/*! 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 12 Find all real numbers that satis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 12

Find all real numbers that satisfy each equation. $$ \tan (3 x)=-1 $$

Short Answer

Expert verified
x = \(\frac{\text{pi}}{4} + k\text{pi}\).

Step by step solution

01

Identify the tangent function properties

The tangent of an angle in the unit circle can be expressed as \(\tan x = \frac{\text{opposite}}{\text{adjacent}}\). The tangent function is periodic with a period of \(\frac{\text{pi}}{n}\) , which means it repeats its values every \(\text{pi}\).
02

Find the principal angle

Identify the angle where the tangent value is 1. From the known values of the tangent function, \(\tan\frac{\text{pi}}{4} = 1\).
03

Determine the general solution

Since the tangent function is periodic with a period of \(\text{pi}\), the general solution for \(\tan x=1\) is given by \[x = \frac{\text{pi}}{4} + k\text{pi}\] for any integer \k\.

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.

tangent function properties
The tangent function, denoted as \(\tan x\), measures the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. In terms of the unit circle, this means it's the ratio of the y-coordinate to the x-coordinate of a point on the circle.
A few important properties of the tangent function include:
  • It is undefined at \(\frac{\text{pi}}{2} + n\text{pi}\) where n is an integer.
  • The tangent function has no amplitude, as it can take any real value.
  • Its range is the set of all real numbers, from negative infinity to positive infinity.
Understanding these properties helps in solving equations like \(\tan x = 1\).
periodic functions
Many trigonometric functions are periodic, meaning they repeat their values at regular intervals. The tangent function, in particular, has a period of \[ \text{pi} \].
This periodicity can be written mathematically as \(\tan(x + \text{pi}) = \tan x\). This means that if we know the value of \( \tan x \) at one angle, we know it at every other angle which differs by an integer multiple of \(\text{pi}\).
  • This is why the general solution to \(\tan x = 1\) includes an infinite number of solutions: \[x = \frac{\text{pi}}{4} + k\text{pi} \] for any integer \(\text{k}\).
  • It also underscores how periodic functions bring structure to seemingly complex problems, offering us a repeating pattern that simplifies the solving process.
Embracing periodicity in trigonometry can help you solve a variety of equations more quickly and efficiently.
trigonometric equations
Trigonometric equations involve trigonometric functions like sine, cosine, and tangent. Solving these requires understanding both the properties of the functions and their graphs. Key steps include:
  • Identifying a principal angle that satisfies the equation. For example, the principal angle where \(\tan x = 1\) is \( \frac{\text{pi}}{4} \).
  • Recognizing the function鈥檚 periodicity to find all possible solutions. With the tangent function, this means adding multiples of \(\text{pi}\) to the principal angle.
  • Using these insights to write a general solution. For \(\tan x = 1\), the general solution is \[x = \frac{\text{pi}}{4} + k\text{pi} \] where \(k \) is any integer.
These steps showcase the blend of algebraic manipulation and function properties in solving trigonometric equations. They also demonstrate the power of understanding periodic functions in uncovering all possible solutions.

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

Find all real numbers that satisfy each equation. Round approximate answers to 2 decimal places. $$ \frac{\sin 33.2^{\circ}}{a}=\frac{\sin 45.6^{\circ}}{13.7} $$

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

Find all real numbers that satisfy each equation. Round approximate answers to 2 decimal places. $$ 4=6 \cos ^{-1}(x)+1 $$

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

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied 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.