x is the cosine and y is the sine. The point sits in the first quadrant, with a reference angle of 30°.
The dashed line is the sine — the height of the point above the axis. The distance from the centre to the foot of it is the cosine. That is all the two functions are, and every identity between them is a statement about this picture.
x = cos θ
√3/2
= 0.8660254
y = sin θ
1/2
= 0.5
In radians
π/6
= 0.52359878
Reference angle
30°
the acute angle to the x-axis
Quadrant
Q1
the first quadrant
x² + y²
1
always 1 — the Pythagorean identity
The coordinates ARE the cosine and sine: a point on the unit circle at angle θ is (cos θ, sin θ). That is the definition the functions come from, rather than a consequence of them.
x² + y² = 1 at every point, which is the Pythagorean identity sin²θ + cos²θ = 1 written geometrically.
The reference angle here is 30° — the acute angle to the x-axis. Every trigonometric value at 30° equals the value at 30° up to a sign the quadrant decides.
Exact surd coordinates at the sixteen special angles, with the radian measure written as a multiple of π.
What this tool shows
A point at 30° on the unit circle sits at (√3/2, 1/2). That x is cos 30° and that y is sin 30° — not because of a formula, but because that is what the functions are defined to be. Every trigonometric identity is a statement about this one picture.
Where any angle lands on the unit circle
The exact coordinates, in surd form
The radian measure as a multiple of π
The reference angle and the quadrant
x² + y², which is always 1
Why the coordinates and the trig values are the same thing
Drawn for your angle Exact coordinates Reference angle and quadrant Radians as a multiple of π
Exact surd coordinates at the sixteen special angles.
Updated 7 September 2026 · Works in any browser, no installation
A point at angle θ on a circle of radius 1 has coordinates (cos θ, sin θ). At 30° that is (√3/2, 1/2). The x-coordinate is the cosine and the y-coordinate is the sine — not a rule to apply, but the definition the two functions come from.
At a glance
Formula shown
A point at angle \u03b8 on the circle of radius 1 has coordinates (cos \u03b8, sin \u03b8). Because the radius is 1, x\u00b2 + y\u00b2 = 1 \u2014 which is the Pythagorean identity sin\u00b2\u03b8 + cos\u00b2\u03b8 = 1 written as geometry.
Scenario support
Learning where the trig values come from; finding exact coordinates for a special angle; checking a sign without reciting a mnemonic.
Educational estimate
Planning support from the values you enter — not professional advice.
Why it is called the unit circle
Radius 1. That is the whole reason for the name, and it is also the reason the circle is worth drawing at all.
On a circle of radius r, a point at angle θ sits at (r cos θ, r sin θ). Setting r = 1 makes the r disappear, and the coordinates become the trigonometric values themselves.
It also makes the Pythagorean identity obvious. Every point on the circle is distance 1 from the centre, so x² + y² = 1. Substituting the coordinates gives cos²θ + sin²θ = 1 — the most-used identity in trigonometry, and it is just the distance formula on a circle of radius one.
Reading the values off it
Cosine is horizontal. How far right or left the point is. Positive on the right half, negative on the left.
Sine is vertical. How far up or down. Positive on the top half, negative on the bottom.
Tangent is the slope. y divided by x, which is the gradient of the line from the centre to the point. It is undefined where the line is vertical — at 90° and 270°, where x is zero.
A way to keep cosine and sine apart that survives exam pressure: cos comes before sin alphabetically, and x comes before y. Cosine is x.
The sixteen angles
Sixteen angles have coordinates you can write exactly: the four axes, and the multiples of 30° and 45° in between.
They come from two triangles. The 30-60-90 gives √3/2 and 1/2; the 45-45-90 gives √2/2 for both. Every one of the sixteen is one of those three numbers with a sign.
So the “unit circle chart” people photograph before an exam contains three values and a sign rule. Drawing the two triangles reconstructs it in about ten seconds, which is more reliable than remembering which of √2/2 and √3/2 goes with 45°.
