Unit Circle

An interactive unit circle: drag to any angle for exact sin, cos and tan, read the full chart, or print a blank one to practise.

Snap to standard angles

Angle

Any angle, negative or past 360

Point on the circle

(1/2, √3/2)

60° = π/3

Function	Exact	Decimal
sin 60°	√3/2	≈ 0.8660254038
cos 60°	1/2	0.5
tan 60°	√3	≈ 1.732050808
csc 60°	2√3/3	≈ 1.154700538
sec 60°	2
cot 60°	√3/3	≈ 0.5773502692

Quadrant: I: All six are positive
Reference angle: 60°
Coterminal: 60°
Coterminal (signed): 60°

Find the angles with a value

Function

Value

√3/2, -0.5, 1/√3 or 0.3

The unit circle chart, live. The 16 standard angles are marked in degrees and radians with their points, and the point drags to any angle, or steps with the arrow keys. For the angle shown, all six trigonometric values appear exactly where an exact value exists, at every multiple of 15°, beside their decimals, with the quadrant, the reference angle and the coterminal angles.

The tool also works backwards: choose a function and a value (sin θ = √2/2) to see every angle in one turn that gives it, marked on the circle. The chart prints in full or blank for practice, and downloads as an SVG.

How to use it

Type an angle in degrees or in radians. Radians take π (5π/6, 5pi/6) and keep the value exact; a decimal such as 2.6 is read as it is. Negative angles and angles past one turn are taken to the same point, and the coterminal angles say which.

Snap to standard angles starts on, so the point jumps between the 16 chart angles. Turned off, it moves in whole degrees, or in steps of 0.01 radians when the angle is in radians.

How it works

A circle of radius 1 around the origin turns an angle into a point. Measure θ anticlockwise from the positive x-axis: the point where the angle's side crosses the circle is (cos θ, sin θ). The right triangle it makes with the x-axis has a hypotenuse of 1, so its horizontal leg is cos θ and its vertical leg is sin θ. Then tan θ = sin θ ÷ cos θ, and csc, sec and cot are the reciprocals of sin, cos and tan.

The signs follow the quadrant, remembered as ASTC, anticlockwise from the first: All six are positive in I, Sin (and csc) in II, Tan (and cot) in III, and Cos (and sec) in IV. The reference angle, the acute angle to the x-axis, gives the size of every value; the quadrant gives the sign.

A radian is the angle that cuts an arc as long as the radius, so one turn is 2π radians and 180° is π. At multiples of 15° the values are sums of square roots, such as sin 15° = (√6 − √2)/4; the tool keeps them exact, and gives decimals for other angles.

Point at θ: (cos θ, sin θ)

tan θ = sin θ ÷ cos θ

csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ

Radians = degrees × π ÷ 180

Examples

210°

Type 210. The point is (−√3/2, −1/2) in quadrant III, where only tan and cot are positive: tan 210° = √3/3. The reference angle is 30°, so every value is a 30° value with its sign.

5π/4 in radians

With the angle in radians, type 5π/4. It is 225°, the point is (−√2/2, −√2/2), and sec 5π/4 = −√2. The same point is also at −3π/4 measured clockwise.

Every angle with sin θ = −½

Under Find the angles with a value, choose sin and type -1/2. Sine is negative below the x-axis, in quadrants III and IV, so there are two: 210° (7π/6) and 330° (11π/6).

The standard angles

Exact sin, cos and tan of the 16 angles on the chart

Degrees	Radians	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined
120°	2π/3	√3/2	−1/2	−√3
135°	3π/4	√2/2	−√2/2	−1
150°	5π/6	1/2	−√3/2	−√3/3
180°	π	0	−1	0
210°	7π/6	−1/2	−√3/2	√3/3
225°	5π/4	−√2/2	−√2/2	1
240°	4π/3	−√3/2	−1/2	√3
270°	3π/2	−1	0	undefined
300°	5π/3	−√3/2	1/2	−√3
315°	7π/4	−√2/2	√2/2	−1
330°	11π/6	−1/2	√3/2	−√3/3

Frequently asked questions

Learn the first quadrant and reflect it. There, the sines of 0°, 30°, 45°, 60° and 90° are √0/2, √1/2, √2/2, √3/2 and √4/2: 0, ½, √2/2, √3/2 and 1. The cosines are the same list backwards. The hand trick holds the same pattern: spread one hand, palm up, and let the fingers from the little finger to the thumb stand for 0°, 30°, 45°, 60° and 90°. Fold down a finger: the fingers below it count n for sin = √n/2, and those above it count n for cos. Every other angle copies one of these with the sign of its quadrant.

At 90° the point is (0, 1), and tan θ = sin θ ÷ cos θ would divide by 0. Close to 90° the tangent grows without limit, positive just before and negative just after, so no value fits. The same holds for sec at 90° and 270°, and for csc and cot at 0° and 180°.