Trig calculator
Any angle in degrees or radians (−30, 400, 5π/6): exact and decimal sin, cos, tan, sec, csc and cot, and the point on the unit circle.
How the values are found
Every value in the table comes from one picture. Turn from the positive x-axis through the angle θ (counter-clockwise for positive angles, clockwise for negative ones) and stop on the circle of radius 1. The point you reach is (cos θ, sin θ)[1]. The other four functions are built from those two coordinates:
A function is undefined where its denominator is zero: tangent and secant at 90° and 270°, where the point is on the y-axis and cos θ = 0; cotangent and cosecant at 0° and 180°, where sin θ = 0. The table says “undefined” there rather than printing an enormous number.
Negative angles and angles beyond a full turn
Adding or removing a whole turn (360°, or 2π radians) brings you back to the same point, so it changes none of the six values. Angles that differ by whole turns are called coterminal[1]. The calculator reduces your angle to the coterminal one between 0° and 360° and says so: type −30 and it reports that −30° lands on 330°, the same point. Type 400 and it lands on 40°. Nothing is rounded away: 47.5° stays 47.5°.
When is the answer exact?
At every multiple of 15° (π/12 radians) sine and cosine are short expressions in square roots, such as √3/2 or (√6 − √2)/4. They come from an equilateral triangle (30° and 60°), a square cut along its diagonal (45°), and the angle-difference identity for 15° and 75°[2]. Reciprocals are written with the square root on top: sec 30° = 1/(√3/2) = 2√3/3. At any other angle the values are irrational numbers with no such closed form, so the table gives ten decimal places (15 significant digits for very large values) and says “not a surd”.
Worked examples
θ = 5π/6. That is 150°. The point is a reflection of the 30° point in the y-axis, so cos 150° = −√3/2 and sin 150° = 1/2. Then tan 150° = (1/2)/(−√3/2) = −√3/3, sec 150° = −2√3/3, csc 150° = 2 and cot 150° = −√3.
θ = −30°. Turn 30° clockwise; you land on the same point as 330°, in quadrant IV, with reference angle 30°. Cosine is positive there and sine negative: cos(−30°) = √3/2, sin(−30°) = −1/2, tan(−30°) = −√3/3.
θ = 2 (radians). No π, so this is 2 × 180/π ≈ 114.59°, in quadrant II. sin 2 ≈ 0.9092974268 and cos 2 ≈ −0.4161468365. Most calculators expect degrees unless told otherwise; sin 2° is a very different number (≈ 0.0349).
Precision and range
Decimals are computed in double-precision floating point, which carries about 16 significant digits. They are shown to ten decimal places, or to 15 significant digits when the value is so large that ten places would need more (near a pole, tan 89.9999999° ≈ 572,957,795.130823), and every digit shown is correct to within one unit in the last place. That holds right up to the poles because the angle is first reduced exactly, using the decimal you typed, to the nearest multiple of 90° plus a small remainder, and the functions are computed from that remainder. Exact forms are used only within a billionth of a degree of a multiple of 15°; they are looked up in a table that the site checks against the decimals at every multiple of 15°. The box accepts angles up to a million degrees either way; anything it cannot read gets a message beside the box, and the figure keeps the last angle it could read.
Sources
- (2021). Precalculus 2e (§5.1 Angles, §5.2 Unit circle, §5.3 The other trigonometric functions, §6.1–6.2 Graphs, §6.3 Inverse trigonometric functions, §7.2–7.3 Identities, §8.1–8.2 Non-right triangles, §8.3–8.4 Polar coordinates). OpenStax, Rice University. textbook
- (2026). NIST Digital Library of Mathematical Functions, Chapter 4: Elementary Functions (§4.14 definitions, §4.21 identities). National Institute of Standards and Technology, Release 1.2.8. reference