0.7071 |
| tan | 1 | 1 |
| csc | √2 | 1.4142 |
| sec | √2 | 1.4142 |
| cot | 1 | 1 |
The point on the circle is (cos θ, sin θ): cos θ is its horizontal position and sin θ its height.
Common angles
Sine & cosine waves
The height of the point is sin θ; its horizontal position is cos θ. As θ goes from 0° to 360°, the height traces the sine wave and the horizontal position traces the cosine wave.
About the Unit Circle Calculator
The unit circle is the circle of radius 1 centred on the origin, and it is the single most useful picture in all of trigonometry. Any point on it can be written as (cos θ, sin θ), where θ is the angle measured counter-clockwise from the positive x-axis. That is why cosine is just the point's horizontal position and sine is its height — no memorising required. This interactive tool lets you drag the point (or use the slider and preset buttons) and instantly read the angle in degrees, radians and gradians, see which quadrant you are in, and get all six trig functions: sine, cosine, tangent, and their reciprocals cosecant, secant and cotangent. At the special angles (0°, 30°, 45°, 60°, 90° and their reflections) you also get the exact values such as √3/2 and √2/2, not just decimals. A reference right triangle and a live sine/cosine wave strip connect the picture to the formulas. Everything runs in your browser with exact trigonometry, so it is the visual companion to any trig textbook.
How to use the unit circle
- 1 Set the angle: drag the point around the circle, slide the angle slider, or tap a common-angle button (0°, 30°, 45°, 60°, …).
- 2 Turn on "Snap to 15°" if you want the angle to jump to clean values, or leave it off to explore freely.
- 3 Read the angle in degrees, radians (with the exact value like π/3 at special angles) and gradians, and check which quadrant the point is in.
- 4 Read the six trig functions: each row shows the exact value at special angles (such as √3/2 or undefined) alongside the 4-decimal approximation. Watch the sine and cosine waves track the point's height and horizontal position.
The math behind it
On a circle of radius 1, a point at angle θ has coordinates (cos θ, sin θ). The other four functions are defined from these: tan θ = sin θ ⁄ cos θ — the slope of the radius line. csc θ = 1 ⁄ sin θ, sec θ = 1 ⁄ cos θ, cot θ = cos θ ⁄ sin θ — the reciprocals. Angles can be measured in degrees, in radians (where π rad = 180°, so a radian is the angle whose arc length equals the radius), or in gradians (400 gon = a full turn). The exact values at 30°, 45° and 60° come from two special right triangles: the 45-45-90 triangle gives sin 45° = cos 45° = √2⁄2, and the 30-60-90 triangle gives sin 30° = 1⁄2, sin 60° = √3⁄2. A function is undefined wherever you would divide by zero: tan θ and sec θ are undefined when cos θ = 0 (at 90° and 270°), and csc θ and cot θ are undefined when sin θ = 0 (at 0° and 180°).
Frequently asked questions
What is the unit circle?
The unit circle is the circle of radius 1 centred at the origin. A point on it at angle θ (measured counter-clockwise from the positive x-axis) has coordinates (cos θ, sin θ). This makes the unit circle the easiest way to define and visualise the trig functions for any angle, not just the angles inside a right triangle.
How are degrees and radians related?
A full turn is 360° or 2π radians, so π radians = 180° and 1 radian ≈ 57.2958°. A radian is the angle for which the arc length equals the radius. To convert, multiply degrees by π⁄180 to get radians, or multiply radians by 180⁄π to get degrees. This tool shows both, plus gradians (400 gon = a full turn).
Why is tan 90° undefined?
Tangent is defined as tan θ = sin θ ⁄ cos θ. At 90° (and 270°) the cosine is 0, so the formula divides by zero and the value is undefined — the radius line is vertical and has no finite slope. For the same reason secant (1⁄cos θ) is undefined there, while cosecant and cotangent are undefined at 0° and 180° where sine is 0.
What are the exact values at 30°, 45° and 60°?
sin 30° = 1⁄2, cos 30° = √3⁄2, tan 30° = √3⁄3. sin 45° = cos 45° = √2⁄2, tan 45° = 1. sin 60° = √3⁄2, cos 60° = 1⁄2, tan 60° = √3. These come from the 45-45-90 and 30-60-90 special triangles, and the tool shows the exact value next to the decimal for every special angle.
Related tools
Keep exploring math on UnitConv. Use the triangle solver for right and oblique triangles, plot sine and cosine with the graph/function plotter, convert between degrees, radians and gradians in the unit converter's angle category, and browse the formulas library for the full set of trig identities and special-angle values.