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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Verified step by step guidance

1. Identify the original trigonometric function from the given graph. It will be either sine or cosine, as their reciprocal functions are cosecant and secant respectively.
2. Recall that the reciprocal function of sine is cosecant, defined as \(\csc x = \frac{1}{\sin x}\), and the reciprocal function of cosine is secant, defined as \(\sec x = \frac{1}{\cos x}\).
3. Analyze the original graph to find where the function crosses the x-axis (zeros). These points correspond to vertical asymptotes in the reciprocal function's graph because division by zero is undefined.
4. Sketch the reciprocal function by plotting the reciprocal values of the original function's points, noting that the reciprocal function will have vertical asymptotes where the original function is zero, and will approach zero where the original function has large magnitude.
5. Write the equation of the reciprocal function based on the original function identified: if the original is \(y = \sin x\), then the reciprocal is \(y = \csc x\); if the original is \(y = \cos x\), then the reciprocal is \(y = \sec x\).

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal functions like cosecant (csc) and secant (sec) are defined as the reciprocals of sine and cosine, respectively. Specifically, csc(x) = 1/sin(x) and sec(x) = 1/cos(x). Understanding these relationships is essential to transform sine or cosine graphs into their reciprocal counterparts.

Graphing Reciprocal Functions

To graph cosecant or secant, start with the sine or cosine graph and identify points where the original function is zero, as these correspond to vertical asymptotes in the reciprocal graph. The reciprocal graph has branches that approach these asymptotes and reflect the peaks and troughs of the original function inversely.

Equation Identification from Graphs

Determining the equation from a graph involves recognizing the function type, amplitude, period, phase shift, and vertical shift. For reciprocal functions, these parameters come from the original sine or cosine function before taking the reciprocal, allowing you to write the exact equation of the cosecant or secant graph.