The Unit Circle

The Unit Circle

Sine and cosine are the coordinates of a point going around a circle. Drag it, and the rest of trigonometry follows.

An interactive explanation · Updated November 2026

Type any angle (−30, 400, 5π/6) or drag the point. Drawn as a plate from Byrne’s Euclid: the blue bar is cos θ, the vermilion bar is sin θ, the yellow plane is the right triangle under the point, and the yellow bar on the tangent line is tan θ. Switch layers on and off under the figure; the sine wave beside the circle (on wide screens) is the height of the point, unrolled.

The figure above is the whole subject in one picture. A point goes around a circle of radius 1; its two coordinates are the cosine and the sine of the angle it has turned through. The tour below follows that one point through eight steps, from turning it to the derivative of sine, and six chapters then go deeper. Each figure on this site is the same point: set an angle once and every figure you meet starts from it.

Figure 1 The same point as the figure at the top of the page. Scroll the steps, use the arrows, or take hold of the point at any time; once you have, the steps change only what is drawn, never your angle.#

1. Turn the point

Start at (1, 0), on the right of the circle, and turn counter-clockwise. The black wedge at the center is the angle θ you have turned through. Turn past a full circle and you are back where you began: 390° lands on the same point as 30°. Turn the other way and the angle is negative: −30° is the same point as 330°. Type any of those into the box at the top of the page and watch where the point goes.

2. Name the point by its coordinates

Every point of the plane has an address: how far across, how far up. For the point on the circle those two numbers depend only on the angle, so they deserve names. The first, the distance across, is the cosine of θ. The second, the height, is the sine. The point is (cos θ, sin θ), and that sentence is the definition modern textbooks use[1].

3. Cosine and sine as lengths

Draw the two coordinates as bars: blue along the x-axis for cosine, vermilion straight up for sine, with the yellow right triangle between them. The bars are signed. Past 90° the blue bar points left, so the cosine is negative; past 180° the vermilion bar points down. Neither bar can be longer than the radius, which is why both functions stay between −1 and 1. and compare the heights.

4. Radians: the angle as a length

The green arc is the distance the point has walked along the rim. On a circle of radius 1 that distance is the angle in radians: an arc one radius long is one radian, about 57.3°, and the ticks mark each whole radian. A full turn is the whole rim, 2π ≈ 6.283 radii. Calculus measures angles this way because then arc, angle and (for small angles) sine are the same number[2].

5. The special angles

At sixteen angles, the multiples of 30° and 45°, the coordinates are short exact surds instead of endless decimals. They come from two shapes: an equilateral triangle gives sin 30° = 1/2, and a square cut on its diagonal gives √2/2 at 45°. Reflect those into the other quadrants and the signs change, the sizes do not. Drag near a dot and the point snaps onto it and shows the exact pair.

6. The Pythagorean identity

The blue bar, the vermilion bar and the radius make a right triangle whose hypotenuse is 1. Pythagoras’s theorem therefore says cos²θ + sin²θ = 1, at every angle, because the point never leaves a circle of radius 1[3]. The readout adds the two squares for your angle; whatever the angle, the total is 1. The identities chapter draws the squares.

7. Circular motion becomes oscillation

Now let the point move at a steady rate, θ = ωt: it turns through the same angle every second. Two ringed dots sit on the axes, the point’s shadows, one under it and one beside it.

Question 1 Prediction before the motion: where does the shadow move fastest?#

Press Play and check. Each shadow slides back and forth along its axis, fastest in the middle and pausing at the ends, while the point itself never speeds up or slows down. The panel under the circle plots those shadows against time: a cosine and a sine wave, repeating once per turn, every T = 2π/ω seconds. Change the speed with the Speed menu beside Play: the waves keep their shape, but the seconds marked under them change, and at 5 s a turn instead of 10 s the period halves. This is why steady rotation, in a generator or a turning Earth, produces sine waves[4][5].

8. Derivatives and phase

Which way is the point heading at each moment? Drag it slowly round the circle, or press Play, and watch the black arrow before you read on.

The arrow is the point’s velocity. It always runs along the circle, at right angles to the radius: it is the radius turned a quarter turn (the gray dot marks θ + 90°), and at one radian per second it is exactly one radius long. Read off its parts. Its rise, in blue, is cos θ; its run, in vermilion, is −sin θ. So the rate at which the height sin θ changes is cos θ: the derivative of sine is cosine, and the derivative of cosine is −sine, when θ is in radians[6]. On the graph, the slope of the sine curve at the cursor equals the height of the cosine curve, and the cosine wave is the sine wave a quarter turn early: cos θ = sin(θ + π/2).

The six chapters take each step further, with the history and the proofs:

And the tools, each on its own page, with the working shown:

Why this site exists

Most people meet the unit circle as a chart to memorize before a test. That is a pity, because the chart is the least interesting thing about it. The circle is the reason sine and cosine work for angles bigger than 90°, the reason a full turn is 2π, the reason the identities hold and the waves have the shape they do. This site tries to show those reasons, in the way a good geometry book does: with careful diagrams, except that these move. The look borrows from Oliver Byrne’s 1847 edition of Euclid, which replaced letters with colored lines, angles and flat colored planes (the about page has the reference). Here every figure is drawn as one of his plates: blue always means cosine, vermilion sine, and yellow tangent, the triangle under the point is a flat yellow plane, and the angle is a black sector.

How it is checked

Every figure is driven by a small, separate mathematical engine. Its exact table of special values is compared with computed sines and cosines at all 24 multiples of 15°; every identity in the site’s tester is checked at 4,096 angles; the triangle solver’s ambiguous-case logic is tested at each boundary. An independent reviewer then rechecks the engine against published references without reading its code, and every historical and numerical claim on the site is listed with its source in a claims register. Sources for each chapter are at its foot, and all of them are on the sources page.

Sources for the tour

  1. Abramson, J., et al. (2021). Precalculus 2e (§5.1 Angles, §5.2 Unit circle, §5.3 The other trigonometric functions, §6.1–6.2 Graphs, §6.3 Inverse trigonometric functions, §7.2–7.3 Identities, §8.1–8.2 Non-right triangles, §8.3–8.4 Polar coordinates). OpenStax, Rice University. textbook
  2. Strang, G., Herman, E. (2016). Calculus Volume 1, §2.3 The Limit Laws (sin θ < θ < tan θ and lim sin θ/θ = 1). OpenStax, Rice University. textbook
  3. Roy, R., Olver, F. W. J. (chapter authors); Olver, F. W. J., et al. (eds.) (2026). NIST Digital Library of Mathematical Functions, Chapter 4: Elementary Functions (§4.14 definitions, §4.21 identities). National Institute of Standards and Technology, Release 1.2.8. reference
  4. Ling, S. J., Sanny, J., Moebs, W. (2016). University Physics Volume 1, §15.1 Simple Harmonic Motion and §15.3 Comparing Simple Harmonic Motion and Circular Motion. OpenStax, Rice University. textbook
  5. Ling, S. J., Sanny, J., Moebs, W. (2016). University Physics Volume 2, §15.1 AC Sources (v = V₀ sin ωt; 60 or 50 Hz; amplitude 170 or 311 V). OpenStax, Rice University. textbook
  6. Strang, G., Herman, E. (2016). Calculus Volume 1, §3.5 Derivatives of Trigonometric Functions (d/dx sin x = cos x, d/dx cos x = −sin x, x in radians). OpenStax, Rice University. textbook