📐 Unit Circle Visualizer

A unit circle is a circle with a radius of exactly 1, centered at the origin (0,0) of a coordinate plane. It's the fundamental tool for defining trigonometric functions. On the unit circle, any point on the circumference has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. This elegant relationship makes the unit circle essential for understanding sine, cosine, and all other trigonometric functions geometrically.

For any angle θ measured from the positive x-axis:
• sin θ = the y-coordinate of the point on the unit circle
• cos θ = the x-coordinate of the point on the unit circle
• tan θ = sin θ / cos θ = the length of the tangent line segment from (1,0) to where the radius line (extended) intersects the vertical line x = 1
This geometric interpretation helps visualize why sin²θ + cos²θ = 1 (the Pythagorean theorem on the unit circle).

The most important special angles (in degrees) are: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, and 360°. These correspond to radian measures that are simple fractions of π, such as π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), and so on. At these angles, the sine and cosine values take on exact forms like 1/2, √2/2, and √3/2, making them easy to memorize and essential for exact calculations.

Degrees divide a full circle into 360 equal parts (360° = one full rotation). Radians measure angles by the length of the arc on the unit circle. One full rotation equals 2π radians (≈ 6.283 radians). The conversion formula is: radians = degrees × π / 180. Radians are the natural unit for calculus and higher mathematics because they simplify derivative and integral formulas for trigonometric functions.

Tangent is defined as tan θ = sin θ / cos θ. At 90° (π/2 radians), cos(90°) = 0 and sin(90°) = 1, so tan(90°) = 1/0, which is undefined (division by zero). Geometrically, at 90° the radius line points straight up and never intersects the vertical tangent line at x = 1, so no tangent segment exists. The same applies at 270° (3π/2). As θ approaches 90° from below, tan θ → +∞; from above, tan θ → −∞.

The six fundamental trigonometric functions are:
• sin θ (sine) — y-coordinate on the unit circle
• cos θ (cosine) — x-coordinate on the unit circle
• tan θ (tangent) — sin/cos, slope of the radius line
• csc θ (cosecant) — 1/sin, the reciprocal of sine
• sec θ (secant) — 1/cos, the reciprocal of cosine
• cot θ (cotangent) — 1/tan or cos/sin, the reciprocal of tangent
All six can be visualized geometrically on the unit circle using various line segments.

The unit circle provides a visual reference for understanding trigonometric relationships. It helps you:
• Quickly determine the sign of sin, cos, and tan in each quadrant
• Memorize special angle values through geometric patterns
• Understand periodicity (sin and cos repeat every 2π)
• Visualize identities like sin²θ + cos²θ = 1
• Solve equations by seeing where angles intersect the circle
• Convert between degrees and radians intuitively

The coordinate plane is divided into four quadrants:
• Quadrant I (0°–90°): sin > 0, cos > 0, tan > 0 — All positive
• Quadrant II (90°–180°): sin > 0, cos < 0, tan < 0 — Only sine positive
• Quadrant III (180°–270°): sin < 0, cos < 0, tan > 0 — Only tangent positive
• Quadrant IV (270°–360°): sin < 0, cos > 0, tan < 0 — Only cosine positive
A common mnemonic is "All Students Take Calculus" (All, Sine, Tangent, Cosine positive in Q1–Q4 respectively).