Unit Circle Explorer

What is the unit circle?

The unit circle is the circle of radius 1 centred on the origin, and it defines sine and cosine for every angle. Measure an angle θ anticlockwise from the positive x axis, and the point where it meets the circle is (cos θ, sin θ): cos θ is the x coordinate, sin θ is the y coordinate, and tan θ = sin θ / cos θ.

That is the right-triangle definition with the hypotenuse set to 1. SOH CAH TOA divides the opposite and adjacent sides by the hypotenuse, and dividing by 1 changes nothing, so the two legs of the triangle in the scene are sin θ and cos θ themselves. What the circle adds is everything past 90°. A right triangle cannot hold an obtuse angle, but the point can keep going round, and its coordinates, negative ones included, are the values of the functions there.

Sine, cosine and tangent as lengths

Each of sin θ, cos θ and tan θ is a length you can see. cos θ is the horizontal leg, from the centre along the x axis to directly below or above the point. sin θ is the vertical leg, from the x axis to the point. Both are signed, so a leg that points left or down is negative.

tan θ lives on a line of its own. The vertical line x = 1 touches the circle at (1, 0), and the radius, extended, meets it at a height of exactly tan θ. That is similar triangles: the small one has legs cos θ and sin θ, the large one has legs 1 and tan θ, so tan θ / 1 = sin θ / cos θ. In quadrants II and III the radius points away from that line, so it is extended backwards through the centre instead, which is where the sign of tan θ comes from there.

Because the hypotenuse is 1, Pythagoras gives the identity that holds at every angle: cos²θ + sin²θ = 1. The point is on the circle, and that equation is the circle.

Why tan 90° is undefined

At 90° the radius is vertical, parallel to the tangent line, and parallel lines never meet, so there is no height to read. The algebra says the same: cos 90° is 0, and 1 / 0 has no value. Step towards 90° with the arrow keys and the tangent segment grows until it runs off the top of the panel: tan 89° is already 57.29. Step past it and the segment comes back from the bottom, with tan 91° at −57.29. The two sides head for opposite infinities, so no single number could stand for tan 90°. The same happens at 270°, the other point on the y axis.

Degrees and radians

To convert, multiply degrees by π/180: θ in radians = θ in degrees × π/180. A full turn is 360°, or 2π ≈ 6.283 radians, and one radian is about 57.30°.

The radian is not an arbitrary unit. It is the angle subtended at the centre of a circle by an arc equal in length to the radius, which is how NIST Special Publication 330 defines it. On a circle of radius 1 that makes an angle in radians exactly the length of arc from (1, 0) round to the point, and the scene draws that arc in a thicker line so you can see it. At 30° the arc is π/6 ≈ 0.5236 long. For other conversions there is the angle conversion table.

Unit circle chart: exact values at the special angles

The multiples of 30° and 45° are the special angles. At each of them sin and cos have exact values, written with at most a square root of 2 or 3, and so does tan wherever it is defined, which is everywhere but 90° and 270°. They all come from two triangles with a hypotenuse of 1. Half a square has angles of 45°, 45° and 90° and both legs √2/2. Half an equilateral triangle has angles of 30°, 60° and 90° and legs of 1/2 and √3/2. Learn the first quadrant and the rest of the circle follows from the reference angle and the quadrant’s signs.

tan 30° is written √3/3 here, rationalised, as the NIST Digital Library of Mathematical Functions prints it. It is the same number as 1/√3, the form many textbooks use.

Which functions are positive in each quadrant

Sine is positive above the x axis and cosine to the right of the y axis, because they are the y and x coordinates. Tangent is their quotient, so it is positive where the two share a sign. That gives the pattern all, sin, tan, cos, anticlockwise from quadrant I:

• Quadrant I, 0° to 90°: all three positive.
• Quadrant II, 90° to 180°: sin positive, cos and tan negative.
• Quadrant III, 180° to 270°: tan positive, sin and cos negative.
• Quadrant IV, 270° to 360°: cos positive, sin and tan negative.

The CAST diagram writes the same pattern starting from quadrant IV, and All Students Take Calculus starts from quadrant I. Neither is needed once you can see where it comes from: with the quadrant signs shown, the shaded quarter always matches the signs in the readouts. On an axis the point is in no quadrant at all: on the x axis sin θ and tan θ are 0, and on the y axis cos θ is 0 and tan θ is undefined.

Reference angles: every angle has a twin in the first quadrant

The reference angle is the angle between the radius and the nearer half of the x axis, from 0° to 90°. sin, cos and tan of any angle have the same size as those of its reference angle and differ at most in sign, which is why the first quadrant is all there is to learn. For θ between 0° and 360°, after adding or taking away whole turns if it is not:

• Quadrant I: the reference angle is θ itself.
• Quadrant II: the reference angle is 180° − θ.
• Quadrant III: the reference angle is θ − 180°.
• Quadrant IV: the reference angle is 360° − θ.

