Problem 27 - Thomas Calculus

Question

a. Graph \(y=\cos x\) and \(y=\sec x\) together for \(-3 \pi / 2 \leq x\) \(\leq 3 \pi / 2\) . Comment on the behavior of sec \(x\) in relation to the signs and values of \(\cos x\) b. Graph \(y=\sin x\) and \(y=\csc x\) together for \(-\pi \leq x \leq 2 \pi\) Comment on the behavior of \(\csc x\) in relation to the signs and values of \(\sin x .\)

Step-by-Step Solution

Answer

\( \sec x \) and \( \csc x \) mirror \( \cos x \) and \( \sin x \), respectively, with reciprocal behavior, and are undefined where the originals are zero.

Step 1: Understanding the Functions

The function \( y = \cos x \) is the cosine function, which oscillates between -1 and 1 with a period of \( 2\pi \). The function \( y = \sec x \) is the secant function, which is the reciprocal of the cosine function. For \( y = \sec x \), it is undefined where \( \cos x = 0 \), which occurs at odd multiples of \( \pi/2 \).

Step 2: Graphing \( y = \cos x \) and \( y = \sec x \)

Graph \( y = \cos x \) from \(-3\pi/2\) to \(3\pi/2\). The cosine curve will have intercepts at \( -3\pi/2, -\pi/2, \pi/2, \) and \( 3\pi/2 \) where \( y = 0 \), peaks at \( -\pi \) and \( \pi \) where \( y = 1 \), and troughs at \( 0 \) where \( y = -1 \). Now, graph \( y = \sec x \), noting that it will have vertical asymptotes at \( -\pi/2, \pi/2, \) and \( 3\pi/2 \) where \( \cos x = 0 \). \( \sec x \) takes positive values when \( \cos x > 0 \), negative values when \( \cos x < 0 \), and \( \sec x \rightarrow \infty \) or \( -\infty \) near asymptotes.

Step 3: Commenting on \( \sec x \) in relation to \( \cos x \)

The secant curve \( y = \sec x \) is defined where the cosine is not zero and extends to infinity at points where \( \cos x = 0 \). When \( \cos x > 0 \), \( \sec x \) is positive and mirrors \( \cos x \) growing towards infinity near \( \cos x = 0 \). Similarly, when \( \cos x < 0 \), \( \sec x \) is negative.

Step 4: Understanding the Sine and Cosecant Functions

The function \( y = \sin x \) is the sine function, which oscillates between -1 and 1 with a period of \( 2\pi \). The function \( y = \csc x \) is the cosecant function, which is the reciprocal of the sine function. For \( y = \csc x \), it is undefined where \( \sin x = 0 \), occurring at multiples of \( \pi \).

Step 5: Graphing \( y = \sin x \) and \( y = \csc x \)

Graph \( y = \sin x \) from \(-\pi\) to \(2\pi\), finding intercepts at \( -\pi, 0, \pi, \) and \( 2\pi \), peaks at \( \pi/2, 5\pi/2, \) and troughs at \( 3\pi/2 \). Now, graph \( y = \csc x \), noting vertical asymptotes at \( -\pi, 0, \pi, \) and \( 2\pi \). \( \csc x \) takes positive values when \( \sin x > 0 \), negative values when \( \sin x < 0 \), and tends towards infinity as it approaches these asymptotes.

Step 6: Commenting on \( \csc x \) in relation to \( \sin x \)

The cosecant curve \( y = \csc x \) is defined where the sine is not zero and heads to infinity at points where \( \sin x = 0 \). \( \csc x \) follows the behavior of \( \sin x \) but inversely extends towards infinity or negative infinity where \( \sin x \) is small and non-zero.

Key Concepts

Cosine Function

The cosine function, represented by the equation \( y = \cos x \), is a fundamental trigonometric function that exhibits periodic behavior. It oscillates between -1 and 1 as \( x \) varies. This predictable wave-like structure has a period of \( 2\pi \), meaning it completes one full cycle every \( 2\pi \) units of \( x \).

Some key points to understand about the cosine graph are:

• The maximum value occurs at \( \cos(0), \cos(2\pi), \cos(-2\pi), ...\) where \( y = 1 \).
• The minimum value of -1 is reached at \( x = \pi, 3\pi, -\pi, -3\pi, ... \).
• Critical points, where \( \cos x = 0 \), appear at odd multiples of \( \pi/2 \), such as \( \pm\pi/2, \pm3\pi/2, ... \).

This sinusoidal shape is smoothly continuous and is symmetrical around the y-axis (even function). Understanding the cosine function is essential for comprehending its reciprocal counterpart, the secant function.

Secant Function

The secant function \( y = \sec x \) is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). It shares some traits with cosine but diverges at points where the cosine equals zero. At those points, the secant function becomes undefined, creating vertical asymptotes in its graph. These asymptotes occur at odd multiples of \( \pi/2 \), just where \( \cos x = 0 \).

Important properties of the secant function include:

• When \( \cos x > 0 \) (for example, from \(-\pi\) to \(0\)), \( \sec x \) is positive.
• It takes negative values wherever \( \cos x < 0 \) (such as from \( \pi \) to \( 2\pi \)).
• \( \sec x \) approaches infinity as it gets closer to \( \cos x = 0 \).

Since \( \sec x \) is not defined at the zeros of \( \cos x \), this creates gaps in the graph. Observation of this behavior can help students grasp the interplay between a function and its reciprocal.

Sine Function

The sine function \( y = \sin x \) is another primary trigonometric function. Like the cosine function, it oscillates between -1 and 1. However, its graph initiates at the origin and has a period of \( 2\pi \), completing a wave every \( 2\pi \) units of \( x \).

Notable features of the sine wave include:

• The function hits its peak value of 1 at \( \pi/2, 5\pi/2, ... \).
• It reaches its lowest value of -1 at \( 3\pi/2, 7\pi/2, ... \).
• \( \sin x = 0 \) at all integer multiples of \( \pi \), such as \( 0, \pi, 2\pi, ...\).

The sine function is an odd function, reflected as symmetrical around the origin. This quality distinguishes it and provides a basis for understanding its reciprocal, the cosecant function.

Cosecant Function

The cosecant function \( y = \csc x \) is defined as the reciprocal of the sine function: \( \csc x = \frac{1}{\sin x} \). This function is significant in trigonometry due to its relationship with sine. It's undefined wherever \( \sin x = 0 \), resulting in vertical asymptotes at those points: \( 0, \pi, 2\pi, ... \).

Key aspects of the cosecant graph include:

• \( \csc x \) attains positive values where \( \sin x > 0 \), for instance, from \( 0 \) to \( \pi \).
• It takes negative values when \( \sin x < 0 \), such as from \( \pi \) to \( 2\pi \).
• \( \csc x \) trends towards either infinity or negative infinity as \( \sin x \) approaches zero.

Understanding cosecant and secant functions helps students deeply explore how functions behave under reciprocal transformations, particularly near zero-crossings, where their original function is close to zero.