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!

Cosine
cos = 0.7071
x-coordinate
Sine
sin = 0.7071
y-coordinate
Tangent
tan = 1.0000
sin/cos

🧮 Quick Calculator

📊 Common Angles

DegreesRadianscos(θ)sin(θ)tan(θ)
0°0100
30°π/6√3/21/2√3/3
45°π/4√2/2√2/21
60°π/31/2√3/2√3
90°π/201∞
120°2π/3-1/2√3/2-√3
135°3π/4-√2/2√2/2-1
150°5π/6-√3/21/2-√3/3
180°π-100

🎯 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:

x² + y² = 1

Since x = cos θ and y = sin θ, we get the fundamental identity:

cos²θ + sin²θ = 1
Unit Circle Equation
x² + y² = 1

📈 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:

3cos(30°) = √3/2
2cos(45°) = √2/2
1cos(60°) = 1/2

For Sine:

1sin(30°) = 1/2
2sin(45°) = √2/2
3sin(60°) = √3/2

Quick Tip:

Count your fingers! For 30°, there are 3 fingers above → cos = √3/2

Remember: Just 3 numbers: 1/2, √2/2, and √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

Quadrant I
0° to 90°
All positive
Quadrant II
90° to 180°
Sin positive
Quadrant III
180° to 270°
Tan positive
Quadrant IV
270° to 360°
Cos positive
🎯
Interactive Learning
Visualize angles and coordinates in real-time
📊
Common Values
Memorize key angles and their trig values
🧮
Dual Units
Switch between degrees and radians
🎮
Practice Quiz
Test your knowledge with interactive questions

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

Unit Circle • Trigonometric Functions • Sine Cosine Tangent •Radians and Degrees • Unit Circle Chart • Trig Identities •Unit Circle Values • Unit Circle Quadrants • Unit Circle Table •Interactive Unit Circle • Unit Circle Practice • Unit Circle Quiz •Unit Circle Coordinates • Unit Circle Angles • Unit Circle Learning

Values are calculated to 4 decimal places. Perfect for learning and exploration!