Unit Circle Calculator

Enter an angle to find its coordinates and trigonometric function values on the unit circle.

Our calculator simplifies trigonometry by providing detailed results for any angle.

Practical Examples

Explore common scenarios to see how the calculator works.

Angle in Degrees (45°)

Angle in Degrees

Calculate trigonometric values for a 45-degree angle.

Angle: 45

Unit: Degrees (°)

Angle in Radians (π/3)

Angle in Radians

Calculate trigonometric values for an angle of π/3 radians.

Angle: 1.04719755

Unit: Radians (rad)

Quadrant Angle (90°)

Quadrant Angle

Observe the values for a 90-degree angle, where some functions are undefined.

Angle: 90

Unit: Degrees (°)

Negative Angle (-30°)

Negative Angle

See how the calculator handles negative angles.

Angle: -30

Unit: Degrees (°)

Other Titles
Understanding the Unit Circle: A Comprehensive Guide
The unit circle is a fundamental concept in trigonometry, providing a bridge between angles and the coordinate plane. This guide explores its definition, applications, and the mathematics behind it.

What is the Unit Circle?

  • Definition and Core Concepts
  • The Role of Radians and Degrees
  • Visualizing Trigonometric Functions
The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) of the Cartesian coordinate system. Its simplicity is its power, as it allows for the direct definition of trigonometric functions like sine, cosine, and tangent for all real numbers.
Key Properties
For any point (x, y) on the unit circle, the coordinates are defined by an angle θ (theta), measured counter-clockwise from the positive x-axis. The core relationships are x = cos(θ) and y = sin(θ). The equation of the unit circle is x² + y² = 1, which is a direct result of the Pythagorean theorem.
Angles: Degrees vs. Radians
Angles on the unit circle can be measured in degrees or radians. A full circle is 360°, which is equivalent to 2π radians. Radians are often preferred in higher mathematics because they are a more natural measure of an angle, based on the radius of the circle itself.

Common Angles and Their Coordinates

  • Angle: 30° (π/6 rad) -> Coordinates: (√3/2, 1/2)
  • Angle: 45° (π/4 rad) -> Coordinates: (√2/2, √2/2)
  • Angle: 60° (π/3 rad) -> Coordinates: (1/2, √3/2)

Step-by-Step Guide to Using the Unit Circle Calculator

  • Inputting Your Angle
  • Interpreting the Results
  • Using the Examples
Our calculator is designed for ease of use. Follow these steps to get your results instantly.
Step 1: Enter the Angle
In the 'Angle' input field, type the numerical value of your angle. This can be positive, negative, or zero.
Step 2: Select the Unit
Choose whether the angle you entered is in 'Degrees (°)' or 'Radians (rad)' from the dropdown menu. This is a critical step as it determines how the input is interpreted.
Step 3: Calculate and Review
Click the 'Calculate' button. The tool will display the (x, y) coordinates, the angle in both units, and the values for all six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. If a function is undefined for a given angle (e.g., tan(90°)), it will be indicated.

Example Calculation

  • Input: Angle = 150, Unit = Degrees
  • Result: Coordinates ≈ (-0.866, 0.5), sin(150°) = 0.5, cos(150°) ≈ -0.866

Real-World Applications of the Unit Circle

  • Physics and Engineering
  • Computer Graphics and Game Development
  • Navigation and Astronomy
The unit circle is not just an abstract mathematical tool; it has numerous applications in science, technology, and engineering.
Modeling Periodic Phenomena
In physics, the unit circle is essential for describing oscillations and waves. Sound waves, light waves, and the motion of a pendulum can all be modeled using sine and cosine functions, which are geometrically defined by the unit circle.
Digital Graphics and Animation
In computer graphics, the unit circle is used to calculate rotations of objects, determine positions in 2D and 3D space, and create circular or spherical effects. Game developers rely on it for character movement, camera control, and more.

Application Scenario

  • A Ferris wheel with a radius of 1 unit can be modeled with the unit circle. The height of a passenger at any angle θ is given by y = sin(θ).

Common Misconceptions and Correct Methods

  • Confusing Radians and Degrees
  • Signs of Functions in Quadrants
  • Handling Undefined Values
Understanding common pitfalls can help in mastering the unit circle.
The Four Quadrants
A frequent point of confusion is the sign (positive or negative) of trigonometric functions in each of the four quadrants. A helpful mnemonic is 'All Students Take Calculus': Quadrant I (All positive), Quadrant II (Sine positive), Quadrant III (Tangent positive), Quadrant IV (Cosine positive).
Dealing with Undefined Functions
Some functions are undefined at certain angles. For example, tan(θ) = sin(θ)/cos(θ) is undefined when cos(θ) = 0, which occurs at 90° (π/2) and 270° (3π/2). This is because division by zero is not possible. Our calculator correctly identifies these cases.

Sign Convention Example

  • Angle: 210° (Quadrant III)
  • Sine and Cosine are negative. Tangent is positive.

Mathematical Derivation and Formulas

  • Pythagorean Identity
  • Formulas for All Six Functions
  • Angle Conversion Formulas
The relationships in the unit circle are derived from fundamental geometric and algebraic principles.
The Core Formulas

For a point (x, y) on the unit circle at an angle θ:

  • cos(θ) = x
  • sin(θ) = y
  • tan(θ) = y/x
  • sec(θ) = 1/x
  • csc(θ) = 1/y
  • cot(θ) = x/y
The Pythagorean Identity
From the equation of the circle x² + y² = 1, we can substitute x = cos(θ) and y = sin(θ) to get the most important trigonometric identity: cos²(θ) + sin²(θ) = 1. This identity is true for all angles.
Angle Conversion
To convert from degrees to radians, use the formula: Radians = Degrees × (π / 180). To convert from radians to degrees, use: Degrees = Radians × (180 / π).

Conversion Example

  • Convert 180° to radians: 180 × (π / 180) = π radians.
  • Convert π/2 radians to degrees: (π/2) × (180 / π) = 90°.