Unit Circle Degrees Calculator

The unit circle is a fundamental concept in trigonometry that represents all possible positions of a point moving at a distance of 1 unit from the origin in the Cartesian plane. This calculator helps you find the coordinates (x, y) and trigonometric values (sine, cosine, tangent) for any angle in degrees.

What is the Unit Circle?

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. It's used extensively in trigonometry to define the sine and cosine functions. Every point on the unit circle corresponds to an angle, and its coordinates give the sine and cosine of that angle.

Key properties of the unit circle:

All points satisfy the equation x² + y² = 1
At 0 degrees, the point is (1, 0)
At 90 degrees, the point is (0, 1)
At 180 degrees, the point is (-1, 0)
At 270 degrees, the point is (0, -1)

The unit circle is periodic with a period of 360 degrees, meaning the pattern repeats every full rotation.

How to Use the Calculator

To use the unit circle degrees calculator:

Enter the angle in degrees in the input field
Click "Calculate" to see the results
View the coordinates (x, y) and trigonometric values
Use the chart to visualize the point on the unit circle

The calculator will show you:

The x-coordinate (cosine of the angle)
The y-coordinate (sine of the angle)
The tangent value (y/x)
A visual representation of the point on the unit circle

Formula

The coordinates on the unit circle for an angle θ (in degrees) are given by:

x = cos(θ)

y = sin(θ)

tan(θ) = y/x (when x ≠ 0)

Where:

θ is the angle in degrees
cos(θ) is the cosine of θ
sin(θ) is the sine of θ
tan(θ) is the tangent of θ

All trigonometric functions in this calculator use degrees, not radians.

Worked Examples

Example 1: 30 Degrees

For θ = 30°:

x = cos(30°) ≈ 0.8660
y = sin(30°) = 0.5
tan(30°) ≈ 0.5774

Example 2: 120 Degrees

For θ = 120°:

x = cos(120°) = -0.5
y = sin(120°) ≈ 0.8660
tan(120°) ≈ -1.7321

Example 3: 270 Degrees

For θ = 270°:

x = cos(270°) = 0
y = sin(270°) = -1
tan(270°) is undefined (division by zero)

FAQ

What is the difference between degrees and radians?

Degrees and radians are two different units for measuring angles. A full circle is 360 degrees or 2π radians. This calculator uses degrees, while some other trigonometric calculators use radians.

Why is the tangent undefined at 90° and 270°?

The tangent function is defined as tan(θ) = sin(θ)/cos(θ). At 90° and 270°, the cosine value is 0, which makes the tangent undefined (division by zero).

How can I convert radians to degrees?

To convert radians to degrees, multiply by 180/π. For example, π/2 radians is 90 degrees.