/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 What is a phase shift? Give an e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 3

What is a phase shift? Give an example of a sine function that has a phase shift.

Short Answer

Expert verified
A phase shift refers to the horizontal shift that occurs when a trigonometric function is translated left or right. An example of a sine function with a phase shift is \(y = \sin(x - \pi /2)\), which has a phase shift of \(\pi /2\), shifting the function to the right by \(\pi /2\) units.

Step by step solution

01

Definition of Phase Shift

Phase shift refers to the horizontal shift of a wave, in this case, a trigonometric function, either sine or cosine, that occurs when the function is translated left or right. This shifts the starting point of the periodic function.
02

Basic Form of a Sine Function

The standard form of a sine function is \(y = A \sin(B(x - C)) + D\). In this equation, A denotes the amplitude, B affects the period, C is the phase shift, and D is the vertical shift.
03

Example and Calculation

An example of a sine function with a phase shift could be \(y = \sin(x - \pi /2)\). In this case, the phase shift is \(\pi /2\), which means that the graph of the sine function will be shifted to the right by \(\pi /2\) units compared to the basic sine function \(y = \sin(x)\). So instead of beginning the cycle at \(x = 0\), it would start at \(x = \pi /2\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function
The sine function is one of the fundamental trigonometric functions. It is represented mathematically as \(y = \sin(x)\), which describes waves or oscillations. The standard sine wave starts at the origin, moves up to its maximum value at \(\frac{\pi}{2}\), returns to zero, goes to its minimum at \(\frac{3\pi}{2}\), and completes the cycle at \(2\pi\).
It is often used to model periodic phenomena, such as sound waves or tides.
The function repeats its pattern over a fixed interval, which is crucial for understanding waves in various physical contexts.
Understanding the sine function's behavior is essential in studying how shifts and changes in amplitude affect the graph.
Trigonometric Functions
Trigonometric functions include sine, cosine, and tangent, among others. These functions are essential in mathematics, especially in the study of angles and periodic phenomena.
Each function has its unique properties and uses, but all relate to the angles of a right triangle or the unit circle.
For example, in a right triangle:
  • The sine function represents the ratio of the opposite side to the hypotenuse.
  • Cosine is the adjacent side over the hypotenuse.
  • Tangent is the ratio of the opposite side to the adjacent side.
Beyond triangles, these functions help describe waves and oscillations. Understanding them is vital for analyzing any periodic phenomena.
Horizontal Shift
A horizontal shift in a trigonometric function, often referred to as a phase shift, translates the graph left or right.
In the function \(y = A \sin(B(x - C)) + D\), the term \(C\) represents this shift.
When \(C > 0\), the graph shifts to the right. If \(C < 0\), it shifts to the left. This adjustment modifies where the wave begins its cycle.
_Example:_
  • Consider the function \(y = \sin(x - \pi/2)\). The graph is shifted to the right by \(\pi/2\) units compared to \(y = \sin(x)\).
  • Contrast this with \(y = \sin(x + \pi/2)\), where the graph shifts \(\pi/2\) units to the left.
Mastering how to handle phase shifts is crucial for precisely tuning models to match real-world data.
Periodic Function
Periodic functions are mathematical functions that repeat their values in regular intervals, or periods.
A well-known example is the sine function, \(y = \sin(x)\), which has a period of \(2\pi\).
This means every \(2\pi\) units, the function's values repeat.
  • **Period:** The length of one complete cycle of the function.
  • **Amplitude:** The height from the centerline to the peak or trough of the wave.
Alternating functions, like sine and cosine, are perfect examples of periodic behavior, essential in fields such as physics and engineering.
They provide predictability and consistency, vital for modeling cyclical processes like seasonal temperatures or sound vibrations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

MAKING AN ARGUMENT Your friend claims it is possible to draw a right triangle so the values of the cosine function of the acute angles are equal. Is your friend correct? Explain your reasoning.

Sketch the angle. Then find its reference angle. \(-370^{\circ}\)

The motion of a pendulum can be modeled by the function \(d=4 \cos 8 \pi t\), where \(d\) is the horizontal displacement (in inches) of the pendulum relative to its position at rest and \(t\) is the time (in seconds). Find and interpret the period and amplitude in the context of this situation. Then graph the function.

Use the unit circle to evaluate the six trigonometric functions of \(\theta\). \(\theta=\frac{\pi}{2}\)

Sketch the angle. Then find its reference angle. \(320^{\circ}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.