These are the related-angle rules in another form: sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ, and likewise round the rest of the circle.

Worked example: sin, cos and tan of 150°

150° lies between 90° and 180°, so it is in quadrant II, and its reference angle is 180° − 150° = 30°. In quadrant II only sine is positive, so:

• sin 150° = sin 30° = 1/2 = 0.5
• cos 150° = −cos 30° = −√3/2 ≈ −0.866
• tan 150° = sin 150° / cos 150° = −√3/3 ≈ −0.5774

In radians, 150 × π/180 = 5π/6 ≈ 2.618, which is also the length of arc from (1, 0) round to the point. Going clockwise instead reaches the same point at −210°, one full turn away. Set the angle to 150 and every one of these figures appears in the readouts.

An angle with no exact form works the same way. 200° is in quadrant III with a reference angle of 20°, so sin 200° = −sin 20° ≈ −0.342, cos 200° = −cos 20° ≈ −0.9397 and tan 200° = tan 20° ≈ 0.364, positive, as quadrant III says it should be. In radians it is 10π/9 ≈ 3.491.

The graphs are the circle unrolled

Plot the height of the point against the angle and you get the sine curve; plot its x coordinate and you get the cosine curve. On a wide screen the graph shares the circle’s scale and a level line joins the point to the sine curve, so you can see they are the same height at every angle. The bold part of each curve is the part the point has traced since 0°.

Three facts about the graphs are facts about the circle. Sine and cosine repeat every 360°, or 2π, because a full turn brings the point back where it started, while tan repeats every 180°, because the point and the one opposite it give the same line through the centre. Sine is odd, sin(−θ) = −sin θ, and cosine is even, cos(−θ) = cos θ, because a negative angle reflects the point in the x axis, which flips its height and leaves its x coordinate alone. The graph runs from −360° to 360° so that both symmetries are on screen.

The same two coordinates resolve angled quantities all through physics: a launch at angle θ splits into v₀ cos θ across and v₀ sin θ up in the projectile motion simulator, and the dot and cross products carry cos θ and sin θ of the angle between two vectors in the vector cross product explorer. The derivative and tangent line explorer shows the next fact about these curves, that the gradient of sin x, with x in radians, is cos x.

Common mistakes

• A calculator in radian mode. sin 30 in radians is −0.988, the sine of about 1718.87°. If a familiar value comes out strange, check the mode first.
• Dropping the sign with the reference angle. cos 150° is −√3/2, not √3/2. The reference angle gives the size; the quadrant gives the sign.
• Swapping the coordinates. The point is (cos θ, sin θ), cosine first, because x comes first. At 30° that is (√3/2, 1/2).
• Treating tan 90° as a very large number. It has no value at all. tan 89° is about 57.29 and tan 91° about −57.29, so the two sides do not even agree on a sign.
• Reading sin²θ as sin(θ²). It means (sin θ)², which is why cos²θ + sin²θ = 1 is Pythagoras.
• Treating a negative angle as an error. −30° is measured clockwise and lands on the same point as 330°, with the same sin, cos and tan.

What this does not cover

The angle moves in whole degrees between −360° and 360°, which is what the slider and a shared link can hold. For an angle given in radians, or a fraction of a degree, the graphing calculator plots sin(x) with x in radians and reads its value wherever you tap or hover on the curve. The graph beside the circle draws sin θ and cos θ only. The graphing calculator plots tan(x) as well, with a break at every odd multiple of π/2, where tan has no value.

Only sin, cos and tan are drawn. The other three are their reciprocals, sec θ = 1/cos θ, csc θ = 1/sin θ and cot θ = cos θ / sin θ, and one of them is already in the picture: the distance from the centre to where the radius line meets x = 1 is |sec θ|, since sec²θ = 1 + tan²θ. Exact forms are given at the 16 special angles only. The other multiples of 15° have exact forms as well, such as sin 15° = (√6 − √2)/4, but most whole-degree angles, 20° among them, have none worth writing down, so the readouts give those as decimals to four significant figures. Inverse functions, which turn a value back into an angle, are not covered.

Sources

The definitions follow NCERT’s Class XI Mathematics textbook, chapter 3, which defines cos x and sin x as the coordinates of a point on the unit circle and one radian as the angle an arc of length 1 subtends at its centre. The exact values, the signs in each quadrant and the related-angle rules are those tabulated in chapter 4 of the NIST Digital Library of Mathematical Functions, in Tables 4.16.1, 4.16.2 and 4.17.1, and the degree is π/180 radians as NIST Special Publication 330 gives it. The Department for Education’s GCSE mathematics subject content (2013) expects students to know the exact values of sin θ and cos θ at 0°, 30°, 45°, 60° and 90°, and of tan θ at 0°, 30°, 45° and 60°.