Cosecant (csc)

Cosecant, written $\csc\theta$, is the reciprocal of sine: $\csc\theta = \frac{1}{\sin\theta}$. One of three reciprocal trig functions (along with $\sec\theta = 1/\cos\theta$ and $\cot\theta = 1/\tan\theta$).

Domain: all $\theta$ where $\sin\theta \neq 0$, i.e. $\theta \neq k\pi$ for integer $k$. Range: $|\csc\theta| \geq 1$.

In a right triangle: $\csc\theta = \frac{\text{hypotenuse}}{\text{opposite}}$.

Pythagorean identity: $1 + \cot^2\theta = \csc^2\theta$. Derivative: $\frac{d}{dx}\csc x = -\csc x \cot x$.

Cosecant appears most often in calculus integrals (especially when integrating powers of sin/cos via substitution). In modern practice, students mostly convert csc back to 1/sin to compute, using $\sin$ directly.