Unit Circle Calculator
Visualize and calculate coordinates and trigonometric values on the unit circle.
Unit Circle
Unit Circle
The unit circle is a circle with radius 1 centered at the origin (0,0).
Key Points
- \[ (1,0) \] - 0° or 0 radians
- \[ (0,1) \] - 90° or π/2 radians
- \[ (-1,0) \] - 180° or π radians
- \[ (0,-1) \] - 270° or 3π/2 radians
Trigonometric Functions
- \[ \sin(\theta) = y \]
- \[ \cos(\theta) = x \]
- \[ \tan(\theta) = \frac{y}{x} \]
- \[ \csc(\theta) = \frac{1}{y} \]
- \[ \sec(\theta) = \frac{1}{x} \]
- \[ \cot(\theta) = \frac{x}{y} \]
Common Values
- 30° (π/6):
- \[ \sin(30°) = \frac{1}{2} \]
- \[ \cos(30°) = \frac{\sqrt{3}}{2} \]
- \[ \tan(30°) = \frac{1}{\sqrt{3}} \]
- 45° (π/4):
- \[ \sin(45°) = \frac{1}{\sqrt{2}} \]
- \[ \cos(45°) = \frac{1}{\sqrt{2}} \]
- \[ \tan(45°) = 1 \]
- 60° (π/3):
- \[ \sin(60°) = \frac{\sqrt{3}}{2} \]
- \[ \cos(60°) = \frac{1}{2} \]
- \[ \tan(60°) = \sqrt{3} \]
Properties
- Radius = 1
- Circumference = 2π
- Area = π
- All points satisfy x² + y² = 1
Applications
- Trigonometry
- Calculus
- Physics (wave motion)
- Engineering (signal processing)
- Music and sound analysis
Quadrant Signs
- Quadrant I (0° to 90°): All positive
- Quadrant II (90° to 180°): sin positive, others negative
- Quadrant III (180° to 270°): tan positive, others negative
- Quadrant IV (270° to 360°): cos positive, others negative