/*! 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 62 For \(y=5 \cos (4 x-\pi),\) find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 62

For \(y=5 \cos (4 x-\pi),\) find the amplitude, the period, and the phase shift.

Short Answer

Expert verified
Amplitude: 5, Period: \(\frac{\pi}{2}\), Phase Shift: \(\frac{\pi}{4}\)

Step by step solution

01

- Identify the amplitude

The amplitude of a cosine function of the form \(y = a \, \text{cos}(b x - c)\) is given by the absolute value of the coefficient in front of the cosine function. Here, the coefficient is 5. Therefore, the amplitude is \(|5| = 5\).
02

- Find the period

The general form of the cosine function \(y = a \, \text{cos}(b x - c)\) has a period given by \( \frac{2\pi}{b} \). Here, the value of \(b\) is 4. Thus, the period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).
03

- Determine the phase shift

The phase shift of a cosine function \(y = a \, \text{cos}(b x - c)\) is given by \( \frac{c}{b} \). In this case, \(c\) is \(\pi\) and \(b\) is 4. Therefore, the phase shift is \( \frac{\pi}{4} \).

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.

Amplitude
In trigonometry, the amplitude of a cosine function represents the maximum distance from the midline (or equilibrium position) to the peak (or trough) of the wave.
To find the amplitude of the given function, \(y=5 \cos (4 x-\pi)\), you need to look at the coefficient in front of the cosine function.
The amplitude is given by the absolute value of this coefficient.
  • In our example, the coefficient is 5.
  • The absolute value of 5 is \|5| = 5\.
Therefore, the amplitude is 5. This means that the wave reaches a maximum height of 5 and a minimum height of -5 from its midline.
Period
The period of a cosine function represents the length of one complete cycle of the wave.
For the general form of the cosine function, \(y = a \, \text{cos}(b x - c)\), the period is determined by the value of \(b\).
You can calculate the period using the formula \[ \frac{2\pi}{b} \]
  • In our function, \(b\) is 4.
  • Plugging into the formula, the period is \( \frac{2\pi}{4} = \frac{\pi}{2}\).
This means that the wave completes one full cycle every \( \frac{\pi}{2} \) units along the x-axis.
Phase Shift
The phase shift of a cosine function indicates how much the wave is shifted horizontally from the standard position.
In the general form of the cosine function, \(y = a \cos(b x - c)\), the phase shift is found using the values of \(b\) and \(c\).
It is calculated as \ \frac{c}{b} \.
  • For our function \(y = 5 \cos (4 x-\pi)\), \(c = \pi\) and \(b = 4\).
  • The phase shift, therefore, is \ \frac{\pi}{4} \.
This means that the cosine wave is shifted \ \frac{\pi}{4} \ units to the right.

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

If \(z=\tan \frac{\alpha}{2},\) show that \(\sin \alpha=\frac{2 z}{1+z^{2}}\)

Use the following discussion. The formula $$ D=24\left[1-\frac{\cos ^{-1}(\tan i \tan \theta)}{\pi}\right] $$ Approximate the number of hours of daylight in New York, New York \(\left(40^{\circ} 45^{\prime}\right.\) north latitude \()\), for the following dates: (a) Summer solstice \(\left(i=23.5^{\circ}\right)\) (b) Vernal equinox \(\left(i=0^{\circ}\right)\) (c) July \(4\left(i=22^{\circ} 48^{\prime}\right)\)

Write each trigonometric expression as an algebraic expression containing u and \(v .\) Give the restrictions required on \(u\) and \(v\). $$ \cos \left(\cos ^{-1} u+\sin ^{-1} v\right) $$

Find the value of the number \(C\) : $$ \frac{1}{2} \cos ^{2} x+C=\frac{1}{4} \cos (2 x) $$

Show that \(\sin \left(\sin ^{-1} v+\cos ^{-1} v\right)=1\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.