Unit Circle Calculator
Explore exact Cartesian coordinates (x, y) = (cos θ, sin θ), degree-to-radian conversions, and all 6 trigonometric functions (sin, cos, tan, csc, sec, cot) on an interactive unit circle (r = 1).
Interactive Angle Controller
Step-by-Step Unit Circle Evaluation & ASTC Sign Analysis
How the Unit Circle Defines Trigonometric Functions
On a unit circle of radius r = 1 centered at (0, 0), any angle θ produces a point P(x, y) on the circumference where x = cos θ and y = sin θ. Tangent is the slope tan θ = y / x = sin θ / cos θ. Key identities follow from the circle equation x² + y² = 1, giving sin² θ + cos² θ = 1.
The Unit Circle (x² + y² = 1) & (cos θ, sin θ) Coordinates
The unit circle is the foundational bridge connecting right-triangle geometry to continuous periodic wave functions. Defined algebraically as the circle with radius r = 1 centered at (0, 0):
By constructing a right reference triangle inside the circle with hypotenuse r = 1, the adjacent side along the x-axis equals cos θ, and the vertical opposite side equals sin θ.
Special Reference Triangles (30°-60°-90° and 45°-45°-90°)
Both legs are equal in length: x = y = 1 / √2 = √2 / 2.
Derived by bisecting an equilateral triangle of side length 1.
The 4 Quadrants & The ASTC Sign Rule
Exact Coordinates Table for All 16 Standard Unit Circle Angles
| Degrees | Radians | cos θ (x) | sin θ (y) | tan θ (y/x) |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3 / 2 | 1 / 2 | √3 / 3 |
| 45° | π/4 | √2 / 2 | √2 / 2 | 1 |
| 60° | π/3 | 1 / 2 | √3 / 2 | √3 |
| 90° | π/2 | 0 | 1 | Undefined |
| 180° | π | −1 | 0 | 0 |
| 270° | 3π/2 | 0 | −1 | Undefined |
Real-World Applications of the Unit Circle
AC Electrical Power Grids
Electrical engineers use phasor diagrams rotating at 60 Hz around the unit circle to calculate real power, reactive power, and phase angles.
Digital Signal Processing (DSP)
Fourier Transforms decompose audio, speech, and wireless radio signals using Euler's formula ($e^{i\theta} = \cos \theta + i\sin \theta$) mapped on the complex unit circle.
Game Engine Physics
3D graphics engines use unit vector direction normalization $(v_x / |v|, v_y / |v|)$ for character movement trajectories and raycasting.
Step-by-Step Worked Trigonometric Evaluations
Problem: Evaluate the exact value of tan(150°).
Common Pitfalls & Mistakes
Reversing (x, y) as (sin, cos)
x is ALWAYS cosine, and y is ALWAYS sine: (x, y) = (cos θ, sin θ). Reversing them ruins all downstream calculations.
Forgetting Quadrant Signs
Forgetting that cosine is negative in Quadrants II & III leads to severe errors in physics and engineering.
Dividing by Zero at Asymptotes
tan(90°) and tan(270°) are undefined because cos θ = 0 (vertical division by zero).
Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.
Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.