/*! 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 93 Find all solutions to \((\sin x-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 93

Find all solutions to \((\sin x-1)(\sin x+1)=0\) in the interval \((0,2 \pi)\)

Short Answer

Expert verified
The solutions are \(\pi/2\) and \((3\fin)/2\).

Step by step solution

01

Identify the equation components

Given the equation \( (\sin x - 1)(\fin x + 1) = 0 \) , note it is a product of two terms set to zero. Therefore, either \(\sin x - 1 = 0 \) or \(\fin x + 1 = 0 \) must hold true.
02

Solve \( \sin x - 1 = 0 \)

Solving \(\sin x - 1 = 0 \) gives \(\sin x = 1 \). The solutions to \(\sin x = 1 \) in the interval \((0, 2\pi)\) occur at \(\pi/2\).
03

Solve \(\sin x + 1 = 0 \)

Solving \(\sin x + 1 = 0 \) gives \(\sin x = -1 \). The solutions to \(\sin x = -1 \) in the interval \(0, 2\pi)\) occur at \((3\pi)/2\).
04

Combine the solutions

The combined solutions are \(\pi/2\) and \((3\pi)/2\).

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.

sin function
The sine function, denoted as \(\text{sin}(x)\), is one of the fundamental trigonometric functions. It relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. The sine function is periodic with a period of \(2\text{蟺}\), which means that its values repeat every \(2\text{蟺}\) units along the x-axis.

Important properties of the sine function include:
  • The maximum value of \(\text{sin}(x)\) is 1.
  • The minimum value of \(\text{sin}(x)\) is -1.
  • It is an odd function, i.e., \(\text{sin}(-x) = -\text{sin}(x)\).
The sine function is especially useful in modeling periodic phenomena like sound waves and circular motion. In our given equation, \(\text{sin}(x) - 1 = 0\) and \(\text{sin}(x) + 1 = 0\), we look for points on the sine wave where the sine value is 1 or -1.
unit circle
The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It plays a vital role in understanding trigonometric functions. Each point on the unit circle is associated with an angle \(x \), measured from the positive x-axis, and has coordinates \((\text{cos}(x), \text{sin}(x))\).

Here are some key points about the unit circle:
  • At an angle of \(\text{蟺}/2\), the coordinates are (0, 1), where \(\text{sin}(\text{蟺}/2) = 1\).
  • At an angle of \(3\text{蟺}/2\), the coordinates are (0, -1), where \(\text{sin}(3\text{蟺}/2) = -1\).
In our problem, solving the equation involves finding which angles correspond to a sine of 1 or -1. These angles are found where the terminal side of the angle intercepts the unit circle at these specific points.
solution intervals
Solution intervals are ranges within which we seek to find solutions to an equation. In trigonometric equations, these intervals typically span from \(0\) to \(2\text{蟺}\) to cover one complete cycle of the trigonometric function.

When solving for \(\text{sin}(x) = 1\) and \(\text{sin}(x) = -1\) within the interval \(0, 2\text{蟺}\), we:
  • Identify angles within \(0 \leq x \leq 2\text{蟺}\) that satisfy the equations.
  • Utilize the periodic property of the sine function to determine these angles.
For \(\text{sin}(x) = 1\), we find that the angle is \(\text{蟺}/2\). For \(\text{sin}(x) = -1\), the angle is \(3\text{蟺}/2\). Thus, the solutions in the interval \(0, 2\text{蟺}\) are \(\text{蟺}/2\) and \(3\text{蟺}/2\).

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

Solve each equation. Round approximate answers to the nearest tenth of a degree. $$ 2 \cos (\alpha)-2=0 \text { for }-360^{\circ} \leq \alpha \leq 360^{\circ} $$

Use a graphing calculator to graph \(y=\cos (x) /(x-\pi / 2)\) and determine the number of solutions to \(\cos (x) /(x-\pi / 2)=0\) in the interval \((-2 \pi, 2 \pi) .\) What is the minimum value of this function on this interval?

Use a calculator to find the approximate value of each composition. Round answers to four decimal places. Some of these expressions are undefined. $$ \cot \left(\cos ^{-1}(-1 / \sqrt{7})\right) $$

Find the exact value of each expression without using a calculator or table. a. \(\arcsin (1 / 2)\) b. \(\cos ^{-1}(-1 / 2)\) c. \(\tan ^{-1}(-1)\) d. \(\sin (\pi / 3)\) e. \(\cos (-\pi / 2)\) f. \(\sin ^{-1}(-1)\)

Find the exact value of each composition without using a calculator or table. $$ \tan ^{-1}(\sin (\pi / 2)) $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.