Understanding the Unit Circle Evaluator

The unit circle is a fundamental concept in trigonometry. It is defined as a circle centered at the origin (0,0) with a radius of exactly 1 unit. The equation for the unit circle is x² + y² = 1. This simple circle allows us to extend the definitions of sine, cosine, and tangent to any angle, not just acute angles in right triangles.

Coordinates and Trigonometry

For any angle θ in standard position (starting from the positive x-axis), the point where the terminal side of the angle intersects the unit circle has coordinates (cos θ, sin θ).

• Cosine (x-coordinate): Represents the horizontal distance from the origin.
• Sine (y-coordinate): Represents the vertical distance from the origin.
• Tangent (y/x): Represents the slope of the line (sine divided by cosine).

Why use a Unit Circle Evaluator?

Memorizing the “special angles” (30°, 45°, 60°) is crucial for calculus and physics. However, calculating values for arbitrary angles like 127° or 5π/7 can be difficult without visualization. This tool helps students bridge the gap between the abstract numerical value and the geometric position on the circle. It also handles the conversion between Degrees and Radians instantly.

Degrees vs. Radians

While degrees are common in daily life (360° in a circle), mathematicians prefer radians. A full circle is 2π radians. The conversion factor is π radians = 180°.