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Chapter 4: Problem 21

Sketch the graph of the function. (Include two full periods.) $$y=\csc \pi x$$

Short Answer

Expert verified
The graph of \(y=\csc \pi x\) is a series of 鈥楿鈥� shapes coinciding with peaks of the sine function graph, approaching but never reaching the x-values where the sine function equals zero.

Step by step solution

01

Recall the Definition of Cosecant Function

The cosecant function is the reciprocal of the sine function, that is, \(\csc(\pi x)=1/\sin(\pi x)\). Therefore, it's undefined where \(\sin(\pi x)\) equals 0.
02

Identify the Period of the Function

In this case, the period of the sine function is \(2\pi/(b)\), where \(b\) is the coefficient of x. Here, \(b=\pi\), so the period of the sine function (and hence the cosecant function) is 1. Thus, \(\sin(\pi x)\) and consequently \(\csc(\pi x)\) are undefined at integer values of x.
03

Sketch the Basic Sine Function

To help plot \(\csc(\pi x)\), first sketch the graph of \(y=\sin(\pi x)\) over two periods (from x=-2 to x=2). This sine function oscillates between -1 and 1, crossing the x-axis at integer values.
04

Sketch the Cosecant Function

The graph of the cosecant function is a series of 鈥楿鈥� shaped parts above each peak and below each valley of the sine function, approaching but not reaching the zeros of the sine function. Thus, for each peak of the sine function, there is a downward-opening 'U' from \(-(1/\sin(\pi x))\) and for each valley, there is an upward-opening 'U' from \((1/\sin(\pi x))\). Each 'U' grows closer to an integer x but never reaches it, as \(\csc(\pi x)\) is undefined at these points.

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