Introduction to the Unit Circle in Trigonometry
The unit circle is one of the most important conceptual structures in trigonometry. It is a circle with a radius of exactly one (r = 1), centered at the origin (0, 0) in the Cartesian plane. While basic trigonometry defines sine, cosine, and tangent using right triangles (restricted to angles between 0° and 90°), the unit circle extends these definitions to any real angle, positive or negative, in degrees or radians.
On the unit circle, any angle θ corresponds to a coordinate point (x, y) where the terminal ray intersects the circle. By definition, x = cos(θ) and y = sin(θ). The tangent is the ratio of these coordinates: tan(θ) = y/x. This circle underlies graphing trigonometric functions, solving trigonometric equations, and analyzing wave harmonics in physics and engineering.
This calculator evaluates trigonometric values for any target angle. Enter the angle in degrees or radians, and it computes the coordinate values, reference angle, quadrant, and the reciprocal trig functions (cosecant, secant, cotangent), returning exact fraction/radical values for the 16 standard multiples of 30° and 45°.