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Unit Circle Calculator

Find the coordinates x (cos), y (sin), or tan values of an angle on the unit circle.

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Unit Circle Calculator

This unit circle calculator allows you to calculate trigonometric values (sine, cosine, and tangent) for any angle. Simply enter the angle in degrees, radians, or π radians and get:

The coordinates (sine and cosine) of the point on the unit circle
The tangent value

What Is a Unit Circle?

A unit circle is a circle centered at the origin (0,0) with a radius of 1.

Unit Circle - Sine and Cosine

The endpoint of a rotation from the positive x-axis (1, 0) on the unit circle determines the cosine, sine, and tangent values of an angle θ. The unit circle is essential for visualizing and calculating trigonometric functions.

The equation of a circle with center (x₁, y₁) and radius r is:

(x - x₁)² + (y - y₁)² = r²

For a unit circle (radius = 1, center = (0,0)):

x² + y² = 1

Or equivalently in trigonometric terms:

cos²θ + sin²θ = 1

Trigonometric Ratios on the Unit Circle

Sine and Cosine

Sine = y-coordinate of the point
Cosine = x-coordinate of the point

Unit Circle - Sine and Cosine

Point P is on the unit circle
Horizontal side (adjacent to angle) = x-coordinate
Vertical side (opposite angle) = y-coordinate
Hypotenuse = radius = 1

sin α = y

cos α = x

Unit circle equation: cos²θ + sin²θ = 1

Tangent

Tangent is the ratio of sine to cosine:

Tangent formula

tan α = opposite / adjacent = y / x

How to Find Trig Ratios From an Angle

Identify the point on the unit circle corresponding to the angle θ
Cosine = x-coordinate
Sine = y-coordinate
Tangent = y / x (undefined if x = 0)

Examples

1. Angle = 60°

sin 60° = √3/2 ≈ 0.866, cos 60° = 1/2 = 0.5

tan 60° = sin/cos = √3 ≈ 1.732

2. Angle = π/3 radians

sin π/3 = √3/2 ≈ 0.866, cos π/3 = 1/2 = 0.5

tan π/3 = sin/cos = √3 ≈ 1.732

3. Angle = π radians

sin π = 0, cos π = -1

tan π = 0 / -1 = 0

Unit Circle Chart

Unit Circle Chart with Radians and Degrees

Angles wrap every 360° (2π radians). Subtract 360° until the angle is within 0–360°.

Angle (°)	Angle (rad)	Coordinates (cos, sin)
30°	π/6	(√3/2, 1/2)
45°	π/4	(√2/2, √2/2)
60°	π/3	(1/2, √3/2)
90°	π/2	(0, 1)
120°	2π/3	(-1/2, √3/2)
135°	3π/4	(-√2/2, √2/2)
150°	5π/6	(-√3/2, 1/2)
180°	π	(-1, 0)
210°	7π/6	(-√3/2, -1/2)
225°	5π/4	(-√2/2, -√2/2)
270°	3π/2	(0, -1)
300°	5π/3	(1/2, -√3/2)
315°	7π/4	(√2/2, -√2/2)
330°	11π/6	(√3/2, -1/2)
360°	2π	(1, 0)

FAQs

Applications of the Unit Circle

Modeling periodic phenomena
Engineering calculations
Computer graphics and animation
Navigation and positioning
Signal processing and data analysis

Positive Angles

Measured counterclockwise from the positive x-axis. They lie between 0° and 360° (0 to 2π radians).

Special Right Triangles and the Unit Circle

Triangles 30-60-90 and 45-45-90 scaled to radius 1 give sine and cosine for key angles. These angles help determine coordinates for other angles on the circle.

References

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