Unit Circle Calculator

Angle θ → (cos θ, sin θ) coordinates on the unit circle. Reference angle + quadrant.

Inputs

Result

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General calculation reads

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Jordan Ellenberg - math thinking
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Steven Strogatz - accessible math
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Scientific calculator (Casio fx-991EX)
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How to use this calculator

  • Enter angle and pick unit.
  • Read (cos, sin), quadrant, and reference angle.

About this calculator

The unit circle is the circle of radius 1 centered at origin. Any angle θ corresponds to a point (cos θ, sin θ) on the circle. Quadrant I (0-90°): both positive. II (90-180°): sin+, cos−. III (180-270°): both−. IV (270-360°): sin−, cos+. Reference angle = the acute angle to the x-axis. Memorizing "famous" angles (30°, 45°, 60°) and their exact cos/sin values is foundational trig.

Frequently asked

What's a "reference angle"?+
The acute angle between the terminal side and the x-axis. Always between 0° and 90°. Used to find trig values via known quadrant rules.
Famous unit-circle values?+
0°: (1, 0). 30°: (√3/2, 1/2). 45°: (√2/2, √2/2). 60°: (1/2, √3/2). 90°: (0, 1).
Why radians?+
Natural angle unit for calculus. 2π rad = 360°. The arc length on a unit circle equals the radian angle.
Tangent at 90°?+
tan 90° = sin 90° / cos 90° = 1 / 0 = undefined. Asymptote in the tangent function.
How is unit circle used?+
Foundation for periodic functions, complex numbers (Euler's formula e^(iθ) = cos θ + i sin θ), and 2D rotations.

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