ConverterHub

Arccosine Calculator

Get the angle from its cosine.

How it is calculated

You enter a cosine value v, a unitless ratio. The tool returns θ = arccos(v), the single angle between 0° and 180° whose cosine equals v. It is computed in radians and shown to 6 decimal places, then converted with θdeg = θrad × 180 ÷ π and shown to 4 decimal places. Cosine never leaves the range −1 to 1, so a value such as 1.01 or −3, and an empty box, give no result. The ends behave as expected: v = 1 gives 0°, v = 0 gives 90° and v = −1 gives 180°. A positive v always gives an acute angle and a negative v an obtuse one. Unlike arcsine, the result is never negative.

Worked example

Two vectors have a normalized dot product of −0.5, meaning the dot product divided by the product of their lengths. The angle between them is arccos(−0.5). The tool returns 2.094395 radians, and 2.094395 × 180 ÷ π gives 120.0000 degrees. The negative cosine already tells you the angle is obtuse, and 120° is greater than 90°. Compare arccos(0.5), which gives the acute 60°.

Try this example →

Common mistakes

  • Entering the angle where the cosine value belongs. The box wants a ratio from −1 to 1, not 60.
  • Dropping the minus sign. arccos(0.5) is 60° but arccos(−0.5) is 120°; the sign decides whether the angle is acute or obtuse.
  • Forgetting to normalize a dot product. For vectors you must divide by both lengths first; the raw dot product is usually larger than 1 and will be refused.
  • Expecting a negative angle like arcsine gives. The output here always runs from 0° to 180°.
  • Reading the radians line as degrees. 1.047198 is 60° expressed in radians, not about one degree.

About this calculator

The arccosine function, written cos⁻¹(x) or arccos(x), is the inverse of cosine: given a ratio between −1 and 1, it returns the angle (in the range 0° to 180°, or 0 to π radians) whose cosine produces that ratio. Because cosine itself repeats every 360°, arccos is defined only on this restricted range so that each input has exactly one output — this is the "principal value" convention used throughout mathematics and engineering.

Arccos shows up constantly wherever an angle needs to be recovered from a dot product or a ratio of sides. In physics and 3D graphics, the angle between two vectors is found by taking arccos of their normalized dot product; in surveying and navigation, it appears in bearing and triangulation formulas; in mechanical engineering, it is used to solve for joint angles in linkages and robotic arms from known link lengths. It is also the natural complement to arcsine — the two are related by arccos(x) = 90° − arcsin(x).

This calculator takes a cosine value between −1 and 1 and returns the corresponding angle in both degrees and radians, handling the domain restriction automatically so you don't have to consult a unit circle by hand.

Was this helpful?

You can pick up to 3 reactions.

Comments (0)

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators