Problem 42 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=\csc \left(x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The graph of \(y = \csc(x-\pi/2)\) has peaks and valleys at every maximum and minimum point of the related sine function and has vertical asymptotes wherever the sine function is zero. The curve repeats every \(2\pi\), with asymptotes at \(x = 0, \pi, 2\pi\) and \(3\pi\).

Step 1: Identify the Related Sine Function

The cosecant function is the reciprocal of the sine function. Thus, the graph of \(y=\csc \left(x-\frac{\pi}{2}\right)\) can be obtained by graphing the related sine function \(y=\sin \left(x-\frac{\pi}{2}\right)\). The sine function has a period of \(2\pi\), therefore it repeats its values every \(2\pi\). The graph for two periods of this function will span the interval \([-\pi, 3\pi]\).

Step 2: Determine the Critical Points of the Sine Function

The critical points of the sine function are at its maximum, minimum, and zero crossing points. For \(y=\sin(x-\frac{\pi}{2})\), these points occur every \(\pi/2\). So, for two periods spanning the interval \([-\pi, 3\pi]\), the critical points will be at \(x = -\pi, -\pi/2, 0, \pi/2, \pi, 3\pi/2, 2\pi, 5\pi/2\) and \(3\pi\). The corresponding \(y\) values can be computed as \(\sin(-\pi), \sin(-\pi/2), \sin(0), \sin(\pi/2), \sin(\pi), \sin(3\pi/2), \sin(2\pi), \sin(5\pi/2)\) and \(\sin(3\pi)\) respectively.

Step 3: Draw the Graph of the Sine Function

Plot the critical points from step 2 on the graph and draw the period of the sine curve by smoothly connecting the points.

Step 4: Draw the Graph of the Cosecant Function

Now, draw the graph of the cosecant function using the sine graph as a guide. The cosecant function has asymptotes wherever the related sine function is 0, and the value of the cosecant function is the same as the sine function at their maximum and minimum points. Thus, plot the asymptotes at \(x = 0, \pi, 2\pi\) and \(3\pi\) and use the maximum and minimum points from the sine graph to draw the curve of the cosecant function. The curve will be a series of ‘U’ shapes that touch the maximum and minimum points and approach the asymptotes.