Problem 53 - Precalculus

Question

How does the period of \(y=\csc x\) compare with the period of \(y=\sin x ?\)

Step-by-Step Solution

Answer

The periods of both \( y = \sin x \) and \( y = \csc x \) are the same: \( 2\pi \).

Step 1: Understanding the Functions

The function \( y = \sin x \) is the basic sine function, while \( y = \csc x \) is its reciprocal, given by \( \csc x = \frac{1}{\sin x} \). It's important to recognize these functions to understand their periodic nature.

Step 2: Period of \( y = \sin x \)

The sine function, \( y = \sin x \), is periodic with a period of \( 2\pi \). This means that the waveform repeats itself every \( 2\pi \) units along the x-axis.

Step 3: Determining the Period of \( y = \csc x \)

Since \( \csc x = \frac{1}{\sin x} \), it will share the same periodicity as \( y = \sin x \) because the repetition of \( \sin x \) after each \( 2\pi \) will cause \( \csc x \) to also repeat. Therefore, the period of \( y = \csc x \) is \( 2\pi \).

Step 4: Compare the Periods

Both \( y = \sin x \) and \( y = \csc x \) have the same period of \( 2\pi \). Thus, there is no difference in the period between these functions.

Key Concepts

Sine Function

The sine function, denoted as \( y = \sin x \), is a fundamental trigonometric function that maps any real number to the interval \([-1, 1]\). It originates from the ratio in a right triangle, opposite side over hypotenuse.

Key characteristics include:

• Waveform Shape: It creates a smooth, wave-like pattern typical of trigonometric functions.
• Cycle Repeats: The sine function is periodic, completing one full cycle every \(2\pi\) along the x-axis, which means its values repeat regularly.
• Zeros: It crosses the x-axis at integer multiples of \(\pi\), such as \(0, \pi, 2\pi, \) etc.
• Amplitude: The maximum and minimum values are 1 and -1, respectively.

Understanding the sine function's periodic behavior helps when analyzing its reciprocal and comparing the sine with other trigonometric functions.

Cosecant Function

The cosecant function is the reciprocal of the sine function, expressed as \( y = \csc x = \frac{1}{\sin x} \). This means wherever the sine function exists, the cosecant value is derived by taking the reciprocal of that sine value.

• Domain: It is not defined when \(\sin x = 0\), such as when \(x\) is an integer multiple of \(\pi\), because division by zero is undefined.
• Range: It extends to all real numbers except between -1 and 1, as these are excluded by the reciprocal nature.
• Graph Characteristics: Consists of alternating upward and downward curves corresponding to the peaks and troughs of the sine function with vertical asymptotes at each zero of the sine function.

When comparing cosecant to sine, it's pertinent to analyze how these graphs complement each other, with cosecant mirroring the peaks and troughs of the sine function by shooting towards infinity around the zeros of the sine.

Periodicity

Periodicity refers to the repeating nature of a function across its domain. Both the sine and cosecant functions exhibit this behavior.

• Sine Periodicity: As stated, the sine function has a fixed period of \(2\pi\), meaning the pattern repeats every \(2\pi\) units.
• Cosecant Periodicity: The cosecant function also possesses the same periodicity of \(2\pi\). This is because its behavior is directly dependent on the sine function.

Therefore, every characteristic of the sine function repeats after this interval, causing the cosecant function to also repeat at the same interval because it's the sine's reciprocal. Understanding periodicity in these functions assists in predicting values and behavior over extended x-values.