/*! 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 50 Use a graph to solve the equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 50

Use a graph to solve the equation on the interval \(-2 \pi, 2 \pi\). $$\tan x=\sqrt{3}$$

Short Answer

Expert verified
The solutions to the equation \( \tan x = \sqrt{3} \) in the interval \( -2\pi \) to \( 2\pi \) are \( x = -\frac{2\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \) and \( \frac{5\pi}{3} \).

Step by step solution

01

Draw the Tangent Function

Begin by sketching the graph of \( y = \tan x \). Tangent has a period of \( \pi \), with vertical asymptotes at \( x = \frac{\pi}{2} \) + n\( \pi \), where n is an integer. Plot this in a graph.
02

Draw the Line \( y = \sqrt{3} \)

Next, draw a horizontal line at \( y = \sqrt{3} \). This line will intersect with the graph of \( y = \tan x \) at the solutions to the equation \( \tan x = \sqrt{3} \).
03

Find Intersection Points

Determine where the line \( y = \sqrt{3} \) intersects with the graph of \( y = \tan x \). These intersection points are the solutions to the equation \( \tan x = \sqrt{3} \).
04

Identify Relevant Intersection Points

Identify the intersection points that fall within the interval \( -2\pi \) to \( 2\pi \).
05

Determine the X-values of the Intersection Points

The \( x \)-values of these intersection points correspond to values of \( x \) which satisfy the equation \( \tan x = \sqrt{3} \). From the knowledge of trigonometric values, we know the positive solutions are \( x = \frac{\pi}{3} + n\pi \), and the negative solutions are \( x = -\frac{\pi}{3} + n\pi \), where \( n \) is an integer.
06

List all Possible Solutions Within the Interval

Within the interval \( -2\pi \), \( 2\pi \), the solutions are \( x = -\frac{2\pi}{3}, -\frac{\pi}{3}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \) and \( \frac{5\pi}{3} \).

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!

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

Graph the functions \(f\) and \(g .\) Use the graphs to make a conjecture about the relationship between the functions. $$f(x)=\cos ^{2} \frac{\pi x}{2}, \quad g(x)=\frac{1}{2}(1+\cos \pi x)$$

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\csc x$$

Sketch a graph of the function. $$f(x)=\arccos \frac{x}{4}$$

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$h(x)=2^{-x^{2} / 4} \sin x$$

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\cot x$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.