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Calculator Answers

Unit Circle Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A unit circle calculator is a trigonometric tool that calculates the coordinates and trigonometric values associated with an angle on a circle of radius one centered at the origin. By entering an angle in degrees or radians, the calculator computes the precise coordinates on the circle perimeter, where the x-coordinate corresponds to cosine and the y-coordinate to sine. It outputs cosine, sine and tangent, exactly for the common angles and as decimals otherwise, and indicates the reference angle and quadrant. The reciprocal functions secant, cosecant and cotangent are not included. Mathematics students use this utility to study circle geometry and memorize unit circle values.

Pick degrees or radians, enter the angle, and the calculator returns the unit-circle coordinate (cos θ, sin θ), the reference angle, the quadrant, sin, cos, and tan. Exact values are shown for the 17 common angles.

See also: Block Circle Generator for Minecraft, Circle Calculator – Area & Circumference, and Time Unit Converter.

Quick Answer

Find trigonometric values using the unit circle. Enter an angle in degrees or radians to calculate sine, cosine, tangent, and coordinates.

Angle unit

Negative angles and values above 360° (or 2π) are automatically wrapped to the standard 0 to 360 range. · e.g. 60

Common angles

Result

Unit-circle coordinate

(1/2, √3/2)

Angle 60° = 1.047198 rad; Quadrant I

Reference angle60°
QuadrantQuadrant I
cos1/2 (= 0.5)
sin√3/2 (= 0.866025)
tan√3

The unit-circle coordinate is (cos θ, sin θ). tan θ = sin θ / cos θ is undefined where cos θ = 0 (at 90° and 270°). The reference angle is the acute angle between the terminal side and the x-axis.

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Examples

30°

(√3/2, 1/2), tan = √3/3, Q1

120°

(-1/2, √3/2), tan = -√3, Q2

225°

(-√2/2, -√2/2), tan = 1, Q3

315°

(√2/2, -√2/2), tan = -1, Q4

How it works

The unit-circle coordinate of any angle θ is (cos θ, sin θ). Tangent is the slope of the terminal side, sin θ / cos θ. The calculator finds these directly and looks the angle up against a 17-entry exact-value table.

Coordinate · (cos θ, sin θ)

Tangent · tan θ = sin θ / cos θ

Common exact values: 30 → √3/2, 1/2; 45 → √2/2, √2/2; 60 → 1/2, √3/2.

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Frequently asked questions

The unit circle is a circle of radius 1 centered at the origin. Any angle θ measured counterclockwise from the positive x-axis lands at the point (cos θ, sin θ) on this circle, which makes it the standard reference for the values of sine, cosine, and tangent.

The reference angle is the acute angle between the terminal side of θ and the x-axis. It is always between 0 and 90 degrees and is used to find the trig values of any angle by combining the value of the reference angle with the sign in the angle's quadrant.

Quadrant I (0 to 90), Quadrant II (90 to 180), Quadrant III (180 to 270), Quadrant IV (270 to 360). Angles outside 0 to 360 are wrapped first. Each quadrant has a known sign pattern for sin, cos, and tan.

tan θ = sin θ / cos θ. At 90 and 270 degrees, cos θ = 0, so the ratio is undefined (the function has a vertical asymptote there).

Switch the unit toggle to Radians and enter a decimal value. π is approximately 3.14159, so π/3 is about 1.047 and π/4 is about 0.785. The calculator converts to degrees internally to find exact values when the angle matches a common one.

Only when the angle exactly matches one of 17 common positions (every multiple of 30 or 45 degrees from 0 to 360). For other angles, sin, cos, and tan are reported as decimals to 6 places.