Unit Circle Calculator

Find coordinates and trigonometric function values for any angle on the unit circle

Calculate Unit Circle Values

Angle

Enter the angle to find its position on the unit circle

Common Angles

Unit Circle Coordinates

(1, 0)

Exact Coordinates

(1.000000, 0.000000)

Decimal Coordinates

Quadrant:I

Reference Angle:0.000°

Trigonometric Function Values

Primary Functions

sin(0°)0

cos(0°)1

tan(0°)0

Reciprocal Functions

csc(0°)undefined

sec(0°)1

cot(0°)undefined

Decimal Values

sin0.000000

cos1.000000

tan0.000000

Pythagorean Identity Verification

sin²θ + cos²θ = 0² + 1² = 1.000000 ≈ 1 ✓

Example: Finding sin(30°)

Step-by-Step Solution

1. Locate 30° on unit circle: First quadrant

2. Find y-coordinate: On the unit circle, sine equals the y-coordinate

3. Special angle value: sin(30°) = 1/2

4. Coordinates: (√3/2, 1/2)

Key Concepts

• Sine = y-coordinate of point on unit circle

• Cosine = x-coordinate of point on unit circle

• Tangent = y/x = sin/cos (when cos ≠ 0)

• All points satisfy x² + y² = 1

Special Angles Chart

Anglesincostan

0°010

30°1/2√3/2√3/3

45°√2/2√2/21

60°√3/21/2√3

90°10∞

180°0-10

270°-10∞

Unit Circle Properties

•

Radius = 1 unit

•

Center at origin (0, 0)

•

Equation: x² + y² = 1

•

Sine = y-coordinate

•

Cosine = x-coordinate

•

Complete revolution = 360° = 2π radians

Memory Tips

✓

SOHCAHTOA: Sin = Opposite/Hypotenuse

✓

All Students Take Calculus (Quadrant sign pattern)

✓

Special triangles: 30-60-90 and 45-45-90

✓

Reference angles help find function values

Understanding the Unit Circle

What is the Unit Circle?

The unit circle is a circle with radius 1 centered at the origin (0, 0) of a coordinate system. It's fundamental in trigonometry because it provides a geometric interpretation of trigonometric functions.

Key Relationships

•Sine: y-coordinate of the point
•Cosine: x-coordinate of the point
•Tangent: sin/cos = y/x (slope of the line)
•Point coordinates: (cos θ, sin θ)

Fundamental Identity

sin²θ + cos²θ = 1

This identity comes directly from the Pythagorean theorem, since every point on the unit circle satisfies x² + y² = 1.

Quadrant Signs

Quadrant I: sin(+), cos(+), tan(+)

Quadrant II: sin(+), cos(-), tan(-)

Quadrant III: sin(-), cos(-), tan(+)

Quadrant IV: sin(-), cos(+), tan(-)

Special Angles and Exact Values

Certain angles have exact trigonometric values that can be expressed using simple fractions and radicals. These special angles (0°, 30°, 45°, 60°, 90°, etc.) are derived from special right triangles and are essential for understanding trigonometry.

30-60-90 Triangle

Sides in ratio 1 : √3 : 2
Gives exact values for 30° and 60°

45-45-90 Triangle

Sides in ratio 1 : 1 : √2
Gives exact values for 45°