/*! 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 11 In Exercises 5-18, find the peri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 11

In Exercises 5-18, find the period and amplitude. \(y\ =\ -4\ sin\ x\)

Short Answer

Expert verified
The amplitude is \(4\) and the period is \(2\pi\).

Step by step solution

01

Identify the Amplitude

The amplitude of a function is the absolute value of its highest peak or lowest trough from the middle of its graph. For a sine or cosine function, this is the absolute value of the coefficient of the function. For the given equation \(y = -4 \sin x\), the amplitude is the absolute value of \(-4\), which is \(4\).
02

Calculate the Period

The period of a function is the horizontal length of one complete cycle. For a sine or cosine function, the period is \(2\pi/B\), where \(B\) is the coefficient of \(x\). In the given equation \(y = -4 \sin x\), \(B = 1\), so the period is \(2\pi/1 = 2\pi\).

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
Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In trigonometric terms, amplitude can be thought of as the height of the wave from the center line to the peak. This is an important concept for understanding the behavior of periodic functions like sine and cosine.

To find the amplitude of a sine function such as \(y = -4 \sin x\), we look at the coefficient in front of the sine term. The amplitude is the absolute value of this coefficient. So even if the coefficient is negative, like in this case, we take its absolute value. For \(y = -4 \sin x\), the amplitude is \(\lvert -4 \rvert = 4\).

Always remember:
  • The amplitude tells you how far up or down the wave goes from the center axis.
  • It determines the peak and trough heights of the wave.
  • It is always a positive number, no matter the sign of the coefficient in front of the sine or cosine function.
Period
The period of a trigonometric function is the length of one complete cycle on a graph. For sine and cosine functions, the period tells us how long it takes for the function to repeat its values. This is crucial for predicting the behavior of waves over time.

To find the period of a standard sine function \(y = A \sin(Bx)\), use the formula \(2\pi / B\), where \(B\) is the coefficient of \(x\). In the function \(y = -4 \sin x\), \(B = 1\). Therefore, the period is \(2\pi/1 = 2\pi\). This means the curve completes a full cycle every \(2\pi\) radians on the x-axis.

Key points to understand:
  • The period is always measured in the same units as the x-variable, typically radians or degrees.
  • The standard period of \(\sin x\) and \(\cos x\) is \(2\pi\).
  • Changes in the coefficient \(B\) will change the length of the period.
Sine Function
The sine function, often written as \(\sin(x)\), is a fundamental trigonometric function that describes a smooth, periodic oscillation. It is essential in various fields such as physics, engineering, and mathematics due to its ability to model wave-like phenomena.

The basic form of the sine function is \(y = A \sin(Bx + C) + D\):
  • \(A\) determines the amplitude, affecting how "tall" the wave appears.
  • \(B\) affects the period, or how "long" the wave takes to repeat.
  • \(C\) represents the horizontal shift, moving the wave left or right.
  • \(D\) represents the vertical shift, moving the wave up or down.
For \(y = -4 \sin x\), the sine wave:
  • Has an amplitude of 4, meaning it oscillates between -4 and 4.
  • Has a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.
  • Starts from the origin since there is no horizontal or vertical shift.
The sine function is beautifully symmetric and periodic, making it simple yet powerful, perfect for modelling any periodic process.

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

In Exercises 67-76, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) sec(arctan \(3x\))

In Exercises 23-40, use a calculator to evaluate the expression. Round your result to two decimal places. \(arctan\ 2.8\)

In Exercises 23-40, use a calculator to evaluate the expression. Round your result to two decimal places. \(tan^{-1}\ (-\sqrt{372})\)

In Exercises 97 and 98, write the function in terms of the sine function by using the identity $$ A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right) $$ Use a graphing utility to graph both forms of the function. What does the graph imply? $$ f(t)=3 \cos 2 t+3 \sin 2 t $$

HARMONIC MOTION In Exercises 57-60, for the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c ) the value of \(d\) when \(t=5\), and (d) the least positive value of \(d\) for which \(t=5\). Use a graphing utility to verify your results. \(d\ =\ \dfrac{1}{2} cos\ 20 \pi t\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.