/*! 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 82 Find all solutions of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 82

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\tan (x+\pi)+2 \sin (x+\pi)=0$$

Short Answer

Expert verified
The solutions to the equation \(\tan (x+\pi) + 2 \sin (x+\pi) = 0\) in the interval \([0,2 \pi)\) are \(x = 0, \pi, \pi/3, 5\pi/3\).

Step by step solution

01

Simplify the Equation

First, the equation can be simplified using trigonometric identities. It is known that \(\tan (x + \pi) = \tan x\) and \(\sin (x + \pi) = - \sin x\). Therefore, the original equation \(\tan (x + \pi) + 2 \sin (x + \pi) = 0\) can be simplified to \(\tan x - 2 \sin x = 0\).
02

Transform the Equation

Next, transform \(\tan x\) to \(\sin x / \cos x\). The formula now looks like: \(\sin x / \cos x - 2 \sin x = 0\). Rearranging the terms gets \( \sin x (1 - 2 \cos x) = 0\).
03

Solve the Equation

This equation can be solved by setting each factor to 0 and solve for x: \(\sin x = 0\) or \( 1 - 2 \cos x = 0\). The first equation \(\sin x = 0\) has solutions of \(x = 0, \pi\) in the given interval. The second equation \(\cos x = 1/2\) leads to \(x = \pi/3, 5\pi/3\) in the given interval. Therefore, the solution for the original equation are \(x = 0, \pi, \pi/3, 5\pi/3\).
04

Check the Solutions

Substitute these solutions back into the original equation to check. They should all satisfy the equation \(\tan (x + \pi) + 2 \sin (x + \pi) = 0\) as well as fall 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!

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

The monthly sales \(S\) (in thousands of units) of a seasonal product are approximated by $$S=74.50+43.75 \sin \frac{\pi t}{6}$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January. Determine the months in which sales exceed 100,000 units.

The table shows the average daily high temperatures in Houston \(H\) (in degrees Fahrenheit) for month \(t,\) with \(t=1\) corresponding to January. (Source: National Climatic Data Center) $$ \begin{array}{|c|c|} \hline \text { Month, } t & \text { Houston, } \boldsymbol{H} \\ \hline 1 & 62.3 \\ 2 & 66.5 \\ 3 & 73.3 \\ 4 & 79.1 \\ 5 & 85.5 \\ 6 & 90.7 \\ 7 & 93.6 \\ 8 & 93.5 \\ 9 & 89.3 \\ 10 & 82.0 \\ 11 & 72.0 \\ 12 & 64.6 \\ \hline \end{array} $$ (a) Create a scatter plot of the data. (b) Find a cosine model for the temperatures in Houston. (c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data? (d) What is the overall average daily high temperature in Houston? (e) Use a graphing utility to describe the months during which the average daily high temperature is above \(86^{\circ} \mathrm{F}\) and below \(86^{\circ} \mathrm{F}\).

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec ^{2} x-4 \sec x=0$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$3 \tan ^{2} x+5 \tan x-4=0, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot ^{2} x-6 \cot x+5=0$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Calculus

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.