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Chapter 4: Problem 26

Sketch the graph of the function. (Include two full periods.) $$y=\csc \frac{x}{3}$$

Short Answer

Expert verified
The graph of the function \(y = \csc \frac{x}{3}\) is derived from its corresponding \(\sin\) function. It includes vertical asymptotes at \(x = n(3\pi)\) where \(n\) is an integer, and has maxima and minima derived from the points where the \(\sin\) function reaches its peaks. The graph periodically goes from a minimum trough, reaches the maximum peak, and then descends back to the next minimum trough.

Step by step solution

01

Define the basic shape

Start with a basic sine function \( y = \sin x \). It follows the standard sine wave shape, which goes between -1 and 1 with a period of \(2\pi\). However, because of the division by 3, this period will be scaled by a factor of 3 and now be \(6\pi\). There will be three critical points during a period: at 0, at midpoint of the period, and at the end of the period.
02

Identify the asymptotes

The \(\csc\) function will have vertical asymptotes where the corresponding \(\sin\) function is 0. For the function \(y = \sin \frac{x}{3}\), it is zero at multiples of \(3\pi\). So, for the function \(y = \csc \frac{x}{3}\), there will be vertical asymptotes at multiples of \(3\pi\).
03

Identify the maxima and minima

The \(\csc\) function will have maxima and minima at points where \(\sin\) function reaches its peaks. In addition, at these points, the \(\csc\) function is negative where \(\sin\) function is negative and the \(\csc\) function is positive where \(\sin\) function is positive. The maxima occurs when \(\sin x = 1\), which for \(\sin \frac{x}{3}\) will happen when \(\frac{x}{3} = \frac{\pi}{2} + n\pi\) where \(n\) is an integer. For similarly finding minima, solve the equation \(\frac{x}{3} = \frac{3\pi}{2} + n\pi\).
04

Sketch the graph

Sketch the \(\csc\) function between the vertical asymptotes. During the period, the graph will go from an a trough at the left asymptote, rise to a maximum, then descend to a trough at the right asymptote for regions where \(\sin\) function is positive and vice-versa for regions where \(\sin\) function is negative.

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