Unit Circle Calculator — Trig Values
Enter an angle in degrees or radians to find sin, cos, tan, csc, sec, cot — with exact values for special angles, reference angle, quadrant, and unit-circle coordinates. Converts degrees to radians and back.
What Is the Unit Circle?
The unit circle is a circle with radius 1 centered at the origin of the Cartesian plane. Every point on the circle has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. The unit circle provides a geometric way to define all six trigonometric functions for any angle — not just acute angles in a right triangle.
Trig Definitions on the Unit Circle
sin θ = y-coordinate
cos θ = x-coordinate
tan θ = sin θ / cos θ
csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = cos θ / sin θ
Conversion: radians = degrees × π / 180
Special Angles
The unit circle has well-known exact values at the "special angles": 0°, 30°, 45°, 60°, and 90° (and their equivalents in all four quadrants). For example:
- sin(30°) = 1/2, cos(30°) = √3/2
- sin(45°) = √2/2, cos(45°) = √2/2
- sin(60°) = √3/2, cos(60°) = 1/2
- sin(90°) = 1, cos(90°) = 0
This calculator recognises all special angles and shows their exact symbolic values alongside decimal approximations.
Reference Angles
A reference angle is the acute angle between the terminal side of your angle and the x-axis. It is always between 0° and 90°. The trigonometric function values for any angle equal those of its reference angle, with the sign determined by the quadrant (CAST rule).
The CAST Rule
CAST tells you which functions are positive in each quadrant. Reading counterclockwise from Q4: Cosine (Q4), All (Q1), Sine (Q2), Tangent (Q3). All other functions in that quadrant are negative. This rule, combined with reference angles, lets you evaluate trig functions without memorizing the entire unit circle.
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Applications
The unit circle and trigonometric functions are essential in physics (waves, oscillations, circular motion), engineering (signal processing, AC circuits), computer graphics (rotations, transformations), navigation (bearings, GPS), music (sound waves), and architecture (structural angles). Understanding the unit circle is a prerequisite for calculus, differential equations, and Fourier analysis.