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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The cosecant function, $\csc(x)$, has vertical asymptotes at points where $\sin(x) = 0$, specifically at integer multiples of $\pi$.
The range of the cosecant function is $(-\infty, -1] \cup [1, \infty)$.
The period of the cosecant function is $2\pi$, meaning it repeats every $2\pi$ units.
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.
$\csc(x)$ 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?
Related terms
Sine Function: A fundamental trigonometric function defined as $\sin(x)$ which gives the y-coordinate on the unit circle for a given angle x.
Secant Function: The reciprocal of the cosine function, defined as $\sec(x) = \frac{1}{\cos(x)}$ for all values where $\cos(x) \ne 0$.
Cotangent Function: $\cot(x)$ is defined as $\cot(x) = \frac{\cos(x)}{\sin(x)}$ for all values where $\sin(x) \ne 0$, representing another trigonometric ratio.