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The unit circle

The unit circle is the circle of radius 1 centred on the origin. Measure an angle counter-clockwise from the positive x-axis and the point where its terminal side crosses the circle has coordinates (cos, sin). Because the radius is exactly 1, no division is needed: the x-coordinate is the cosine of the angle and the y-coordinate is its sine.

Drag the slider below, or tap one of the 16 standard angles, to watch the coordinates, the quadrant and the reference angle update together. The full chart of exact values is printed underneath.

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Angle: 30° (π/6)

sin θ1/2 ≈ 0.5

cos θ√3/2 ≈ 0.866

tan θ√3/3 ≈ 0.5774

QuadrantQ1

Reference angle30°

The green leg is cos θ (x), the amber leg is sin θ (y). Drag the slider or pick a special angle. This explorer is free, forever.

Unit circle chart: exact values at every standard angle

These are the 16 angles a trigonometry course expects you to know cold. Sine is the y-coordinate, cosine is the x-coordinate, and tangent is the quotient of the two.

Exact sine, cosine and tangent values at the 16 standard unit-circle angles
Degrees	Radians	sin	cos	tan	Quadrant
0°	$0$	$0$	$1$	$0$	on an axis
30°	$\dfrac{\pi}{6}$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{3}}{3}$	Q1
45°	$\dfrac{\pi}{4}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$	Q1
60°	$\dfrac{\pi}{3}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$	Q1
90°	$\dfrac{\pi}{2}$	$1$	$0$	$\text{undefined}$	on an axis
120°	$\dfrac{2\pi}{3}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{-1}{2}$	$-\sqrt{3}$	Q2
135°	$\dfrac{3\pi}{4}$	$\dfrac{\sqrt{2}}{2}$	$-\dfrac{\sqrt{2}}{2}$	$-1$	Q2
150°	$\dfrac{5\pi}{6}$	$\dfrac{1}{2}$	$-\dfrac{\sqrt{3}}{2}$	$-\dfrac{\sqrt{3}}{3}$	Q2
180°	$\pi$	$0$	$-1$	$0$	on an axis
210°	$\dfrac{7\pi}{6}$	$\dfrac{-1}{2}$	$-\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{3}}{3}$	Q3
225°	$\dfrac{5\pi}{4}$	$-\dfrac{\sqrt{2}}{2}$	$-\dfrac{\sqrt{2}}{2}$	$1$	Q3
240°	$\dfrac{4\pi}{3}$	$-\dfrac{\sqrt{3}}{2}$	$\dfrac{-1}{2}$	$\sqrt{3}$	Q3
270°	$\dfrac{3\pi}{2}$	$-1$	$0$	$\text{undefined}$	on an axis
300°	$\dfrac{5\pi}{3}$	$-\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$-\sqrt{3}$	Q4
315°	$\dfrac{7\pi}{4}$	$-\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$-1$	Q4
330°	$\dfrac{11\pi}{6}$	$\dfrac{-1}{2}$	$\dfrac{\sqrt{3}}{2}$	$-\dfrac{\sqrt{3}}{3}$	Q4

How to read the circle instead of memorising it

Every row above is recoverable from three first-quadrant facts plus a sign. At 30 degrees the sine is 1/2 and the cosine is the square root of 3 over 2; at 45 degrees both are the square root of 2 over 2; at 60 degrees the pair swaps back, with sine the square root of 3 over 2 and cosine 1/2. Written as square roots over 2, the sines of 0, 30, 45, 60 and 90 degrees run through the square roots of 0, 1, 2, 3 and 4, and the cosines run through the same list backwards.

Outside Quadrant I, take the reference angle, look up its first-quadrant value, then attach the sign of the quadrant. Sine follows the sign of y, cosine follows the sign of x, and tangent is positive exactly where the two signs agree.

Quadrant I, 0 to 90 degrees: sine, cosine and tangent are all positive.
Quadrant II, 90 to 180 degrees: only sine is positive.
Quadrant III, 180 to 270 degrees: only tangent is positive.
Quadrant IV, 270 to 360 degrees: only cosine is positive.

Why the circle satisfies the Pythagorean identity

A point on the unit circle is one radius from the origin, so its coordinates obey x squared plus y squared equals 1. Substituting the coordinates gives the identity every later formula leans on.

\sin^{2}\theta + \cos^{2}\theta = 1

That single line is why the circle is worth more than a chart: it is the geometry behind the Pythagorean identities, and dividing it by cosine squared or sine squared produces the other two.

Frequently asked questions

What is the unit circle?

The unit circle is the circle of radius 1 centred on the origin. For an angle measured counter-clockwise from the positive x-axis, the point where the circle meets the terminal side has coordinates (cos, sin). Because the radius is 1, the x-coordinate is the cosine and the y-coordinate is the sine, with no division needed.

How many angles do I actually need to memorise?

Three. Once you know the first-quadrant values at 30, 45 and 60 degrees, the other 13 standard angles follow from the reference angle and the sign rules of the quadrant you are in. That is why this page lists all 16 rows but the pattern only has three ingredients.

How do I find a reference angle?

Reduce the angle to the range 0 to 360 degrees, then measure the acute angle it makes with the x-axis: the angle itself in Quadrant I, 180 minus the angle in Quadrant II, the angle minus 180 in Quadrant III, and 360 minus the angle in Quadrant IV. The sine, cosine and tangent of the original angle equal those of the reference angle, up to sign.

Which functions are positive in which quadrant?

All three are positive in Quadrant I. Only sine (and its reciprocal cosecant) is positive in Quadrant II, only tangent (and cotangent) in Quadrant III, and only cosine (and secant) in Quadrant IV. The sign follows from the sign of the coordinates: sine tracks y, cosine tracks x, and tangent is y over x.

Why is tangent undefined at 90 and 270 degrees?

Tangent is sine divided by cosine. At 90 and 270 degrees the point on the circle sits on the y-axis, so the cosine is exactly 0 and the quotient has no value. Those are the vertical asymptotes of the tangent graph.

How do degrees and radians line up on the circle?

A full turn is 360 degrees or 2 pi radians, so 180 degrees equals pi radians and one radian is 180 divided by pi degrees. The standard angles land on neat fractions of pi, which is why the table below lists both measures side by side.

Keep going

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