What does cosecant mean?
Cosecant is one of the six fundamental trigonometric functions, defined as the reciprocal of the sine: for any angle θ, csc θ = 1/sin θ. In a right triangle, this means the ratio of the hypotenuse to the side opposite the angle. Its graph is a series of U-shaped curves with vertical asymptotes wherever sine crosses zero, making it undefined at multiples of 180 degrees. The word comes from New Latin 'cosecans', built on the same root as 'secant' ('cutting'), reflecting how these ratios were historically represented by lines cutting through a circle. Though less commonly used than sine and cosine — most expressions can be rewritten without it — the cosecant remains standard notation in trigonometry, calculus, and physics curricula, typically abbreviated csc. It is chiefly encountered in academic and technical contexts rather than everyday language.
nounIn trigonometry, the reciprocal of the sine: for an acute angle in a right triangle, the ratio of the hypotenuse to the side opposite the angle, written csc θ = 1/sin θ.
- A trigonometric function equal to the reciprocal of sine: csc θ = 1/sin θ.
"To evaluate the cosecant of thirty degrees, take the reciprocal of its sine, one half, giving two."
"The cosecant function has vertical asymptotes wherever the sine equals zero."
"Since sin 30° = 0.5, the cosecant of 30 degrees is 2."
Standard plural, used when referring to multiple values of the function or several instances in discussion.
"The table lists the cosecants of all angles in fifteen-degree increments."
Cosecant was born from Latin 'secans' meaning 'to cut' — because a line cutting through a circle is exactly what these trig ratios were first drawn as.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of cosecant
Cosecant derives from New Latin 'cosecans', formed from the prefix 'co-' plus 'secans', present participle of Latin 'secare' ('to cut'). It entered English mathematical usage in the late sixteenth or early seventeenth century alongside related trigonometric terms such as cosine and cotangent, which were developed to simplify calculations in astronomy and navigation. The underlying root also gives us 'secant' itself and ultimately words like 'section' and 'dissect' from the same Latin verb.
Related word forms
How cosecant is actually used
Primarily used in mathematics education and technical writing; abbreviated csc. In modern practice it appears far less often than sine and cosine, since any expression involving it can be rewritten using 1/sin θ. Plural is cosecants.
Easily confused with cosecant
The secant is the reciprocal of the cosine (1/cos θ), while the cosecant is the reciprocal of the sine (1/sin θ); they are distinct trigonometric functions despite their similar names.
Cosine is a primary trigonometric ratio of adjacent over hypotenuse, whereas the cosecant is the reciprocal of sine — the shared prefix 'co-' does not make them the same function.