Everything else — 1°, 17°, 2 radians — has no exact surd form, and the decimal is the answer. The page says which case yours is.
Radians, and why they are natural here
On the unit circle, the angle in radians IS the arc length. Walk 1 unit round the edge and you have turned through 1 radian.
That is not a coincidence; it is the definition. A radian is the angle whose arc equals the radius, and on a circle of radius 1 the arc and the angle are the same number.
It makes the fractions of a turn read cleanly: a quarter turn is π/2, a third is 2π/3, a full turn is 2π. The page writes the radian measure in that form whenever the denominator is small enough to be useful.
It is also why calculus insists on radians. The derivative of sin x is cos x only because of this identification of angle with arc; in degrees the same derivative carries a factor of π/180.
Signs, without memorising them
ASTC works, and looking at the picture works better, because the picture explains the mnemonic rather than replacing it.
In the second quadrant the point is up and to the left. Up means positive y, so sine is positive. Left means negative x, so cosine is negative. Tangent is their ratio, so it is negative too. That is “S” in ASTC, derived rather than recalled.
In the third quadrant both coordinates are negative, so sine and cosine are both negative — and their ratio is positive. That is why tangent is the survivor there, and it takes two seconds to work out from the diagram.
The reciprocal functions follow their partners exactly, since a reciprocal never changes a sign.
What it does not show
The circle is a picture of one turn. Angles beyond that wrap onto it, and the picture cannot distinguish 30° from 390° — which is correct, because the trigonometric values are the same.
It also cannot show how many times you went round, which matters for anything cumulative: a rotating shaft, a phase that has advanced through several cycles, a winding number.
And it says nothing about the graph shape — period, amplitude, phase shift. For a wave you need the curve rather than the circle, though the curve is generated by walking round the circle and plotting the height.
Sources and methodology
The unit-circle definition is the standard one; these are the references.
Method. The point is computed from the angle you entered rather than snapped to a nearest preset, and the exact surd coordinates come from a table matched against the reduced angle. The radian measure is expressed as a multiple of π when the denominator is small enough to be readable. x² + y² is displayed as a computed number rather than asserted, so the Pythagorean identity is visible on the page. That engine is verified on every change against 71 hand-written assertions, including that the point lies on the circle to twelve decimal places and that its coordinates equal cosine and sine across two hundred and fifty generated angles. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Trigonometric FunctionsAll six functions at once with exact surd values at the sixteen special angles — and tan 90° reported as undefined rather than as the 1.6 × 10¹⁶ a double returns.
Inverse Trigonometric FunctionsThe principal value AND the general solution, because a calculator hands you one member of an infinite family and the context decides which one your problem wants.
Reference AngleThe acute angle to the x-axis — never the y-axis — with the quadrant rule written out using your own numbers and the sign of each trig function beside it.
Coterminal AngleThe principal angle between 0° and 360°, the nearest negative one, and the family θ + 360n they all belong to — reduced by exact division, not by looping.
Special Right Triangles30-60-90 and 45-45-90 from any one side, in exact surd form as well as decimals — with the warning that the long leg is √3 times the short, not twice.
CircleRadius, diameter, circumference and area from any one of them — and why a 16-inch pizza is four times an 8-inch one rather than twice.
For all six functions at once, including the reciprocals and the undefined cases, the Trigonometric Functions Calculator covers them; the Reference Angle Calculator handles the acute-angle-to-the-axis step on its own.
Educational use disclaimer
This is an educational tool. Exact coordinates are exact; the decimals beside them are rounded, and the circle is drawn to the angle you entered rather than to a nearest preset.
Published the unit circle page treating the coordinates as the definition rather than as an illustration: a point at angle θ is (cos θ, sin θ), and every identity between the two is a statement about that picture.
Exact surd coordinates at the sixteen special angles, with the radian measure written as a multiple of π rather than as a decimal.
Reports x² + y² so the Pythagorean identity is visible as a number on the page rather than quoted in prose.
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