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Unit Circle Calculator

Enter an angle in degrees to find its unit-circle coordinates.

How it is calculated

The angle is read in degrees only. The engine converts it to radians with θ(rad) = θ(deg) × π / 180, then computes x = cos θ, y = sin θ and tan θ = y / x. The point is shown as (x, y) and every value is rounded to 4 decimals. Angles above 360 or below 0 need no special handling because the functions repeat every 360 degrees, so 720 gives the same point as 0 and -45 the same as 315. When |cos θ| is smaller than 10-12, which happens at 90, 270 and every angle 180 degrees apart from them, the tangent is shown as undefined instead of a huge number.

Worked example

Angle 210 degrees. It lies in the third quadrant, 30 degrees past the negative x-axis, so its reference angle is 30. Cosine: cos 210 = -cos 30 = -0.8660. Sine: sin 210 = -sin 30 = -0.5000. Both are negative in the third quadrant. Tangent: -0.5000 / -0.8660 = 0.5774 (positive). The point on the circle is (-0.8660, -0.5000).

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Common mistakes

  • Entering an angle in radians (for example 3.1416 for a half turn); the field is in degrees, so that value is read as just over 3 degrees.
  • Swapping the coordinates: the point is (cos, sin), with cosine as the x value and sine as the y value.
  • Expecting a sign-free answer for angles past 90 degrees; the signs of x and y change by quadrant, and the tangent is negative in the second and fourth.
  • Reading a displayed 0.0000 as exactly zero: values are rounded to 4 decimals, so a very small non-zero number shows the same way.
  • Looking for a tangent at 90 or 270 degrees; it does not exist there and the tool reports it as undefined.

About this calculator

The unit circle is a circle of radius 1 centered at the origin of a coordinate plane, and it is the geometric foundation for defining sine and cosine: for any angle θ measured from the positive x-axis, the point where the angle’s ray crosses the circle has coordinates (cos θ, sin θ) by definition, and the tangent follows as tan θ = sin θ / cos θ (undefined where cos θ = 0).

Because the unit circle makes trigonometric values visual and geometric rather than abstract ratios memorized from a right triangle, it is a core teaching tool in trigonometry and precalculus for understanding why sine and cosine repeat every 360°, why they range only between −1 and 1, and how special angles like 30°, 45°, 60° and 90° produce clean, memorable coordinate values.

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Comments (1)

  • Editorial team

    Use the output as a self-check on exam problems: squaring the cosine and the sine and adding them must give 1, up to rounding. For 45 degrees both coordinates read 0.7071, and 0.7071 squared twice is 0.5 plus 0.5. The tool gives decimals, not exact forms like root 3 over 2, so convert back by hand when a course asks for exact values. Would radian input or exact fractions be useful to you?

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