Unit Circle Calculator
Find the exact (cos θ, sin θ) point on the unit circle for any angle in degrees.
How it works
- 1Every angle is normalised into [0°, 360°) so the result is independent of how many full turns you add or subtract.
- 2cos θ and sin θ are the x- and y-coordinates of the point where the angle’s terminal ray meets the unit circle (radius 1).
- 3The reference angle is the acute angle (0°–90°) between the terminal ray and the nearest part of the x-axis.
Use cases
- Quickly verify exact trig values for standard angles like 30°, 45°, 60°, 90°, 120°, and 270°.
- Check which quadrant an angle lands in and confirm the expected signs of cos and sin.
- Visualise the link between an angle, its reference angle, and its unit-circle coordinates.
Frequently asked questions
What is the unit circle?
A circle of radius 1 centred at the origin. For any angle θ measured counter-clockwise from the positive x-axis, the terminal ray meets the circle at (cos θ, sin θ) — the geometric definition of cosine and sine.
What is a reference angle?
The smallest positive acute angle between the terminal ray and the x-axis, always in [0°, 90°]. It gives the magnitude of cos and sin; the quadrant then fixes the signs.
Why does it sometimes show “On an axis”?
At 0°, 90°, 180°, and 270° the terminal ray lies exactly on a coordinate axis. These boundary angles belong to no quadrant, so the tool shows “On an axis”.
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