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Cosecant function

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Algebra and Trigonometry

Definition

The cosecant function is the reciprocal of the sine function. It is defined as $\csc(x) = \frac{1}{\sin(x)}$ for all values of $x$ where $\sin(x) \ne 0$.

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5 Must Know Facts For Your Next Test

  1. The cosecant function, $\csc(x)$, has vertical asymptotes at points where $\sin(x) = 0$, specifically at integer multiples of $\pi$.
  2. The range of the cosecant function is $(-\infty, -1] \cup [1, \infty)$.
  3. The period of the cosecant function is $2\pi$, meaning it repeats every $2\pi$ units.
  4. The graph of the cosecant function consists of a series of curves that approach but never touch its vertical asymptotes and have local minima and maxima corresponding to the zeros of the sine function.
  5. $\csc(x)$ is an odd function, which means $\csc(-x) = -\csc(x)$.

Review Questions

  • What is the relationship between the sine and cosecant functions?
  • At what points does the graph of the cosecant function have vertical asymptotes?
  • What is the period of the cosecant function?

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