What is the unit circle?
The unit circle is a circle of radius 1 centred at the origin. For an angle θ measured anticlockwise from the positive x-axis, the point where the angle's arm meets the circle has coordinates (cos θ, sin θ).
// maths › Trigonometric Functions & Graphs
The unit circle
Explore the unit circle: for any angle it gives the point (cos theta, sin theta), the quadrant and reference angle, and exact surd values at the special angles, in degrees or radians, with an animated point whose sine and cosine projections are drawn live alongside the right-triangle interpretation.
P = (cos θ, sin θ) on the circle of radius 1
Angle unit
Frequently asked questions
Why are the coordinates exactly cosine and sine?
Because the radius is 1, the right triangle drawn from the point has a hypotenuse of 1. The adjacent side is then cos θ and the opposite side is sin θ, so the point sits at (cos θ, sin θ) with no scaling needed.
Can you give a worked example?
At θ = 60°, the point on the unit circle is (cos 60°, sin 60°) = (0.5, 0.866). It lies in Quadrant I, where both coordinates are positive. The same angle in radians is 60° = π/3 ≈ 1.0472 rad — flip the Degrees/Radians toggle to enter and read it either way.
How does the unit circle extend trigonometry past 90°?
Right-angle trig only defines ratios for acute angles, but the unit circle defines sine and cosine for every angle, including obtuse, reflex, and negative ones. The sign of each coordinate depends on the quadrant, which is why the sine and cosine graphs rise and fall.
Why are radians natural on the unit circle?
On a circle of radius 1, the arc length swept equals the angle measured in radians. A full turn is 2π, a right angle is π/2, and a straight angle is π — which is why higher mathematics measures angles this way.
Where is this used in real life?
Engineers and physicists model anything that oscillates or rotates — alternating current, sound and light waves, springs, and orbits. Computer graphics and game developers use it for rotation and circular motion, and navigators use it for circular bearings.