Cosecant
Cosecant is the reciprocal of sine, defined as the ratio of the hypotenuse to the opposite side in a right triangle.
Formula
\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}
Definition
Cosecant (written "csc") is the flip of sine: $\csc(\theta) = 1/\sin(\theta) = \text{hypotenuse}/\text{opposite}$. It is undefined when $\sin(\theta) = 0$ (at $0^\circ$, $180^\circ$, $360^\circ$), and because $|\sin(\theta)| \le 1$, $|\csc(\theta)| \ge 1$ for all defined values, so it can never fall between $-1$ and $1$. Its graph has $2\pi$-periodic poles at every integer multiple of $\pi$, with Laurent expansion $\csc(x) = 1/x + x/6 + 7x^3/360 + \ldots$ near $x = 0$.
Example
If $\sin(30^\circ) = 1/2$, then $\csc(30^\circ) = 1/(1/2) = 2$; and $\csc(45^\circ) = 1/\sin(45^\circ) = 2/\sqrt{2} = \sqrt{2} \approx 1.414$. The integral of $\csc(x) = -\ln|\csc(x) + \cot(x)| + C$, a result used in integrating rational expressions involving $\sqrt{1 - x^2}$.
Key Insight
The graph of cosecant forms U-shaped curves between vertical asymptotes; where sine reaches its maximum of $1$, cosecant reaches its minimum of $1$, and they touch at those points. The Weierstrass product $\sin(x) = x \prod (1 - x^2/(n^2\pi^2))$ over $n \ge 1$ implies a partial fraction (Mittag-Leffler) expansion for cosecant, $\csc(\pi x) = 1/(\pi x) + \sum (2x(-1)^n / (\pi(x^2 - n^2)))$, connecting cosecant to complex analysis and the theory of partial fractions over poles.