What does arccosine mean?
Arccosine is the inverse of the trigonometric cosine function. Where cosine takes an angle and returns a ratio between −1 and 1, arccosine performs the reverse operation: given such a ratio, it returns the angle whose cosine produces it. The result is conventionally confined to the principal range of 0 to π radians (0° to 180°), since many angles share the same cosine and a unique answer must be chosen. Mathematicians often abbreviate the function as arccos or cos⁻¹, while programmers encounter it as acos() in standard libraries. It appears routinely in geometry, vector analysis, and signal processing — for example, when recovering an angle from a dot product or determining a phase shift. Though strictly a technical term, arccosine is indispensable wherever angles must be reconstructed from measured ratios, making it a cornerstone of applied mathematics alongside its siblings arcsine and arctangent.
nounThe arccosine is the inverse trigonometric function of the cosine: for a given value between −1 and 1, it returns the angle whose cosine is that value, expressed in radians or degrees.
- The principal-value inverse function of cosine, returning an angle in [0, π] radians (or [0°, 180°]) whose cosine equals the given argument.
- In computing and programming libraries, the function acos() implementing this inverse relationship numerically.
"To recover the angle from its cosine of 0.5, the calculator evaluates arccosine(0.5), yielding 60 degrees."
"The engineer used the arccosine of the dot product to find the angle between the two vectors."
"Since the cosine of the phase difference was 0.866, taking the arccosine gave a phase shift of thirty degrees."
Rarely pluralized; the term almost always refers to the single mathematical function, though plural forms may appear when discussing different branches or conventions.
"Different software libraries implement their arccosines with slightly different handling of edge cases."
Type arccosine(0.5) into any calculator and you get exactly 60° — but ask for arccosine(2) and math itself refuses to answer, because no real angle has a cosine greater than one.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of arccosine
The word arccosine combines 'arc', referring to the arc of a circle that subtends an angle, with 'cosine', itself derived from Latin 'complementi sinus' ('sine of the complement'), a term developed by European mathematicians in the early modern period. Inverse trigonometric functions were formalized during the seventeenth and eighteenth centuries as mathematicians sought to undo the circular functions, and the 'arc-' prefix was adopted across the family — giving arcsine, arccosine, and arctangent — because these functions return the length of an arc, and hence an angle, rather than a ratio. The notation cos⁻¹ arose later with functional notation, though it can mislead readers into thinking a reciprocal is meant.
How arccosine is actually used
Arccosine is a technical term used almost exclusively in mathematics, physics, engineering, and computer science; it is written formally as arccos or cos⁻¹ in textbooks and as acos() in most programming languages. Because cosine is not one-to-one, arccosine is conventionally restricted to its principal range of 0 to π radians, so results outside that interval require adjustment by the user.
Easily confused with arccosine
Cosine maps an angle to a ratio between −1 and 1, whereas arccosine does the reverse, mapping a ratio back to an angle.
Arcsine returns the angle whose sine equals a given value, while arccosine returns the angle whose cosine equals that value.