The Unit Circle
The unit circle is the circle of radius 1 centred at the origin. The point at any angle on it is (cos, sin), so one chart holds every special value of sine, cosine and tangent.
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This chart gives sine and cosine at all sixteen special angles, in degrees and in radians. Switch it to show tan too, or to a blank circle you can print and fill in.
Unit circle chart
Unit circle values
Every value on the chart, in one table. Tan is sin divided by cos, so it has no value where cos is 0.
| Degrees | Radians | cos | sin | tan |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
| 120° | 2π/3 | −1/2 | √3/2 | −√3 |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 |
| 150° | 5π/6 | −√3/2 | 1/2 | −√3/3 |
| 180° | π | −1 | 0 | 0 |
| 210° | 7π/6 | −√3/2 | −1/2 | √3/3 |
| 225° | 5π/4 | −√2/2 | −√2/2 | 1 |
| 240° | 4π/3 | −1/2 | −√3/2 | √3 |
| 270° | 3π/2 | 0 | −1 | undefined |
| 300° | 5π/3 | 1/2 | −√3/2 | −√3 |
| 315° | 7π/4 | √2/2 | −√2/2 | −1 |
| 330° | 11π/6 | √3/2 | −1/2 | −√3/3 |
Why the point is (cos, sin)
Drag the point round the circle and watch the two dashed legs.
Drag the point. The green leg is cos and the red leg is sin, which is why the point is (cos, sin).
The circle has radius 1, and that is the whole trick. Draw a line from the centre out to the circle at some angle, then drop a vertical line from that point to the x-axis. You now have a right triangle whose hypotenuse is the radius, so the hypotenuse is 1.
In that triangle, cosine is adjacent over hypotenuse. The adjacent side is the horizontal leg, and the hypotenuse is 1, so cosine is the horizontal leg. The same argument makes sine the vertical leg. So the point is (cos(x), sin(x)): horizontal distance first, then vertical, x before y.
The three numbers you actually need
The first quadrant holds every value the circle uses:
| angle | cos | sin |
|---|---|---|
| 30° | √3/2 | 1/2 |
| 45° | √2/2 | √2/2 |
| 60° | 1/2 | √3/2 |
Three distinct values: 1/2, √2/2 and √3/2. Notice that 30° and 60° are the same pair swapped. The two angles add to 90°, so their triangles are the same triangle seen from the other end.
Signs in each quadrant
Every angle outside the first quadrant uses one of those three numbers with a sign. To find it, take the reference angle, the angle between the line and the x-axis: 150° is 30° short of 180°, so its reference angle is 30° and its values are 30°'s values with new signs.
The signs come from where the point is. Cos is the horizontal distance, so it is negative on the left half of the circle. Sin is the vertical distance, so it is negative on the bottom half. The chart marks each quadrant with its signs of (cos, sin):
| quadrant | angles | cos | sin | tan |
|---|---|---|---|---|
| I | 0° to 90° | + | + | + |
| II | 90° to 180° | − | + | − |
| III | 180° to 270° | − | − | + |
| IV | 270° to 360° | + | − | − |
So 150° is (−√3/2, 1/2) and 210°, whose reference angle is also 30°, is (−√3/2, −1/2).
The unit circle in radians
A full turn is 360°, and it is also 2π radians, because the circumference of a circle of radius 1 is 2π: one radian is an arc one radius long. Half a turn is 180° or π.
From there the special angles are fractions of a half turn. 30° is a sixth of 180°, so it is π/6; 45° is a quarter, π/4; 60° a third, π/3. The denominator tells you how many of that angle fit into a half turn, and the numerator counts them: 150° is five sixths of a half turn, 5π/6.
Tangent on the unit circle
Tan is the ratio sin(x)/cos(x), which is the slope of the line from the centre to the point. At 45° sin and cos are equal, so tan is exactly 1. At 30° it is √3/3 and at 60° it is √3.
At 90° the point is straight up at (0, 1), so cos is 0 and tan is undefined: there is no number equal to 1 divided by 0, and a vertical line has no slope. The same happens at 270°. Everywhere else tan follows the sign rule: positive where sin and cos agree (quadrants I and III), negative where they differ. The chart's tan version prints every value.
Unit circle quiz and practice
Switch the draggable circle above to Quiz me: it hides the values, you say them, then you check. Or answer this one from memory:
To practise on paper, print the blank chart and fill in all sixteen points. When you can do it in under two minutes without looking, you know the unit circle.
Common questions
What is the unit circle?
It is the circle of radius 1 centred on the origin. Draw a line from the centre at an angle and mark where it crosses the circle: that point's coordinates are exactly (cos of the angle, sin of the angle). So the circle turns a question about triangles into a question about coordinates, which is much easier to picture.
What are the values on the unit circle?
At the sixteen special angles every coordinate is one of 0, 1/2, √2/2, √3/2 or 1, with a sign. In the first quadrant: 30° is (√3/2, 1/2), 45° is (√2/2, √2/2) and 60° is (1/2, √3/2). Every other special angle repeats one of these pairs with the signs of its quadrant. The table on this page lists all sixteen, with tan.
Is there a printable or blank unit circle?
Yes. The chart at the top of this page downloads as a letter-size PDF or a PNG in three versions: with coordinates, with coordinates and tan, and blank. The blank one keeps the circle, the axes and the sixteen points, so you can fill in every label yourself.
Why is the point on the circle (cos, sin) and not (sin, cos)?
Because cosine measures horizontal distance and the x-coordinate is the horizontal one. Drop a vertical line from the point to the x-axis and you get a right triangle whose hypotenuse is 1. The horizontal leg is adjacent to the angle, so it is cos over 1, which is just cos. The vertical leg is opposite, so it is sin.
How do I remember the unit circle?
Learn three numbers and one rule. The numbers are 1/2, √2/2 and √3/2, the first quadrant values. The rule is the sign: cos is negative on the left half of the circle and sin is negative on the bottom half. Every other angle is a first quadrant value with those signs, found from its reference angle.
How do I convert between degrees and radians?
A full turn is 360 degrees and also 2π radians, so half a turn is 180 degrees and π radians. Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. So 30° is π/6, 45° is π/4, 60° is π/3 and 90° is π/2.
Why is tan undefined at 90 degrees?
Tan is sin divided by cos. At 90° the point is straight up at (0, 1), so cos is 0, and dividing by 0 has no answer. The same happens at 270°. On the circle you can see it too: tan is the slope of the line from the centre, and a vertical line has no slope.
This and every related formula, on one printable sheet:
Trigonometry Formulas