Reference · 17 angles, exact values only
The unit circle
Take a circle of radius 1 centred at the origin. The point where the terminal side of an angle meets it is at (cos θ, sin θ) — that is the whole definition, and it is why sin²θ + cos²θ = 1is not a trick: the point sits one radius from the origin, so Pythagoras forces the squares of its coordinates to add to 1. Every value in the table below comes out of that picture.
Dotted segments are the two projections: the horizontal one is cos θ, the vertical one is sin θ. The three rays shown are the first-quadrant angles whose exact values everyone ends up memorising — 30°, 45° and 60°.
Exact values
The full set. undefined is printed where the function genuinely has no value — tangent is sin θ / cos θ, so it stops existing where cosine is 0.
| Angle | Degrees | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0 | 0° | 1 | 0 | 0 |
| π/6 | 30° | √3/2 | 1/2 | √3/3 |
| π/4 | 45° | √2/2 | √2/2 | 1 |
| π/3 | 60° | 1/2 | √3/2 | √3 |
| π/2 | 90° | 0 | 1 | undefined |
| 2π/3 | 120° | −1/2 | √3/2 | −√3 |
| 3π/4 | 135° | −√2/2 | √2/2 | −1 |
| 5π/6 | 150° | −√3/2 | 1/2 | −√3/3 |
| π | 180° | −1 | 0 | 0 |
| 7π/6 | 210° | −√3/2 | −1/2 | √3/3 |
| 5π/4 | 225° | −√2/2 | −√2/2 | 1 |
| 4π/3 | 240° | −1/2 | −√3/2 | √3 |
| 3π/2 | 270° | 0 | −1 | undefined |
| 5π/3 | 300° | 1/2 | −√3/2 | −√3 |
| 7π/4 | 315° | √2/2 | −√2/2 | −1 |
| 11π/6 | 330° | √3/2 | −1/2 | −√3/3 |
| 2π | 360° | 1 | 0 | 0 |
Which sign, in which quadrant
Only two values per angle are needed to place a point: a size and a sign. The size comes from the reference angle below; the sign comes from the quadrant. These three columns are computed from the same engine the verifier uses, at 45°, 135°, 225° and 315° — one sample inside each quadrant.
| Quadrant | Range | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| I | 0°–90° | + | + | + |
| II | 90°–180° | + | − | − |
| III | 180°–270° | − | − | + |
| IV | 270°–360° | − | + | − |
Read it as: sine keeps the vertical coordinate, so it is positive while the point is above the x-axis; cosine keeps the horizontal one, so it is positive to the right; tangent is their ratio and flips sign whenever exactly one of the two does.
Angles off the first quadrant
Every special angle outside 0°–90° reuses a first-quadrant value with a sign change — this table was built by matching the exact values, not by asserting the pattern. So there is really one small set to remember, and four sign rules to apply.
| Angle | Reference angle | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 2π/3 (120°) | π/3 (60°) | √3/2 | −1/2 | −√3 |
| 3π/4 (135°) | π/4 (45°) | √2/2 | −√2/2 | −1 |
| 5π/6 (150°) | π/6 (30°) | 1/2 | −√3/2 | −√3/3 |
| π (180°) | 0 (0°) | 0 | −1 | 0 |
| 7π/6 (210°) | π/6 (30°) | −1/2 | −√3/2 | √3/3 |
| 5π/4 (225°) | π/4 (45°) | −√2/2 | −√2/2 | 1 |
| 4π/3 (240°) | π/3 (60°) | −√3/2 | −1/2 | √3 |
| 3π/2 (270°) | π/2 (90°) | −1 | 0 | undefined |
| 5π/3 (300°) | π/3 (60°) | −√3/2 | 1/2 | −√3 |
| 7π/4 (315°) | π/4 (45°) | −√2/2 | √2/2 | −1 |
| 11π/6 (330°) | π/6 (30°) | −1/2 | √3/2 | −√3/3 |
What the circle already gives you
- sin²θ + cos²θ = 1 — the point is one radius away from the origin.
- sin θ = cos(90° − θ) — swapping the two coordinates is reflecting in the line y = x.
- sin(−θ) = −sin θ and its cosine partner — turning the other way flips the vertical coordinate only.
- The cofunction family and the Pythagorean family are the identity pages built on this picture.
What this page does not do
It is a table, not a calculator. Exact values exist for the angles above; anything else (sin 41°, say) is a decimal and belongs to a calculator, which this site deliberately does not fake. Two things you can do instead: check whether an equation is an identity at the verifier, or convert an angle between degrees and radians in the converter. The same numbers fit on one page for printing in the cheat sheet.