What it is
The unit circle is the circle in the coordinate plane with its centre at the origin and radius 1. Its equation is x² + y² = 1: a point (x, y) is on it exactly when x² + y² = 1.
The angle v is measured from the positive x-axis, counter-clockwise. The ray from the origin at angle v meets the circle in one point, and that point’s coordinates define cosine and sine:
(x, y) = (cos v, sin v)
Examples:
- v = 0°: the point is (1, 0), so cos 0° = 1 and sin 0° = 0.
- v = 90°: the point is (0, 1), so cos 90° = 0 and sin 90° = 1.
- v = 60°: the point is (1/2, √3/2) ≈ (0.5, 0.866).
- v = 180°: the point is (−1, 0), so cos 180° = −1. The definition therefore works for obtuse angles, where a right triangle is not enough.
- v = 120°: the point is (−1/2, √3/2), so cos 120° = −0.5 and sin 120° = sin 60° ≈ 0.866.
What is not on the unit circle:
- (0.6, 0.6) is not a point on it: 0.6² + 0.6² = 0.72, not 1. (0.6, 0.8) is on it, since 0.36 + 0.64 = 1.
- A circle of radius 2 is not a unit circle. There the point at v is (2 cos v, 2 sin v).
- Tangent is not a coordinate on the circle. It is the quotient sin v / cos v, and undefined where cos v = 0.
All the values are in the unit circle table.
Why it matters
In a right triangle, sine and cosine can only describe angles below 90°. The unit circle extends them to every angle, negative ones and ones beyond a full turn, which is what makes sine on the unit circle and cosine on the unit circle meaningful. The Pythagorean identity follows directly: cos² v + sin² v = 1, because the point lies on x² + y² = 1.
It also gives the symmetries, such as sin (180° − v) = sin v, and the periodicity: after a full turn you are back where you started. Angles are often given in radians. The right-triangle definitions of sine and cosine give the same values for acute angles. The curves can be plotted in the graphing calculator.
Common misconceptions
- “Sine is the x-coordinate.” It is the other way round: cosine is the x-coordinate and sine the y-coordinate.
- “Sine and cosine are always positive.” In a right-angled triangle they are, because sides are positive. On the unit circle cosine is negative left of the y-axis and sine below the x-axis: cos 120° = −0.5.
- “Sine can exceed 1.” On the unit circle y is at most 1 and at least −1, so sin v lies between −1 and 1.
- “Negative angles do not exist.” A negative angle is a turn clockwise: −90° gives the point (0, −1).
Common questions
Why radius 1?
Then the coordinates are cos v and sin v directly, with no extra factor.
What are cos 135° and sin 135°?
The point is (−√2/2, √2/2) ≈ (−0.707, 0.707), so cos 135° ≈ −0.707 and sin 135° ≈ 0.707.
Do I need to memorise the table?
The values for 0°, 30°, 45°, 60° and 90° are enough; the rest follow from the symmetry of the circle.
The whole table is further down this page.