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Cosecant (csc) - Trigonometry function

In a right triangle, the cosecant of an angle is the length of the hypotenuse divided by the length of the opposite side. In a formula, it is abbreviated to just 'csc'.

Of the six possible trigonometric functions, cosecant, cotangent, and secant, are rarely used. In fact, most calculators have no button for them, and software function libraries do not include them.

They can be easily replaced with derivations of the more common three: sin, cos and tan.
cosecant can be derived as the reciprocal of sine:

The inverse cosecant function - arccsc

For every trigonometry function such as csc, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. So the inverse of csc is arccsc etc. When we see "arccsc A", we interpret it as "the angle whose cosecant is A".

csc 30 = 2.000 Means: The cosecant of 30 degrees is 2.000
arccsc 2.0 = 30 Means: The angle whose cosecant is 2.0 is 30 degrees.

Sometimes written as acsc or csc-1

Angles greater than 90°

In a right triangle, the two variable angles are always less than 90° (See Interior angles of a triangle). But we can in fact find the cosecant of any angle, no matter how large, and also the cosecant of negative angles. For more on this see Functions of large and negative angles.

Graph of the cosecant function

Because the cosecant function is the reciprocal of the sine function, it goes to infinity whenever the sine function is zero.

The derivative of csc(x)

In calculus, the derivative of csc(x) is –csc(x)cot(x). This means that at any value of x, the rate of change or slope of csc(x) is –csc(x)cot(x). For more on this see Derivatives of trigonometric functions together with the derivatives of other trig functions. See also the Calculus Table of Contents.

Pages referring to 'cosecant'

This page explores the derivatives of trigonometric functions in calculus. Interactive calculus applet.
Introduction to the 6 trigonometry functions - sine, cosine, tangent, secant, cosecant, cotangent