Csc Calculator

What is the Cosecant (csc) Function?

The cosecant function, abbreviated as $\csc(\theta)$ or $\operatorname{cosec}(\theta)$, is one of the six fundamental trigonometric functions. In trigonometry, cosecant is defined as the reciprocal of the sine function:

In a right-angled triangle, the cosecant of an acute angle $\theta$ is the ratio of the length of the hypotenuse to the length of the opposite side:

Domain, Range, and Asymptotes

Because $\csc(\theta) = \frac{1}{\sin(\theta)}$, the cosecant function is undefined whenever $\sin(\theta) = 0$. This occurs at integer multiples of $\pi$ radians ($180^\circ$), resulting in vertical asymptotes:

• Domain: All real numbers except $\theta = k\pi$ (or $\theta = k \times 180^\circ$) for any integer $k$.
• Range: $(-\infty, -1] \cup [1, \infty)$ (the absolute value of cosecant is always greater than or equal to 1).
• Period: $2\pi$ radians ($360^\circ$).
• Parity: Odd function, satisfying $\csc(-\theta) = -\csc(\theta)$.

Exact Values for Common Angles

Inverse Cosecant (arccsc)

The inverse cosecant function, denoted as $\operatorname{arccsc}(x)$ or $\csc^{-1}(x)$, returns the angle $\theta$ whose cosecant is $x$. It is mathematically defined through the inverse sine function: