/*! 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 4 Solve each equation for exact so... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Solve each equation for exact solutions in the interval \(0 \leq x<2 \pi\) $$\cos x-1=0$$

Short Answer

Expert verified
The exact solution for the equation in the interval \(0 \leq x < 2 \pi\) is \(x = 0\).

Step by step solution

01

Rearranging the equation

First, the equation needs to be rearranged to isolate the cosine function. To accomplish this, add 1 to both sides of the equation to get \(\cos x = 1\).
02

Find solutions in the interval

Next, we need to find the values of \(x\) in the interval \(0 \leq x < 2 \pi \) for which \(\cos x = 1\). The cosine function equals 1 at \(x = 0\) and \(x = 2 \pi\) in the unit circle. However, since our interval does not include \(2 \pi \), our only solution in the interval is \(x = 0\) .

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.

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.

Cosine Function
The cosine function is one of the fundamental trigonometric functions. It relates the adjacent side of a right triangle to its hypotenuse. In the context of a unit circle, the cosine function gives the x-coordinate of a point lying on the circle.

Key aspects of the cosine function include:
  • **Periodicity:** The cosine function has a period of \(2\pi\), meaning it repeats its values every \(2\pi\) interval.
  • **Range:** The values of the cosine function range from \(-1\) to \(1\).
  • **Values:** Cosine is equal to \(1\) at \(x = 0\), \(x = 2\pi\), etc., because these points are where the unit circle intersects the positive x-axis.
Understanding the cosine function can help solve problems involving angles and distances in both the unit circle and real-world contexts.
Unit Circle
The unit circle is a powerful tool in trigonometry. It is defined as a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle helps visualize the behavior of trigonometric functions.
Here's what you need to know about the unit circle:
  • **Coordinates:** Any point on the unit circle is described by coordinates \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle in radians.
  • **Angles:** The angle \(\theta\) is measured from the positive x-axis in a counter-clockwise direction.
  • **Special Angles:** Common angles such as \(0\), \(\pi/2\), \(\pi\), and \(3\pi/2\) have well-known cosine and sine values, making them easy to remember.
Using the unit circle can simplify understanding and solving trigonometric equations by making visual connections between angles and their sine/cosine values.
Interval Notation
Interval notation is a way of expressing a range of numbers, typically used to indicate the set of solutions within a specific domain. This is particularly handy in trigonometry for describing solution sets.
Here's how it works:
  • **Brackets and Parentheses:** Square brackets \([]\) indicate that an endpoint is included, while parentheses \(()\) signify that an endpoint is not included.
  • **Example:** The interval \([0, 2\pi)\) indicates that while \(0\) is included in the solution, \(2\pi\) is excluded. This works well for scenarios like one full revolution around the unit circle.
  • **Application:** In trigonometry, using interval notation helps specify which values of \(x\) are solutions, ensuring clarity when \(x\) can take on multiple equivalents due to periodicity.
Understanding interval notation simplifies interpreting trigonometric solutions and gives precise control over which values are permissible within a set range.

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

In Exercises 73 to \(88,\) verify the identity. $$\sec \left(\frac{\pi}{2}-\theta\right)=\csc \theta$$

Verify the identity. $$\tan \left(\csc ^{-1} x\right)=\frac{\sqrt{x^{2}-1}}{x^{2}-1}, x>1$$

Find \(K\) given \(K=\sqrt{s(s-a)(s-b)(s-c)}\) with \(s=12\) \(a=8, b=6,\) and \(c=10 .[\mathrm{A} .1]\)

Solve \(\frac{42.4}{\sin \theta}=\frac{31.5}{\sin 32.1^{\circ}}\) for \(\theta .\) Round to the nearest tenth of a degree. [5.5]

Solve \(2 x^{2}-2 x=0\) by factoring. [1.1]
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.