Education
Unit Circle Calculator
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.
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
Unit-circle coordinate
(1/2, √3/2)
Angle 60° = 1.047198 rad; Quadrant I
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.
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.
Related geometry and trig calculators
- Circle calculator for radius, diameter, circumference, and area when the radius is not 1.
- Right triangle calculator for solving any right triangle, including with one acute angle given.
- Pythagorean theorem calculator for the a² + b² = c² relationship that underpins the unit circle.
- Scientific calculator for arbitrary trig values in degrees or radians.
- All education calculators.
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.
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