Interactive Unit Circle Explorer
Understand trigonometric functions visually. Explore how angles relate to coordinates, and master sine, cosine, and tangent through hands-on interaction.
Try It Yourself!
Play with the interactive Unit Circle below. See how different angles (in radians or degrees) affect sine, cosine and tangent. Hover over the circle to explore values - the line will automatically follow your mouse!
🧮 Quick Calculator
📊 Common Angles
| Degrees | Radians | cos(θ) | sin(θ) | tan(θ) |
|---|---|---|---|---|
| 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 | ∞ |
| 120° | 2π/3 | -1/2 | √3/2 | -√3 |
| 135° | 3π/4 | -√2/2 | √2/2 | -1 |
| 150° | 5π/6 | -√3/2 | 1/2 | -√3/3 |
| 180° | π | -1 | 0 | 0 |
🎯 What is the Unit Circle?
The Unit Circle is a fundamental concept in trigonometry - a circle with a radius of exactly 1 unit, centered at the origin (0,0) of the coordinate plane.
Key Insights:
- •Every point on the circle represents (cos θ, sin θ)
- •Perfect for visualizing periodic functions
- •Connects geometry with trigonometry
📐 Pythagoras Theorem & Unit Circle
For any point (x, y) on the unit circle:
Since x = cos θ and y = sin θ, we get the fundamental identity:
📈 Trigonometric Functions
Cosine (cos θ) = x-coordinate
The horizontal distance from the origin
Sine (sin θ) = y-coordinate
The vertical distance from the origin
Tangent (tan θ) = sin θ / cos θ
The slope of the radius line
🎯 Memory Aid - Hand Trick
For Cosine:
For Sine:
Quick Tip:
Count your fingers! For 30°, there are 3 fingers above → cos = √3/2
🎮 Interactive Challenge
Can you find an angle where sine and cosine are equal?
Hint: Look for where x = y on the unit circle. Move your mouse over the interactive circle to explore!
Unit Circle Quiz (1/5)
What is cos(0°) on the unit circle?
Unit Circle Quadrants
Master the Unit Circle - Complete Guide
Understanding the Unit Circle
The unit circle is fundamental to trigonometry because it provides a geometric interpretation of the trigonometric functions. Every point on the unit circle corresponds to an angle measured from the positive x-axis, with coordinates (cos θ, sin θ).
Radians and Degrees
Angles can be measured in degrees (0° to 360°) or radians (0 to 2π). The conversion is simple: 180° = π radians. Our interactive tool lets you switch between both measurement systems to build intuition.
Trigonometric Identities
The unit circle demonstrates key trigonometric identities:
- • Pythagorean Identity: cos²θ + sin²θ = 1
- • Periodicity: Functions repeat every 360° or 2π radians
- • Symmetry: cos(-θ) = cosθ, sin(-θ) = -sinθ
Ready to Master the Unit Circle?
Practice with our interactive tools and become a trigonometry expert!
🔍 Unit Circle Topics & Keywords
Values are calculated to 4 decimal places. Perfect for learning and exploration!