/*! 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 17 Find the period of each function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 17

Find the period of each function. \(y=\sin 1.5 x\)

Short Answer

Expert verified
The period of the function is \(\frac{4\pi}{3}\).

Step by step solution

01

Identify the Basic Trigonometric Function

The given function is \( y = \sin 1.5x \). The basic form of a sine function is \( y = \sin bx \), where \(b\) is a constant that affects the period.
02

Understand the Formula for Period

For any sine function of the form \( y = \sin bx \), the period is given by the formula:\[ \text{Period} = \frac{2\pi}{b} \].
03

Substitute the Value of b

In the function \( y = \sin 1.5x \), \(b = 1.5\). Substitute \(b\) into the period formula:\[ \text{Period} = \frac{2\pi}{1.5} \].
04

Simplify the Period Expression

To simplify \( \frac{2\pi}{1.5} \), convert 1.5 to a fraction:\( 1.5 = \frac{3}{2} \). Then the expression becomes:\[ \text{Period} = \frac{2\pi}{\frac{3}{2}} \].
05

Simplify the Fraction

Dividing by a fraction is the same as multiplying by its reciprocal. Hence:\[ \text{Period} = 2\pi \times \frac{2}{3} = \frac{4\pi}{3} \].

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 a fundamental concept in trigonometry. It is part of the family of trigonometric functions that come from studying angles and their relationships. The sine function is often written as \( y = \sin x \). This function maps any angle found in a unit circle to a real number, which represents the vertical position of that point on the circle.

Here are a few key characteristics of the sine function:
  • **Oscillation**: It oscillates, meaning it repeats its pattern over regular intervals.
  • **Range**: The output or value is always between -1 and 1.
  • **Function Shape**: It has a wave-like pattern, often called a sinusoidal wave.
The general form of the sine function can be written as \( y = A \sin(bx + c) + d \), where each parameter modifies the graph:
  • **\( A \):** amplitude, affects the height of the wave.
  • **\( b \):** affects the period.
  • **\( c \):** phase shift, moves the graph left or right.
  • **\( d \):** vertical shift, moves the graph up or down.
Period of a Function
Understanding the period of a function is crucial when working with trigonometric functions like sine. The period is the distance over which the function鈥檚 graph repeats itself.

For the basic sine function \( y = \sin x \), the period is \( 2\pi \). This means it takes \( 2\pi \) units along the x-axis for the wave to complete one full cycle and begin again at the same height and direction.

When the function is modified, as in \( y = \sin bx \), calculating the period changes. You use the formula:\[\text{Period} = \frac{2\pi}{b}\] Example: Calculating the Period
  • For \( y = \sin 1.5x \), the period is \( \frac{2\pi}{1.5} \).
  • Solve \( \frac{2\pi}{1.5} \), which simplifies to \( \frac{4\pi}{3} \).
This means that the sine wave completes a full cycle every \( \frac{4\pi}{3} \) units on the x-axis.
Basic Trigonometric Formulas
Trigonometric functions have essential formulas that make solving problems easier. These formulas provide a quick way to understand relationships in periodic functions.

**Some Basic Formulas to Remember**:
  • **Sine and Cosine Relationship**: \( \sin^2 x + \cos^2 x = 1 \) - This is the Pythagorean identity and always holds true.
  • **Angle Formulas**: - The sum of angles formula, \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
  • **Period Relation**: - For any trigonometric function, if \( y = \sin bx \), it has a period of \( \frac{2\pi}{b} \).
Remembering these will not only help in solving mathematical problems but will also aid in understanding complex trigonometric concepts. By grasping these formulas, you can simplify the process of studying oscillations and waves in various scientific fields.

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

Does \(y=\tan x\) have a maximum and a minimum value? Justify your answer.

Recall from your geometry course that a polygon is circumscribed about a circle if each side of the polygon is tangent to the circle. Since each side is tangent to the circle, the radius of the circle is perpendicular to each side at the point of tangency. We will use the tangent tunction to examine the formula for the perimeter of a circumscribed regular polygon. Let square \(A B C D\) be circumscribed about circle \(O .\) A radius of the circle, \(\overline{O P},\) is perpendicular to \(\overline{A B}\) at \(P .\) (1) In radians, what is the measure of \(\angle A O B ?\) (2) Let \(m \angle A O P=\theta .\) If \(\theta\) is equal to one-half the measure of \(\angle A O B\) , find \(\theta .\) (3) Write an expression for \(A P\) in terms of \(\tan \theta\) and \(r,\) the radius of the circle. (4) Write an expression for \(A B=s\) in terms of \(\tan \theta\) and \(r\) (5) Use part \((4)\) to write an expression for the perimeter in terms of \(r\) and the number of sides, \(n\) . b. Let regular pentagon \(A B C D E\) be circumscribed about circle \(O .\) Repeat part a using pentagon \(A B C D E .\) c. Do you see a pattern in the formulas for the perimeter of the square and of the pentagon? If so, make a conjecture for the formula for the perimeter of a circumscribed regular polygon in terms of the radius \(r\) and the number of sides \(n .\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos 0\)

List two values of \(x\) in the interval \(-2 \pi \leq x \leq 2 \pi\) for which cot \(x\) is undefined.

In \(3-14,\) sketch one cycle of the graph. $$ y=3 \sin 2 x $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

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.