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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Identify the original trigonometric function from the given graph, 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 equals zero, because these points correspond to vertical asymptotes in the reciprocal function's graph (since division by zero is undefined).

For each point on the original graph where the function has a maximum or minimum (peaks and troughs), plot corresponding points on the reciprocal graph by taking the reciprocal of the y-values (i.e., \(y_{reciprocal} = \frac{1}{y_{original}}\)).

Sketch the reciprocal function by drawing curves that approach the vertical asymptotes identified earlier and pass through the reciprocal points, ensuring the shape reflects the behavior of \(\csc x\) or \(\sec x\). Finally, write the equation of the reciprocal function based on the original function identified.

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주요 개념

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Reciprocal Trigonometric Functions

Reciprocal functions are derived by taking the reciprocal of basic trigonometric functions. Specifically, cosecant (csc) is the reciprocal of sine (sin), and secant (sec) is the reciprocal of cosine (cos). Understanding these relationships helps in transforming graphs of sine and cosine into their reciprocal counterparts.

Graphing Reciprocal Functions

To graph cosecant or secant, start with the sine or cosine graph, then plot points where the original function is nonzero by taking their reciprocals. Vertical asymptotes appear where the original function equals zero, since division by zero is undefined. Recognizing these asymptotes and the shape of the graph is essential.

Equation Identification from Graphs

Determining the equation from a graph involves analyzing key features such as amplitude, period, phase shift, and vertical shifts. For reciprocal functions, these features correspond to those of the original sine or cosine function but reflected in the reciprocal form. Accurate interpretation of these parameters allows writing the correct function equation.