Unit Circle Calculator
Find coordinates and trigonometric values on the unit circle
Special Angles on Unit Circle
0° (0)
(1, 0)
sin: 0
cos: 1
30° (π/6)
(√3/2, 1/2)
sin: 1/2
cos: √3/2
45° (π/4)
(√2/2, √2/2)
sin: √2/2
cos: √2/2
60° (π/3)
(1/2, √3/2)
sin: √3/2
cos: 1/2
90° (π/2)
(0, 1)
sin: 1
cos: 0
120° (2π/3)
(-1/2, √3/2)
sin: √3/2
cos: -1/2
135° (3π/4)
(-√2/2, √2/2)
sin: √2/2
cos: -√2/2
150° (5π/6)
(-√3/2, 1/2)
sin: 1/2
cos: -√3/2
How to Use This Calculator
Enter the Angle
Input any angle value. The calculator will find the corresponding point on the unit circle.
Select Unit
Choose whether your angle is in degrees or radians. The calculator shows both.
Calculate
Click "Calculate Unit Circle" to see the coordinates (x, y) on the unit circle and all trigonometric values.
Review Results
See the point coordinates, quadrant location, and verify that sin(θ) = y and cos(θ) = x, and that x² + y² = 1.
Formula
Unit Circle Definition:
cos(θ) = x-coordinate
sin(θ) = y-coordinate
tan(θ) = y / x = sin(θ) / cos(θ)
Unit Circle Properties:
- Radius: Always 1 (unit circle)
- Equation: x² + y² = 1
- Pythagorean Identity: sin²(θ) + cos²(θ) = 1
- Coordinates: (cos(θ), sin(θ)) for any angle θ
Quadrant Rules:
Quadrant I (0°-90°): x > 0, y > 0 → sin > 0, cos > 0, tan > 0
Quadrant II (90°-180°): x < 0, y > 0 → sin > 0, cos < 0, tan < 0
Quadrant III (180°-270°): x < 0, y < 0 → sin < 0, cos < 0, tan > 0
Quadrant IV (270°-360°): x > 0, y < 0 → sin < 0, cos > 0, tan < 0
About Unit Circle Calculator
The Unit Circle Calculator finds the coordinates (x, y) of any point on the unit circle and calculates all trigonometric function values. The unit circle is a fundamental concept in trigonometry - a circle with radius 1 centered at the origin. Understanding the unit circle is essential for mastering trigonometry and extending trigonometric functions beyond right triangles.
What is the Unit Circle?
The unit circle is a circle with radius 1, centered at the origin (0, 0) of a coordinate plane. For any angle θ measured from the positive x-axis, the point on the unit circle has coordinates (cos(θ), sin(θ)). This elegant relationship connects trigonometry to geometry and allows us to define trigonometric functions for all angles, not just those in right triangles.
Key Properties
- Radius = 1: Every point on the unit circle is exactly 1 unit from the origin
- Equation: x² + y² = 1 (Pythagorean theorem on the unit circle)
- Trigonometric Functions: cos(θ) = x-coordinate, sin(θ) = y-coordinate
- Periodicity: Functions repeat every 360° (2π radians)
- Symmetry: The circle exhibits reflection and rotation symmetries
When to Use This Calculator
- Learning Trigonometry: Visualize how angles map to coordinates
- Finding Exact Values: Get coordinates for special angles (30°, 45°, 60°, etc.)
- Quadrant Determination: Determine which quadrant an angle lies in
- Verification: Verify that sin²(θ) + cos²(θ) = 1 for any angle
- Problem Solving: Understand why trigonometric functions have their signs in different quadrants
- Visualization: See the geometric interpretation of trigonometry
Why Use Our Calculator?
- ✅ Complete Information: Shows coordinates, quadrant, and all trig functions
- ✅ Visual Understanding: Helps visualize the unit circle concept
- ✅ Identity Verification: Confirms x² + y² = 1 and sin² + cos² = 1
- ✅ Quadrant Detection: Automatically determines which quadrant the angle is in
- ✅ Multiple Units: Supports both degrees and radians
- ✅ Educational: Perfect for learning and understanding trigonometry
- ✅ 100% Free: No registration required
Understanding Quadrants
- Quadrant I (0°-90°): All trig functions positive (ASTC: All Students Take Calculus)
- Quadrant II (90°-180°): Only sin and csc positive
- Quadrant III (180°-270°): Only tan and cot positive
- Quadrant IV (270°-360°): Only cos and sec positive
Frequently Asked Questions
What is the unit circle?
The unit circle is a circle with radius 1 centered at the origin. For any angle θ, the point on the unit circle has coordinates (cos(θ), sin(θ)). It's fundamental because it extends trigonometric functions to all angles, not just those in right triangles.
Why is it called the "unit" circle?
It's called the unit circle because it has a radius of 1 unit. This makes calculations simple: the distance from the origin to any point on the circle is always 1, and the equation is simply x² + y² = 1.
How does the unit circle relate to right triangles?
When you draw a right triangle with one vertex at the origin and the hypotenuse as a radius of the unit circle, the x-coordinate equals cos(θ) (adjacent/hypotenuse) and y-coordinate equals sin(θ) (opposite/hypotenuse). Since the hypotenuse = 1, these simplify to just the coordinates!
Why is x² + y² always equal to 1?
This is the Pythagorean theorem applied to the unit circle. Since the radius is 1, and the radius is the hypotenuse of the right triangle formed by the point and the axes, we have x² + y² = 1² = 1. This is also the Pythagorean identity: cos²(θ) + sin²(θ) = 1.
How do I use the unit circle to find trig values?
For any angle θ: (1) Find the point where the angle's terminal side intersects the unit circle, (2) The x-coordinate is cos(θ), (3) The y-coordinate is sin(θ), (4) tan(θ) = y/x. The quadrant tells you the signs of these values.
What are the key angles I should know on the unit circle?
Memorize: 0° (1, 0), 30° (√3/2, 1/2), 45° (√2/2, √2/2), 60° (1/2, √3/2), 90° (0, 1), 180° (-1, 0), 270° (0, -1), 360° (1, 0). Use symmetry to find other quadrants.
Why is the unit circle important in calculus?
The unit circle helps derive derivatives of trigonometric functions. For example, the derivative of sin(θ) = cos(θ) becomes intuitive when you visualize how the y-coordinate (sin) changes as you move around the circle, and it equals the x-coordinate (cos).