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Unit Circle Calculator

Calculate trigonometric functions using the unit circle. Find exact values for sin, cos, tan, and other trig functions with our free online calculator. Perfect for students and professionals.

Unit Circle Calculator

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Angle Input
Unit
Display Options
6

Trigonometric Values

Angle Information

Angle:45.00°
Radians:0.7854 rad
Quadrant:Q1
Reference Angle:45.00°

Basic Functions

sin(θ):0.707107
cos(θ):0.707107
tan(θ):1.000000
cot(θ):1.000000

Reciprocal Functions

sec(θ):1.414214
csc(θ):1.414214

Decimal Values

sin(θ):0.707107
cos(θ):0.707107
tan(θ):1.000000

Unit Circle Visualization

11-1-1(0.707, 0.707)
Quadrant 1
Reference Angle: 45.0°

Common Angles and Their Values

Special Angles (Degrees)

0°(1, 0)

sin = 0, cos = 1, tan = 0

  • • Starting point
  • • Positive x-axis
30°(√3/2, 1/2)

sin = 1/2, cos = √3/2, tan = √3/3

  • • π/6 radians
  • • Special triangle
45°(√2/2, √2/2)

sin = √2/2, cos = √2/2, tan = 1

  • • π/4 radians
  • • Isosceles right triangle
60°(1/2, √3/2)

sin = √3/2, cos = 1/2, tan = √3

  • • π/3 radians
  • • Equilateral triangle
90°(0, 1)

sin = 1, cos = 0, tan = undefined

  • • π/2 radians
  • • Positive y-axis

Quadrant Signs

Quadrant I0° - 90°

All positive

  • • sin > 0, cos > 0, tan > 0
  • • sec > 0, csc > 0, cot > 0
Quadrant II90° - 180°

sin positive only

  • • sin > 0, cos < 0, tan < 0
  • • sec < 0, csc > 0, cot < 0
Quadrant III180° - 270°

tan positive only

  • • sin < 0, cos < 0, tan > 0
  • • sec < 0, csc < 0, cot > 0
Quadrant IV270° - 360°

cos positive only

  • • sin < 0, cos > 0, tan < 0
  • • sec > 0, csc < 0, cot < 0

How to Calculate Unit Circle Values

Unit Circle Definition

Unit Circle: Circle with radius 1 centered at origin (0,0)
Coordinates: (cos θ, sin θ) for any angle θ
Pythagorean Identity: cos²θ + sin²θ = 1

Calculation Steps:

  1. 1
    Convert angle to radians if needed
    Multiply degrees by π/180
  2. 2
    Calculate basic trigonometric functions
    sin(θ) = y-coordinate, cos(θ) = x-coordinate
  3. 3
    Calculate tangent and reciprocal functions
    tan(θ) = sin(θ)/cos(θ), sec(θ) = 1/cos(θ), etc.

Important Considerations

⚠️ Mathematical Precision

This calculator provides high-precision trigonometric values. For exact values, use special angles.

📐 Angle Precision

Angles are normalized to 0-360° range

  • • Negative angles converted to positive
  • • Angles > 360° reduced by 360°
  • • Reference angles always 0-90°
🔄 Unit Conversion

Automatic conversion between degrees and radians

  • • Degrees to radians: × π/180
  • • Radians to degrees: × 180/π
  • • 2π radians = 360°
🎯 Exact Values

Special angles have exact fractional values

  • • 30°, 45°, 60° have exact forms
  • • Other angles use decimal approximations
  • • Precision to 6 decimal places
⚠️ Undefined Values

Some trigonometric functions are undefined

  • • tan(90°) and tan(270°) are undefined
  • • sec(90°) and sec(270°) are undefined
  • • csc(0°) and csc(180°) are undefined

Example Cases

Case 1: 45° Angle

Input: 45° (degrees)
Radians: π/4 ≈ 0.7854
Quadrant: I (0° - 90°)
Exact Values:
sin(45°) = √2/2 ≈ 0.7071
cos(45°) = √2/2 ≈ 0.7071
tan(45°) = 1

Use Case: Perfect example of special angle with exact values. Used in 45-45-90 triangles.

Case 2: 120° Angle

Input: 120° (degrees)
Radians: 2π/3 ≈ 2.0944
Quadrant: II (90° - 180°)
Exact Values:
sin(120°) = √3/2 ≈ 0.8660
cos(120°) = -1/2 = -0.5
tan(120°) = -√3 ≈ -1.7321

Use Case: Shows how reference angle (60°) helps find values in Quadrant II. Notice negative cos and tan.

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Frequently Asked Questions

What is the unit circle in trigonometry?
The unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. It's fundamental in trigonometry because it provides a geometric way to define trigonometric functions for any angle. The x-coordinate of a point on the unit circle represents cos(θ), and the y-coordinate represents sin(θ).
How do I find exact values for common angles?
Common angles like 30°, 45°, 60°, and 90° have exact trigonometric values that can be expressed as fractions and radicals. For example, sin(30°) = 1/2, cos(45°) = √2/2, and tan(60°) = √3. These values are derived from special right triangles and are essential for precise calculations.
What are the four quadrants and their signs?
The unit circle is divided into four quadrants: Quadrant I (0°-90°): sin > 0, cos > 0, tan > 0; Quadrant II (90°-180°): sin > 0, cos < 0, tan < 0; Quadrant III (180°-270°): sin < 0, cos < 0, tan > 0; Quadrant IV (270°-360°): sin < 0, cos > 0, tan < 0. This helps determine the sign of trigonometric functions.
How do I convert between degrees and radians?
To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. For example, 90° = 90 × π/180 = π/2 radians. The unit circle uses radians as the natural unit, where 2π radians = 360°.
What is a reference angle and how do I find it?
A reference angle is the acute angle (0° to 90°) formed between the terminal side of an angle and the x-axis. To find it: for angles 0°-90°, the reference angle is the angle itself; for 90°-180°, subtract from 180°; for 180°-270°, subtract 180°; for 270°-360°, subtract from 360°. Reference angles help find exact values for any angle.
Unit Circle Calculator - Free Trigonometry Tool