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.

Verified step by step guidance

1. Identify the original trigonometric function from the graph, either sine or cosine, as the reciprocal function will be cosecant or secant respectively.
2. Determine the key points and intervals of the original function, such as the zeros, maximum, and minimum points, as these will help in sketching the reciprocal function.
3. For the reciprocal function, note that it will have vertical asymptotes at the zeros of the original function, since division by zero is undefined.
4. Sketch the reciprocal function by plotting points where the original function has maximum and minimum values, as these will correspond to the minimum and maximum values of the reciprocal function.
5. Write the equation of the reciprocal function, either \( y = \csc(x) \) for sine or \( y = \sec(x) \) for cosine, based on the original function identified.

Key Concepts

Reciprocal Functions

Reciprocal functions are derived from basic trigonometric functions by taking the reciprocal of their values. For example, the cosecant function is the reciprocal of the sine function, defined as csc(x) = 1/sin(x), and the secant function is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). Understanding how these functions relate to their original counterparts is crucial for graphing and analyzing their behavior.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the values of the function over a specified interval, typically using key points such as maximums, minimums, and intercepts. The shape of the graph is influenced by the function's periodicity and amplitude. For reciprocal functions like cosecant and secant, the graphs will exhibit asymptotes where the original function equals zero, leading to undefined values in the reciprocal.

Asymptotes in Trigonometric Functions

Asymptotes are lines that the graph of a function approaches but never touches. In the context of reciprocal trigonometric functions, vertical asymptotes occur at the points where the original sine or cosine function equals zero, as these points correspond to undefined values for cosecant and secant. Recognizing the locations of these asymptotes is essential for accurately sketching the graphs of these reciprocal functions.