Trigonometry Calculator (sin, cos, tan)
Find sin, cos, tan, and their reciprocals for any angle in degrees or radians, or use the inverse functions to get an angle back from a ratio.
Exact trigonometric values table
The values worth knowing by heart, in exact form as well as decimal. A calculator will tell you sin 60° is 0.8660254, but √3/2 is the answer an exam wants and the one that stays exact when you carry it through the rest of a calculation. Every exact form in this table is checked against the computed value when the site is built.
| Degrees | Radians | sin (exact) | sin | cos (exact) | cos | tan (exact) | tan |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 30° | π/6 | 1/2 | 0.5 | √3/2 | 0.86603 | √3/3 | 0.57735 |
| 45° | π/4 | √2/2 | 0.70711 | √2/2 | 0.70711 | 1 | 1 |
| 60° | π/3 | √3/2 | 0.86603 | 1/2 | 0.5 | √3 | 1.7321 |
| 90° | π/2 | 1 | 1 | 0 | 0.000000000000000061232 | undefined | — |
| 120° | 2π/3 | √3/2 | 0.86603 | −1/2 | -0.5 | −√3 | -1.7321 |
| 135° | 3π/4 | √2/2 | 0.70711 | −√2/2 | -0.70711 | −1 | -1 |
| 150° | 5π/6 | 1/2 | 0.5 | −√3/2 | -0.86603 | −√3/3 | -0.57735 |
| 180° | π | 0 | 0.00000000000000012246 | −1 | -1 | 0 | -0.00000000000000012246 |
| 210° | 7π/6 | −1/2 | -0.5 | −√3/2 | -0.86603 | √3/3 | 0.57735 |
| 225° | 5π/4 | −√2/2 | -0.70711 | −√2/2 | -0.70711 | 1 | 1 |
| 240° | 4π/3 | −√3/2 | -0.86603 | −1/2 | -0.5 | √3 | 1.7321 |
| 270° | 3π/2 | −1 | -1 | 0 | -0.0000000000000001837 | undefined | — |
| 300° | 5π/3 | −√3/2 | -0.86603 | 1/2 | 0.5 | −√3 | -1.7321 |
| 315° | 7π/4 | −√2/2 | -0.70711 | √2/2 | 0.70711 | −1 | -1 |
| 330° | 11π/6 | −1/2 | -0.5 | √3/2 | 0.86603 | −√3/3 | -0.57735 |
| 360° | 2π | 0 | -0.00000000000000024493 | 1 | 1 | 0 | -0.00000000000000024493 |
Decimal columns are rounded for display; tan is undefined at 90° and 270° because cos is zero there and the ratio divides by it.
The four values everything else comes from
There are really only three numbers in the whole table — 1/2, √2/2 and √3/2 — plus zero and one. Every other entry is one of those with a sign attached, decided by which quadrant the angle falls in: sine is positive above the horizontal axis, cosine is positive to the right of the vertical one, and tangent is positive where both agree. Learning the first quadrant and the sign rule is far less work than memorising seventeen rows.
Trigonometric ratios
The three main trig functions relate an angle to the ratios of a right triangle's sides: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent — the classicSOH-CAH-TOA. The other three (cosecant, secant, cotangent) are simply their reciprocals. Enter an angle to get all six, or switch to the inverse mode to go from a ratio back to the angle.
sin θ = opp/hyp · cos θ = adj/hyp · tan θ = opp/adj
csc θ = 1/sin θ · sec θ = 1/cos θ · cot θ = 1/tan θ
Degrees or radians
Angles come in two units and it matters which you use — 30 degreesand 30 radians are wildly different angles. Everyday work uses degrees; higher maths and most programming languages use radians, where a full circle is 2π. Use the angle converterif you need to switch between them.
Worked example
At 30°, sin = 0.5, cos = √3/2 ≈ 0.8660, and tan = 0.5774. Going the other way, the inverse mode answers "what angle has this ratio?" — arcsin(0.5) = 30°, which is how you recover an angle from a measured slope or side ratio.
When a ratio is undefined
Some ratios blow up: tan and sec are undefined at 90° (cos is zero, so you'd divide by zero), and csc and cot are undefined at 0° and 180° (sin is zero). The calculator shows undefined at those points rather than a misleading huge number. For the inverse functions, arcsin and arccos only accept values from −1 to 1, since no real angle has a sine or cosine outside that range.
Common questions
- What is sin 30, cos 30 and tan 30?
- Exactly 1/2, √3/2 and √3/3 respectively — approximately 0.5, 0.866 and 0.577. Those exact forms are worth memorising rather than the decimals, because they stay exact when carried through further working. The full table of standard angles is above.
- My calculator gives a strange answer — what did I do wrong?
- Almost certainly the angle unit. sin(30) is 0.5 in degrees but −0.988 in radians, and they are wildly different angles. Every trigonometric tool defaults to one or the other, and mixing them up is by far the commonest source of a wrong trig answer. Check the unit before checking anything else.
- Why is tan 90 undefined rather than infinite?
- Because tan θ = sin θ / cos θ, and cos 90° is exactly zero — so the ratio divides by zero, which has no value. It is not that the answer is very large; it is that there is no answer. As the angle approaches 90° from below the ratio grows without limit, and from above it grows negatively without limit, so there is no single value it tends towards.
- What is SOH-CAH-TOA?
- A mnemonic for the three ratios in a right triangle: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. "Opposite" and "adjacent" are relative to the angle you are working with, which is the part people trip over — swap the angle and those two sides swap roles.