/*! 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 47 Solve using a calculator, findin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 47

Solve using a calculator, finding all solutions in \([0,2 \pi)\). $$2 \cos ^{2} x=x+1$$

Short Answer

Expert verified
Find solutions in \( [0, 2\pi) \) by solving \( 2 \, \text{cos}^{2} x - 1 - x = 0 \) and taking \( \text{arccos}(y) \) of the solutions.

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in standard form: \[ 2 \, \text{cos}^{2} x - x - 1 = 0 \]
02

Substitute Cosine Function

Let \(y = \text{cos} x\). Then the equation becomes: \[ 2y^{2} - 1 = x \]
03

Solve for y

Using a calculator, solve for \(y\) in the range \([0, 2\pi)\) by setting \[2y^{2} - 1 - y = 0\]
04

Find arccos

For each solution for \(y\), find \( x \) by using \( x = \text{arccos}(y) \). Ensure that \( x \) falls within the interval \([0, 2\pi)\).

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.

Arccosine
Arccosine, denoted as \(\text{arccos}(x)\) or \(\text{cos}^{-1}(x)\), is the inverse function of cosine. It is used to find the angle whose cosine value equals \(x\). The range of \(\text{arccos}(x)\) is between \(0\) and \(\text{蟺}\), which covers all possible angles for the cosine function in one cycle. When solving trigonometric equations, arccosine helps us determine the angle \(x\) for which \(\text{cos}(x) = y\). This is crucial for finding solutions within specified intervals.

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 using a calculator, finding all solutions in \([0,2 \pi)\). $$\cos x-2=x^{2}-3 x$$

Solve, finding all solutions in \([0,2 \pi)\). $$2 \cos x+2 \sin x=\sqrt{6}$$

Nautical Mile. (See Exercise 60 in Exercise Set 7.2 ) In Great Britain, the nautical mile is defined as the length of a minute of arc of the earth's radius. since the earth is flattened at the poles, a British nautical mile varies with latitude. In fact, it is given, in feet, by the function $$N(\phi)=6066-31 \cos 2 \phi$$ where \(\phi\) is the latitude in degrees. At what latitude north is the length of a British nautical mile found to be \(6040 \mathrm{ft} ?\)

Simplify. Check your results using a graphing calculator. $$\frac{\cos ^{2} y \sin \left(y+\frac{\pi}{2}\right)}{\sin ^{2} y \sin \left(\frac{\pi}{2}-y\right)}$$

Evaluate. \(\cos \left(\tan ^{-1} \frac{\sqrt{3}}{4}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.