Problem 55 - Precalculus Mathematics for Calculus

Question

(a) Prove that if \(f\) is periodic with period \(p,\) then \(1 / f\) is also periodic with period \(p\) (b) Prove that cosecant and secant each have period \(2 \pi\).

Step-by-Step Solution

Answer

(a) If \( f(x) \) is periodic, \( \frac{1}{f(x)} \) is periodic with the same period. (b) Both \( \csc(x) \) and \( \sec(x) \) are periodic with period \( 2\pi \).

Step 1: Define a Periodic Function

A function \( f(x) \) is periodic with period \( p \) if for every \( x \) in its domain, \( f(x + p) = f(x) \).

Step 2: Prove 1/f is Periodic with Period p

Given \( f(x) \) is periodic with period \( p \), we have \( f(x + p) = f(x) \). We need to prove that \( \frac{1}{f(x)} \) is also periodic with the same period. Consider \( \frac{1}{f(x+p)} = \frac{1}{f(x)} \). Since \( f(x + p) = f(x) \), it follows that \( \frac{1}{f(x+p)} = \frac{1}{f(x)} \). Thus, \( \frac{1}{f} \) is periodic with period \( p \).

Step 3: Define Cosecant and Secant Functions

The cosecant function is defined as \( \csc(x) = \frac{1}{\sin(x)} \) and the secant function as \( \sec(x) = \frac{1}{\cos(x)} \), where \( \sin(x) \) and \( \cos(x) \) are periodic with period \( 2\pi \).

Step 4: Determine Period of Cosecant and Secant

Since \( \sin(x) \) and \( \cos(x) \) are periodic with period \( 2\pi \), it follows from the earlier proof that \( \csc(x) = \frac{1}{\sin(x)} \) and \( \sec(x) = \frac{1}{\cos(x)} \) inherit the same periodicity. Therefore, both \( \csc(x) \) and \( \sec(x) \) have period \( 2\pi \) since \( 1/f(x) \) is periodic with period \( p \) when \( f(x) \) is periodic with period \( p \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is one of the important trigonometric functions. It is defined as the reciprocal of the sine function:

• \( \csc(x) = \frac{1}{\sin(x)} \)

This definition means that for any angle \( x \), the value of cosecant is the inverse of the value of sine. Due to this relationship, wherever \( \sin(x) = 0 \), the cosecant function becomes undefined, because division by zero is not possible.

The sine function \( \sin(x) \) is known to have a periodicity of \( 2\pi \), meaning it repeats its values every \( 2\pi \) interval. Since \( \csc(x) \) is derived directly from \( \sin(x) \), it inherits this periodicity too. Thus:

• \( \csc(x + 2\pi) = \csc(x) \)

This makes the cosecant function periodic with a period of \( 2\pi \). It's crucial to understand this periodic nature to predict the behavior of the function over different intervals and recognize its patterns.

Secant Function

The secant function, represented as \( \sec(x) \), is another key trigonometric function and the complement of cosine. It is defined as the reciprocal of the cosine function:

• \( \sec(x) = \frac{1}{\cos(x)} \)

Like the cosecant function, the secant becomes undefined at points where \( \cos(x) = 0 \), due to the impossibility of dividing by zero.

The cosine function \( \cos(x) \) also completes a full cycle of its values every \( 2\pi \), establishing its periodicity. Thus, the secant function, being based on \( \cos(x) \), also has the same period:

• \( \sec(x + 2\pi) = \sec(x) \)

This means that secant is periodic with a period of \( 2\pi \). Understanding this makes it easier to analyze its behavior over various ranges of \( x \) and helps in confirming its values at specific intervals.

Trigonometric Functions

Trigonometric functions are fundamental elements in mathematics, relating to the angles and sides of triangles. Key functions such as sine, cosine, tangent, cosecant, secant, and cotangent play crucial roles in various fields like physics, engineering, and computer science.

Here is a brief overview of these functions:

• **Sine (\( \sin \))**: The ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine (\( \cos \))**: The ratio of the adjacent side to the hypotenuse.
• **Tangent (\( \tan \))**: The ratio of the opposite side to the adjacent side; \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• **Cosecant (\( \csc \))**: The reciprocal of sine; \( \csc(x) = \frac{1}{\sin(x)} \).
• **Secant (\( \sec \))**: The reciprocal of cosine; \( \sec(x) = \frac{1}{\cos(x)} \).
• **Cotangent (\( \cot \))**: The reciprocal of tangent; \( \cot(x) = \frac{1}{\tan(x)} \).

The periodic nature of these functions often leads to repeating patterns that can simplify the understanding of complex mathematical concepts. For instance, both sine and cosine have a period of \( 2\pi \), implying that their respective reciprocal functions, cosecant and secant, will also share this periodicity. These relationships are invaluable in solving trigonometric equations and modeling real-world phenomena where periodicity is involved.