Cosecant
TrigonometryCosecant 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. Instead of opposite over hypotenuse, it is hypotenuse over opposite.
Example
If $\sin(30^\circ) = 1/2$, then $\csc(30^\circ) = 1 / (1/2) = 2$.
Key Insight
Cosecant is always $1$ or bigger (or $-1$ or smaller) because you are dividing by the shorter side. It can never be between $-1$ and $1$.
Definition
$\csc(\theta) = 1/\sin(\theta) = \text{hypotenuse}/\text{opposite}$. It is undefined when $\sin(\theta) = 0$ (at $0^\circ$, $180^\circ$, $360^\circ$). Because $|\sin(\theta)| \le 1$, we have $|\csc(\theta)| \ge 1$ for all defined values.
Example
$\csc(45^\circ) = 1 / \sin(45^\circ) = 1 / (\sqrt{2}/2) = 2/\sqrt{2} = \sqrt{2} \approx 1.414$.
Key Insight
The graph of cosecant has U-shaped curves between vertical asymptotes at every multiple of $\pi$. Where sine reaches its maximum of $1$, cosecant reaches its minimum of $1$, and they touch at those points.
Definition
$\csc(x) = 1/\sin(x)$ has period $2\pi$, poles at every integer multiple of $\pi$, and the Laurent expansion $\csc(x) = 1/x + x/6 + 7x^3/360 + \ldots$ near $x = 0$. The Weierstrass product for sin gives: $\sin(x) = x \prod (1 - x^2/(n^2\pi^2))$ over $n \ge 1$, implying the partial fraction expansion of csc.
Example
The integral of $\csc(x) = -\ln|\csc(x) + \cot(x)| + C$. This result is used in integrating rational expressions involving $\sqrt{1 - x^2}$.
Key Insight
The Mittag-Leffler expansion of $\csc(\pi x) = 1/(\pi x) + \sum (2x(-1)^n / (\pi(x^2 - n^2)))$ connects cosecant to complex analysis and the theory of partial fractions over poles.