Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 19: Problem 1876

The number of solution of the equation \(\sqrt{(3) \sin x+\cos x}=4\) is \(x \in[0,2 \pi]\) (a) 1 (b) 2 (c) 0 (d) 3

Short Answer

Expert verified
The number of solutions of the equation \(\sqrt{3 \sin x + \cos x} = 4\) in the interval \(x \in [0, 2 \pi]\) is (c) 0.

Step by step solution

01

Square both sides of the equation

First, we need to get rid of the square root. In order to do this, we square both sides of the equation: \((\sqrt{(3) \sin x + \cos x})^2 = (4)^2\) Simplifying the expression we get: \(3 \sin x + \cos x = 16\)
02

Rearrange the equation

Now we need to express \(\sin x\) as follows: \(\sin x = \frac{16 - \cos x}{3}\)
03

Apply trigonometric identity and solve for \(x\)

Using trigonometric identities, note that \(\sin^2x + \cos^2x = 1\). So we have: \(\frac{16^2 - 32 \cos x + \cos^2x}{3^2} + \cos^2x = 1\) Now let's solve for \(x\) within the interval \(x \in [0, 2\pi]\): \(\frac{256 - 32 \cos x + \cos^2x}{9} + \cos^2x = 1\) To simplify this expression, let's multiply both sides by \(9\): \(256 - 32 \cos x + \cos^2x + 9\cos^2x = 9\) Combine terms: \(10\cos^2x - 32\cos x + 247 = 0\)
04

Determine the number of solutions

Unfortunately, this quadratic equation cannot be factored easily, so we'll proceed by attempting to find the solutions using the quadratic formula: \(\cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In this case, \(a = 10\), \(b = -32\), and \(c = 247\). Compute the discriminant: \(b^2 - 4ac = (-32)^2 - 4(10)(247) = 1024 - 9880 = -8856\) Since the discriminant is negative, the quadratic equation has no real solutions for \(\cos x\). Therefore, there are no solutions for the original equation \(\sqrt{(3) \sin x + \cos x} = 4\) within the interval \(x \in [0, 2\pi]\). The correct answer is: (c) 0

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 \(\left.\tan ^{-1}\left[\left\\{\sqrt{(} 1+x^{2}\right)-1\right\\} / x\right]=(3 / 10)\) then \(x=\) (a) \(\tan (3 / 10)\) (b) \(\tan (4 / 10)\) (c) \(\tan (10 / 3)\) (d) \(\tan (6 / 10)\)

If \(\sin \alpha-\sin \beta=m\) and \(\cos \alpha-\cos \beta=n\) then \(\cos (\alpha-\beta)=\) (a) \(\left[\left(2+m^{2}+n^{2}\right) / 2\right]\) (b) \(\left[\left(2-m^{2}-n^{2}\right) / 2\right]\) (c) \(\left[\left(m^{2}+n^{2}\right) / 2\right]\) (d) \(-\left[\left(m^{2}+n^{2}\right) / 2\right]\)

\(\sin ^{-1}(\sin 4)=\) (a) 4 (b) \(4-2 \pi\) (c) \(\pi-4\) (d) \(4-\pi\)

If \([(\cos A) / 3]=[(\cos B) / 4]=(1 / 5),-(\pi / 2)

The number of values of \(\theta\) in the interval \([0,4 \pi]\) satisfying the equation \(2 \sin ^{2} \theta-\cos 2 \theta=0\) (a) 4 (b) 8 (c) 2 (d) 6
See all solutions

Recommended explanations on English Textbooks

Textual Analysis

Read Explanation

Language Acquisition

Read Explanation

Creative Story

Read Explanation

Pragmatics

Read Explanation

Text Comparison

Read Explanation

Essay Plan

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.

Sign-up for free