How this instrument works
The unit circle is simply the circle of radius exactly 1 centered at the origin of a coordinate plane. Any angle θ, measured counterclockwise from the positive x-axis, corresponds to exactly one point on that circle, and the coordinates of that point ARE, by definition, (cos θ, sin θ). This is not a formula derived from something more basic — it is the actual definition sine and cosine are built from, extended this way precisely so the two functions stay meaningful for angles beyond a right triangle's 0°-to-90° range, including negative angles and angles past a full turn.
Four landmark points anchor the whole circle and are worth knowing by heart: 0° sits at (1, 0), due east; 90° sits at (0, 1), straight up; 180° sits at (−1, 0), due west; and 270° sits at (0, −1), straight down. Every other angle's coordinates interpolate smoothly between these four reference points as the angle sweeps around.
Because the circle has radius 1, the coordinates automatically satisfy x² + y² = 1 for every angle — this IS the Pythagorean trigonometric identity, sin²θ + cos²θ = 1, seen geometrically rather than algebraically. Reading coordinates off the unit circle and confirming the Pythagorean identity are, in a real sense, the same fact viewed from two different angles.
- Enter an angle into the θ (theta) field, measured counterclockwise from the positive x-axis, in whichever unit is convenient.
- Read x = cos(θ): the horizontal coordinate of the point on the unit circle at that angle.
- Read y = sin(θ): the vertical coordinate of that same point.
- Try angles beyond 90° or negative angles to see the coordinates move into other quadrants, including negative x or y values.
Worked example — 60° on the unit circle
At 60°, the point on the unit circle sits at (cos 60°, sin 60°) = (0.5, ≈0.866) — exactly halfway across horizontally, and a bit more than 86% of the way up vertically, well into the first quadrant since 60° is between the 0° and 90° reference points.
Compare 0° and 90° directly: at 0° the point sits at (1, 0), the reference direction itself, and at 90° it has swung a quarter turn counterclockwise to (0, 1) — the x-coordinate dropping from 1 all the way to 0 while the y-coordinate rises from 0 to 1, exactly the trade-off cosine and sine make across that quarter turn.
Questions
What is the unit circle?
It's the circle of radius exactly 1 centered at the origin of a coordinate plane. Any angle measured counterclockwise from the positive x-axis lands on one specific point on that circle, and that point's coordinates are, by definition, (cos θ, sin θ).
Why is the unit circle used to define sine and cosine?
Because it extends sine and cosine cleanly to every possible angle, not just the 0°-to-90° range a right triangle can represent. Angles beyond 90°, negative angles, and angles past a full 360° turn all still correspond to a real point somewhere on the circle, so cos θ and sin θ stay meaningful everywhere.
What are the coordinates at the four quadrant angles?
0° is at (1, 0); 90° is at (0, 1); 180° is at (−1, 0); and 270° is at (0, −1). These four landmark points are worth memorizing, since every other angle's coordinates fall smoothly between whichever pair of these it sits between.
How does the unit circle relate to sin²θ + cos²θ = 1?
Directly — since every point on the unit circle is a distance of exactly 1 from the origin, its coordinates always satisfy x² + y² = 1. Substituting x = cos θ and y = sin θ turns that circle equation directly into the Pythagorean trigonometric identity.
Can the angle be negative or larger than 360°?
Yes — a negative angle simply measures clockwise instead of counterclockwise, and an angle past 360° has wrapped around the circle more than once, landing on the same point a smaller equivalent angle would. Sine and cosine handle both cases without any special adjustment.