Arccot Calculator

What arccot is

The arccotangent reverses the cotangent. If cot θ = x, then arccot x = θ. You will also see it written acot(x), arccotan(x) or cot⁻¹(x). Because cot θ is the ratio adjacent ÷ opposite, arccot turns that ratio into an angle.

• Domain: every real number.
• Range used here: 0° to 180° (0 to π radians), not including either end. The result is therefore always positive.
• Behaviour: arccot is a decreasing function. Large positive inputs give angles close to 0°, zero gives 90° and large negative inputs approach 180°.

The cotangent repeats every 180°, so exactly one angle in a half turn corresponds to each value. Picking the half turn from 0° to 180° gives a single, continuous answer for every x, and it is the convention this calculator follows.

Which range is the right one?

Texts disagree about the principal value. Some choose the interval from 0 to π, as here, others use −π/2 to π/2 (leaving out 0). The two agree whenever x is positive. They differ by 180° when x is negative.

arccot x = 90° − arctan x (range 0° to 180°)

Example: for x = −2, arctan(−2) = −63.434949°, so arccot(−2) = 90° + 63.434949° = 153.434949° (2.677945 radians). Under the −90° to 90° convention the same input would give −26.565051°, which is the same direction modulo 180°. If your textbook or software uses the symmetric range, subtract 180° from this calculator's answer whenever the input is negative.

Most handheld calculators have no cot⁻¹ key. For positive x you can type arctan(1/x). For negative x, take arctan(1/x) and add 180° to land in the 0° to 180° range.

How to use the calculator

Enter the cotangent in the Cotangent box, either as a decimal (0.5, -3) or as a fraction with a slash and no surrounding spaces (20/12, -4/3), and press Calculate. Two lines appear, both labelled arccot(x): the angle in degrees and the angle in radians.

Numbers are rounded to six decimals. Whole numbers appear bare (arccot 0 = 90 degrees), one or two decimals get two, three or four decimals are padded to four (0.0010 radians) and anything longer keeps six, as in 63.434949 degrees.

Unlike arcsine, there are no forbidden inputs, but the box cannot be empty. A zero denominator in a fraction is treated as 0, and text that is not a number is read as 0, giving 90 degrees.

Arccot of common values

Worked examples

Sun angle from a shadow. A 12 m pole casts a 20 m shadow. The cotangent of the sun's elevation is shadow ÷ height = 20 ÷ 12 = 1.666667. Enter 20/12 to find arccot = 30.963757° (0.540420 radians). Cross-check with the cotangent calculator: cot 30.963757° returns about 1.666667.

Direct value. If cot θ = 1.5, then θ = arccot 1.5 = 33.690068°, or 0.588003 radian. The check 1 ÷ tan 33.690068° = 1.5 confirms it.

Large input. For big x, arccot x ≈ 1 ÷ x radians. With x = 1000 the estimate is 0.001, and the exact value is 0.000999999667, which the calculator shows as 0.0010 radians (0.057296°).

Related functions and tools

• For positive x, arccot x = arctan(1/x), so the arctangent calculator gives the same angle for the reciprocal.
• Cotangent is the forward function, and the tangent calculator gives its reciprocal.
• Trigonometry calculator finds missing sides and angles of right triangles, and the trigonometry section lists the other trig tools.

Do not read cot⁻¹ as 1 ÷ cot. The reciprocal of the cotangent is the tangent; the superscript −1 in cot⁻¹ means the inverse function.