What is the Unit Circle Calculator?
A unit circle is a circle of radius 1 centered at the origin. An angle measured counterclockwise from the positive x-axis identifies the point (cos θ, sin θ). This unit circle calculator connects that point to the trigonometric values rather than displaying an unrelated example diagram.
Use degrees for familiar angles such as 30, 45 and 60, or choose radians for inputs such as pi/6, pi/4 and pi/3. The calculator reports sine, cosine, tangent, the normalized angle and a reference angle. The circle always has radius 1; its point and radius direction change with the calculated angle. For solving side and angle measurements, use the Triangle Calculator.
How to use the unit circle calculator with degrees or radians
Choose the angle unit, enter an angle and select Calculate. All result cards and the diagram clear when an input changes, so a previous answer is not mistaken for the current input.
- Select Degrees or Radians.
- Enter a number. Radian mode also accepts pi, pi/3, -2*pi or 5*pi/4.
- Read the sine, cosine and tangent cards and the coordinate point.
- Use the reference angle and quadrant to check signs.
- Review the calculation steps or copy, download or print the result.
The angle selector changes how the entered text is interpreted; it does not automatically convert the text. For example, 60 degrees and 60 radians describe different directions. Optional protractors (paid link) can help compare a paper sketch with the displayed angle.
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Unit circle coordinates and exact trigonometric values
The point has x = cos θ and y = sin θ. Tangent is sin θ / cos θ wherever cos θ is nonzero. Convert degrees to radians using θrad = θdeg × π/180, and radians to degrees using θdeg = θrad × 180/π. Adding a whole turn gives the same point.
For supported multiples of 15° entered in degrees or pi notation, the calculator uses exact expressions such as 1/2, √2/2, √3/2 and (√6 − √2)/4. Numeric radian approximations are evaluated numerically. A rounded value such as 1.57079632679 is not treated as an exact symbolic π/2.
For non-axis angles, the reference angle is the acute angle to the x-axis. On the x-axis this tool reports 0°, and on the y-axis it reports 90°. This convention is stated explicitly because some textbooks reserve “reference angle” for acute angles only.
How to read the circle diagram and result
The headline is labeled as the sine of the entered angle. The point is ordered (cos θ, sin θ), so sine is its vertical coordinate and cosine is its horizontal coordinate. The gold radius runs from the origin to that point; cyan dashed lines show its coordinate projections.
Quadrants I, II, III and IV correspond to (+,+), (−,+), (−,−) and (+,−) coordinates respectively. Exact axis angles are labeled “On an axis.” Tangent is undefined at 90° and 270° plus whole turns; the tool does not display a finite substitute for these exact angles.
The circle is a mathematical radius-1 diagram fitted uniformly to the available screen. It is not a physical-size printout. Optional graph paper (paid link) is useful for drawing the same axes and reference triangle.
Worked unit circle examples: 60° and negative angles
At 60°, or pi/3 radians, the point is (1/2, √3/2). Therefore cos θ = 1/2, sin θ = √3/2 and tan θ = √3. The point lies in Quadrant I, and its reference angle is 60°. Squaring the coordinates confirms 1/4 + 3/4 = 1.
At −450°, adding two whole turns gives 270°. The point is (0, −1), so sine is −1 and cosine is 0. Tangent is undefined. A negative angle means clockwise rotation, not a negative circle radius.
For a nonstandard angle such as 37°, the cards show decimal approximations. Use the Scientific Calculator for other trigonometric expressions. Optional scientific calculators (paid link) can provide an independent numeric check when their degree/radian setting matches.
Accuracy, supported input and limitations
The angle must be finite and represent at most ±1,000,000 degrees. Use zero or an angle magnitude of at least 10⁻⁹ degrees. Supported pi notation is a signed numeric multiple of pi with an optional positive numeric divisor. General expressions such as sin(pi/3), sums of pi terms and variables are not parsed.
Exact output is limited to recognized multiples of 15°; other results use browser floating-point arithmetic and up to 10 significant digits. Very close-to-axis numeric angles can have large finite tangent values. Large-angle normalization and rounding can lose small differences; retain your original input for later calculations.
This tool does not solve trigonometric equations, inverse-function branch problems or triangle geometry. The Complex Number Calculator covers separate complex arithmetic. Optional trigonometry workbooks (paid link) can provide further practice; no purchase is required to use this calculator.
Unit Circle Calculator FAQs
Yes. Select Radians and enter pi/3, 5*pi/4 or another supported numeric multiple of pi. Numeric radians are also accepted.
Recognized multiples of 15 degrees entered in degrees or supported pi notation receive exact radical or rational values. Other angles use approximate decimals.
The exact cosine is zero, and tangent is sine divided by cosine. Division by zero is undefined.
No. A unit circle always has radius 1. The calculated point and radius direction change, while the circle stays fixed.
The text is interpreted in the newly selected unit. The selector does not convert the entered text automatically.
The calculation runs locally in your browser. The site may still load its separately disclosed analytics or advertising services.
Sources and methodology
Standard mathematical definitions independently implemented and reviewed November 3, 2026. Sources explain the mathematics; the linked organizations do not endorse this calculator.