Unit Circle Chart & Calculator
Drag the point to see (cos θ, sin θ) — and print a unit circle chart.
About the Unit Circle Chart & Calculator
Drag the point around the circle — or type any angle in degrees or radians — and watch its coordinates: the x-coordinate is cos θ, the y-coordinate is sin θ, and the height where the extended radius meets the tangent line x = 1 is tan θ. At the standard angles the values are exact, such as (√3/2, 1/2) at 30°, and the quadrant and reference angle are shown for every angle.
Below the circle, a table lists the exact values of cos, sin and tan at the 16 standard angles, or at every multiple of 15°. You can also download a printable unit circle chart as a PDF or an SVG — with degrees, radians and coordinates filled in, or as a blank practice sheet — or print it straight from the page.
How to use it
- Drag the large yellow point around the circle (or tap the circle where you want it), tap a standard angle, or type an angle such as 30, −45 or 5pi/6 in the box.
- Choose how dragging snaps: to the standard angles, to every 15°, 5° or 1°, or to tenths of a degree. With the point focused, the arrow keys move it one step.
- Read (cos θ, sin θ), tan θ, the quadrant and the reference angle. On the circle, the coloured segments are cos θ, sin θ and tan θ.
- For a printed chart, choose the full chart or the blank practice sheet, then download the PDF or SVG, or print it.
Examples
θ = 30° = π/6
(√3/2, 1/2), tan θ = √3/3
A 30-60-90 triangle with hypotenuse 1 has legs √3/2 and 1/2.
θ = 135° = 3π/4
(−√2/2, √2/2), tan θ = −1
In quadrant II cos θ is negative and sin θ positive; the reference angle is 45°.
θ = 240° = 4π/3
(−1/2, −√3/2), tan θ = √3
Both coordinates are negative; the reference angle is 60°.
θ = −30°
(√3/2, −1/2)
Negative angles go clockwise; −30° and 330° are coterminal.
θ = 270° = 3π/2
(0, −1), tan θ undefined
cos θ = 0 there, so tan θ = sin θ / cos θ is undefined.
Common uses
- Learning or revising the exact values of sine, cosine and tangent at the standard angles.
- Seeing why cos is negative in quadrants II and III, and how the reference angle carries the values around the circle.
- Printing a reference chart, or a blank chart to fill in, for a class or for exam revision.
- Checking a value from a calculator against its exact form.
Why the unit circle works
On a circle of radius 1 centred at the origin, the point at angle θ, measured anticlockwise from the positive x-axis, is (cos θ, sin θ) — this is how cosine and sine are defined for every angle, not just acute ones. Because the radius is 1, x² + y² = 1 gives cos² θ + sin² θ = 1. tan θ = sin θ / cos θ is the slope of the radius, and also the height at which the extended radius meets the tangent line x = 1.
Remembering the values
In the first quadrant the coordinates use only 1/2, √2/2 and √3/2: (√3/2, 1/2) at 30°, (√2/2, √2/2) at 45° and (1/2, √3/2) at 60°. Every other standard angle has the same numbers with the signs of its quadrant: x is negative in quadrants II and III, and y is negative in III and IV. To find the point at 210°, take the 30° point (√3/2, 1/2) and make both coordinates negative: (−√3/2, −1/2).
Degrees and radians
A full turn is 360°, or 2π radians, so 180° = π, 90° = π/2, 60° = π/3, 45° = π/4 and 30° = π/6. Multiply degrees by π/180 to get radians, and radians by 180/π to get degrees. The chart lists both for the 16 standard angles.
Limitations
- Exact values are shown at multiples of 15°, 18° and 22.5°; other angles, such as 10°, get decimal values.
- Dragging snaps to the grid you choose; for an angle that is not on the grid, type it in the box.
- The PDF uses the PDF engine and the Noto Sans font files, which are loaded the first time you make one. If the font cannot be loaded, the PDF uses a standard font that writes π as “pi”; the SVG download and printing always work.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
What are the coordinates on the unit circle?
The point at angle θ is (cos θ, sin θ). At the standard angles of the first quadrant they are (1, 0) at 0°, (√3/2, 1/2) at 30°, (√2/2, √2/2) at 45°, (1/2, √3/2) at 60° and (0, 1) at 90°. The other quadrants use the same numbers with different signs.
How do I find tan θ on the unit circle?
Divide the y-coordinate by the x-coordinate: tan θ = sin θ / cos θ. At 60°, tan θ = (√3/2)/(1/2) = √3. Geometrically, it is the height where the extended radius meets the tangent line x = 1; it is undefined at 90° and 270°, where cos θ = 0.
Which functions are positive in each quadrant?
In quadrant I all six are positive; in II only sin and csc; in III only tan and cot; in IV only cos and sec. A common memory aid is “All Students Take Calculus”.
How do I use the blank chart for practice?
Tick “Blank practice sheet” and download the PDF or print it. It keeps the angles in degrees and leaves boxes for the radians and the coordinates. Fill it in, then compare with the full chart or the table on this page.
Why is the point at 210° (−√3/2, −1/2)?
210° is in quadrant III with a reference angle of 210° − 180° = 30°. The 30° point is (√3/2, 1/2); in quadrant III both coordinates are negative, so the point is (−√3/2, −1/2).