Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of \(y=\sec x\) can be obtained on a calculator by graphing a translation of the reciprocal of \(y=\sin x\).

Step-by-Step Solution

Answer

The statement given in the exercise is false.

Step 1: Understanding Sec and Sin Function

The secant function, \(y = sec(x)\), is defined as the reciprocal of the cosine function, not the sine function. Hence, \(y = sec(x) = 1 / cos(x)\). This means the sec(x) is undefined when cos(x) = 0. For sine function, \(y = sin(x)\) it's the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse). Therefore, \(y = sin(x)\) is not equivalent to \(y = sec(x)\).

Step 2: Comparing the Graphs of Sec and Sin Function

If we graph the sine function, we notice that it fluctuates between -1 and 1, crossing the x-axis at \(n \pi\), where n is any integer. On the other hand, \(y = sec(x)\) has vertical asymptotes (undefined points) at \(x = n \pi\), where n is an odd integer. Thus, we cannot obtain the graph of \(y = sec(x)\) by translating the graph of \(y = 1 / sin(x)\), as these functions have different undefined points.

Step 3: Final Analysis

By looking at the definitions and the graphs of these two functions, we cannot obtain the graph of sec(x) from the graph of 1/sin(x) by means of translation. The statement provided in the exercise is incorrect because the sec(x) function is the reciprocal of the cosine function, not the sine function.

Key Concepts

Reciprocal Function

The concept of a reciprocal function is fundamental in understanding many trigonometric identities. A reciprocal function is the inverse of another function. If you have a function, say \( f(x) \), its reciprocal is given by \( \frac{1}{f(x)} \). In trigonometry, some functions have well-defined reciprocal relationships. For example:

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

Understanding reciprocal functions helps in graph analysis. If the original function, say \( f(x) \), is zero at any point \( x \), then its reciprocal \( \frac{1}{f(x)} \) will be undefined, creating vertical asymptotes in its graph. This is crucial when analyzing the graph of the secant function because it will have vertical asymptotes at the zeros of \( \cos(x) \).
Hence, the behavior of reciprocal functions directly informs us about asymptotic behavior and domain restrictions.

Cosine Function

The cosine function \( y = \cos(x) \) is one of the fundamental trigonometric functions. Unlike sine and tangent, cosine describes the x-coordinate of a unit circle at an angle \( x \) from the positive x-axis. The key characteristics of the cosine function include:

• It has a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians.
• Its range is from -1 to 1, indicating the maximum and minimum values the function can attain.
• The graph of \( \cos(x) \) is a wave-like pattern that starts at 1 when \( x = 0 \) and crosses the x-axis at odd multiples of \( \frac{\pi}{2} \) (such as \( \frac{\pi}{2}, \frac{3\pi}{2} \), etc.).

The cosine function is particularly important when considering the secant function because the secant function is its reciprocal. Therefore, it's crucial to note that whenever the cosine function is zero, the secant function becomes undefined. This relationship creates the vertical asymptotes seen in the secant graph.
Understanding the cosine function's behavior is vital in anticipating these asymptotic points.

Trigonometric Graphs

Trigonometric graphs visually represent the relationship between angles and their corresponding trigonometric function values over a set period. They help to visualize functions' properties, such as periodicity, amplitude, and phase shifts. Some important points about trigonometric graphs, using the sine, cosine, and secant functions, include:

• The sine and cosine functions are smooth, continuous wave patterns oscillating between -1 and 1, known as their amplitude. They are periodic with a period of \( 2\pi \).
• The secant function, which is the reciprocal of the cosine function, appears as a series of u-shaped branches and has vertical asymptotes where the cosine function is zero.
• The main challenge when graphing secant is noting that it will have breaks or discontinuities (vertical asymptotes) wherever the cosine graph crosses the x-axis. This occurs at \( x = (2n+1)\frac{\pi}{2} \) where \( n \) is an integer.

A clear understanding of these key features is pivotal. It helps in distinguishing how transformations of these graphs, such as stretching or translating, affect their shape and position on a coordinate plane. For students, grasping these can demystify how different trigonometric functions relate to one another and aid in solving complex trigonometric equations and inequalities. This conceptual clarity supports deeper problem-solving skills in trigonometry.