Question

In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

Accepted Answer

Identify the original trigonometric function from the given graph, which will be either sine or cosine, since their reciprocals are cosecant and secant respectively. Recall that the reciprocal function of sine is cosecant, given by $$y = \csc x = \frac{1}{\sin x}$$, and the reciprocal function of cosine is secant, given by $$y = \sec x = \frac{1}{\cos x}. $$Locate the points on the original graph where the function crosses the x-axis (i.e., where sine or cosine equals zero). These points will correspond to vertical asymptotes in the reciprocal function's graph because division by zero is undefined. For each point on the original graph where the function has a maximum or minimum (peaks and troughs), the reciprocal function will have corresponding minimum or maximum values, since the reciprocal of a number greater than 1 is between 0 and 1, and the reciprocal of a number between 0 and 1 is greater than 1. Sketch the reciprocal function by drawing vertical asymptotes at the zeros of the original function and plotting the reciprocal values of the original function's points, then write the equation of the reciprocal function as either $$y = \csc x or$$ $$y = \sec x$$ depending on the original function.

In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

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Key Concepts

Reciprocal Trigonometric Functions

Graphing Reciprocal Functions

Equation Identification from Graphs