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Cosecant

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Algebra and Trigonometry

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 $\sin(\theta) \neq 0$.

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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?
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