Problem 54 - Precalculus: Functions and Graphs

Question

Find an equation using the cosecant function that has the same graph as \(y=\sec x\)

Step-by-Step Solution

Answer

The equivalent equation using the cosecant function is \( y = -\csc(x - \frac{\pi}{2}) \).

Step 1: Recall the Definitions

The secant function is defined as the reciprocal of the cosine function, i.e., \( y = \sec x = \frac{1}{\cos x} \). The cosecant function is defined as the reciprocal of the sine function, i.e., \( y = \csc x = \frac{1}{\sin x} \).

Step 2: Understand the Goal

We need to find a cosecant equation that results in the same graph as \( y = \sec x \). This means we need to find a function involving \( \csc \) that behaves the same way as \( \sec x \).

Step 3: Identify Key Characteristics

The graph of \( y = \sec x \) has vertical asymptotes where \( \cos x = 0 \). Similarly, the graph of \( y = -\csc x \) has vertical asymptotes where \( \sin x = 0 \). A negative sign in \( \csc x \) can bring the graph to match the appearance of \( \sec x \).

Step 4: Translate Using Symmetry

Since \( \cos x \) and \( -\sin(x - \frac{\pi}{2}) \) share similar properties, we can express \( y = \sec x \) in terms of \( \csc \):\[ y = -\csc(x - \frac{\pi}{2}) \]This manipulates the \( \csc \) function to match the phase and shape of the \( \sec \) function.

Step 5: Confirm with a Graph

To ensure accuracy, compare the graphs of \( y = \sec x \) and \( y = -\csc(x - \frac{\pi}{2}) \). Both should have identical period, range, and amplitude properties, confirming they are equivalent.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in relation to angles and periodic phenomena. They include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), among others. These functions are defined based on a right-angled triangle or a unit circle, where the sine and cosine functions play a central role.

• The sine function gives the ratio of the length of the side opposite the angle to the hypotenuse.
• The cosine function provides the ratio of the length of the adjacent side to the hypotenuse.

These definitions extend beyond right-angled triangles to model waves, oscillations, and circular motion in physics and engineering. The secant function (\( \sec x \)) is one of these, describing the reciprocal of the cosine function (\( \cos x \)). Therefore, \( \sec x = \frac{1}{\cos x} \), allowing us to understand relationships in circles and oscillatory patterns.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric graph along the x-axis. Every periodic function can be shifted left or right, altering its initial placement without changing its shape or period. This concept is crucial when modeling real-world periodic processes, like tides or sound waves.To achieve a phase shift, a constant is added to the variable within the function’s argument. For instance, a phase shift in the function \( \csc(x - \frac{\pi}{2}) \) results in moving the starting position of the graph of \( \csc x \).

• A positive phase shift results in the graph moving to the right.
• A negative phase shift results in the graph moving to the left.

This transformation is evident in the given exercise where the secant function is transformed into a negative cosecant function with a phase shift, specified as\( y = -\csc(x - \frac{\pi}{2}) \). This shift mimics the behavior seen in\( \sec x \), making the graphs identical in behavior.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the primary trigonometric functions - sine, cosine, and tangent. These include the cosecant (\( \csc x \)), secant (\( \sec x \)), and cotangent (\( \cot x \)) functions. Reciprocal functions embody the idea of taking the inverse of these base functions.

• The cosecant function is the reciprocal of the sine function: \( \csc x = \frac{1}{\sin x} \).
• The secant function is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \).
• The cotangent function is the reciprocal of the tangent function: \( \cot x = \frac{1}{\tan x} \).

These functions are critical in trigonometry since they provide alternative ways to solve equations and understand angles in different quadrants of the unit circle. In the provided exercise, transitioning from the secant function to the cosecant function involves understanding these reciprocal properties, highlighting their mathematical flexibility and application.