TeXipedia

cosec

Represents the cosecant trigonometric function, defined as the reciprocal of sine.

Overview

Essential in trigonometry and mathematical analysis for expressing reciprocal relationships and solving triangles.

  • Common in physics and engineering calculations, particularly in wave analysis and mechanics.
  • Often used alongside other trigonometric functions in complex mathematical expressions.
  • Alternative notation to csc, though less commonly used in modern mathematical writing.
  • Particularly useful in calculus when dealing with trigonometric derivatives and integrals.

Examples

Defining the cosecant of angle theta in a trigonometric identity.

cosec⁡θ=1sin⁡θ\cosec \theta = \frac{1}{\sin \theta} 
\cosec \theta = \frac{1}{\sin \theta}

Relationship between cosecant and sine in a double angle formula.

cosec⁡(2θ)=cosec⁡θcot⁡θ2\cosec(2\theta) = \frac{\cosec \theta \cot \theta}{2} 
\cosec(2\theta) = \frac{\cosec \theta \cot \theta}{2}

Derivative of cosecant in calculus.

ddxcosec⁡x=−cosec⁡xcot⁡x\frac{d}{dx} \cosec x = -\cosec x \cot x 
\frac{d}{dx} \cosec x = -\cosec x \cot x