Cosecant

Definition

The cosecant function, denoted as $\csc(\theta)$, is the reciprocal of the sine function. It is defined as $\csc(\theta) = \frac{1}{\sin(\theta)}$ where $\sin(\theta) \neq 0$.

5 Must Know Facts For Your Next Test

1. Cosecant is undefined for angles where the sine value is zero, which occurs at integer multiples of $\pi$ (0°, 180°, 360°, etc.).
2. The period of the cosecant function is $2\pi$, similar to the sine and cosine functions.
3. The graph of the cosecant function has vertical asymptotes where $\sin(\theta) = 0$.
4. The range of the cosecant function is $(-\infty, -1] \cup [1, \infty)$ because it is the reciprocal of sine, which ranges between -1 and 1.
5. Cosecant is an odd function, meaning that $\csc(-x) = -\csc(x)$.

Review Questions

• What is the definition of cosecant in terms of sine?
• For which values of $\theta$ is the cosecant function undefined?
• What are the characteristics of the graph of a cosecant function?

Related terms

Sine: A trigonometric function representing the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

Secant: The reciprocal of cosine; defined as $\sec(\theta) = \frac{1}{\cos(\theta)}$.

Cotangent: The reciprocal of tangent; defined as $\cot(\theta) = \frac{1}{\tan(\theta)}$ or equivalently as $ \cot(θ)= \frac{cos (θ)} {sin (θ)}.$