In Exercises 29–44, graph two periods of the given cosecant or secant function. y = csc(x − π)

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Recall that the cosecant function is the reciprocal of the sine function, so \(y = \csc(x - \pi)\) can be written as \(y = \frac{1}{\sin(x - \pi)}\).

Identify the period of the basic sine function, which is \(2\pi\). Since the argument of the sine function is \((x - \pi)\), the period remains \(2\pi\) because horizontal shifts do not affect the period.

Determine the interval for two periods of the function. Since one period is \(2\pi\), two periods correspond to an interval of length \(4\pi\). For example, you can choose to graph from \(x = \pi\) to \(x = 5\pi\) to cover two full periods starting from the phase shift.

Find the vertical asymptotes of the cosecant function, which occur where the sine function is zero. Solve \(\sin(x - \pi) = 0\) to find these points: \(x - \pi = k\pi\), so \(x = \pi + k\pi\) for all integers \(k\). These asymptotes will help you sketch the graph accurately.

Plot the key points of the sine function shifted by \(\pi\), then take the reciprocal to get the cosecant values. Sketch the graph between the asymptotes, showing the characteristic 'U' and inverted 'U' shapes of the cosecant function over the two periods.

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주요 개념

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Understanding the Cosecant Function

The cosecant function, csc(x), is the reciprocal of the sine function, defined as 1/sin(x). It is undefined where sin(x) equals zero, leading to vertical asymptotes. Recognizing its periodicity and behavior near these asymptotes is essential for graphing.

Phase Shift in Trigonometric Functions

A phase shift occurs when the input variable x is replaced by (x − c), shifting the graph horizontally by c units. For y = csc(x − π), the graph shifts π units to the right, affecting the location of zeros, asymptotes, and peaks.

Period of the Cosecant Function

The period of the basic cosecant function is 2π, meaning the pattern repeats every 2π units. Graphing two periods involves plotting the function over an interval of length 4π, ensuring all key features like asymptotes and peaks are accurately represented.