Reference chart
Unit Circle Chart
By &Pixels, the studio that builds MyCalculator.to. Exact values, written as radicals rather than rounded to decimals.
All sixteen standard angles in degrees and radians with their exact sine, cosine and tangent, plus the pattern that makes the first quadrant worth learning and the rest not.
What is the unit circle?
A circle of radius one, centred on the origin. Its usefulness is that any point on it has the coordinates (cos θ, sin θ), so the trigonometric functions stop being ratios inside a triangle and become positions on a circle. That is what extends them past 90 degrees and keeps them working all the way round.
It also explains the signs. Cosine is the horizontal coordinate and goes negative on the left half of the circle; sine is the vertical one and goes negative on the bottom half. Nothing has to be memorised there, only read off.
Exact values
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 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 |
The coordinates of each point are the cosine and the sine in that order, so 60 degrees sits at (1/2, √3/2). Tangent is the sine divided by the cosine, which is why it is undefined at 90 and 270: the cosine there is zero.
The calculators behind this
The pattern nobody points out
| Degrees | sin, written as | Which is |
|---|---|---|
| 0 | √0/2 | 0 |
| 30 | √1/2 | 1/2 |
| 45 | √2/2 | √2/2 |
| 60 | √3/2 | √3/2 |
| 90 | √4/2 | 1 |
Written that way the first quadrant is one idea rather than five facts: the square roots of 0, 1, 2, 3 and 4, each over two. Cosine is the identical list read backwards, because the cosine of an angle is the sine of its complement. Five values, learned once, and the other eleven rows of the table above are those five with a sign attached.
Signs by quadrant
| Quadrant | Degrees | sin | cos | tan |
|---|---|---|---|---|
| I | 0 to 90 | positive | positive | positive |
| II | 90 to 180 | positive | negative | negative |
| III | 180 to 270 | negative | negative | positive |
| IV | 270 to 360 | negative | positive | negative |
How to read any angle
- Find the reference angle. The acute angle to the horizontal axis. For 210 degrees that is 30, for 135 it is 45.
- Take the value from the first quadrant. Five rows cover every angle on the circle. The reference angle tells you which one.
- Work out the quadrant. Count in nineties anticlockwise from the positive horizontal axis.
- Apply the sign. Sine follows the vertical, cosine the horizontal, and tangent is positive where both agree.
- Read the point as cosine then sine. The coordinates on the circle are (cos, sin) in that order, which is the reverse of how it is usually said aloud.
For an interactive version of the circle itself, the unit circle reference tool shows each angle in place. To move between degrees, radians and gradians the angle converter does it directly, and the triangle calculator solves the triangle these values came out of in the first place.
Formula & Methodology
Did you know? The radian is defined by the circle rather than chosen by convention: it is the angle whose arc is exactly as long as the radius. Because of that, the derivative of sine is cosine only when the angle is in radians. In degrees a factor of π/180 appears and never leaves, which is why calculus abandons degrees entirely.
Sources
- Standard trigonometric identities and exact values for special angles
- The Pythagorean identity, from which every value in the first quadrant follows
- Definition of radian measure as arc length over radius