/*! 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 100 Show that the range of the cotan... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 100

Show that the range of the cotangent function is the set of all real numbers.

Short Answer

Expert verified
The range of the cotangent function is all real numbers, \( (-鈭�, +鈭�). \)

Step by step solution

01

Understand the cotangent function

The cotangent function is defined as \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\). It is the reciprocal of the tangent function.
02

Determine the domain of the cotangent function

The cotangent function is undefined when \(\text{sin}(x) = 0\), that is at \(x = k\frac{\text{蟺}}{2}\) where \(k\) is any integer. Therefore, the domain of \(\text{cot}(x)\) is \(x eq k\frac{\text{蟺}}{2}\).
03

Analyze the behavior of the cotangent function

Observe the behavior of \(\text{cot}(x)\) as \(\text{sin}(x)\) approaches zero. As \(\text{sin}(x)\) approaches zero, the value of \(\text{cot}(x)\) tends to \(\frac{1}{0}\), resulting in vertical asymptotes at \(x = k\frac{\text{蟺}}{2}\). Between these vertical asymptotes at \(x = (k\text{蟺})\) and \(x = (k+1)\frac{\text{蟺}}{2}\), the function \(\text{cot}(x)\) covers all real numbers, descending from \(\text{+鈭瀩\) to \(\text{-鈭瀩\).
04

Conclude the range

Since \(\text{cot}(x)\) decreases continuously from \(\text{+鈭瀩\) to \(\text{-鈭瀩\) within each interval between its vertical asymptotes, the range of \(\text{cot}(x)\) for all \((x eq k\frac{\text{蟺}}{2})\) is \(\text{(-鈭�, +鈭�)}\). Therefore, the range of the cotangent function is all real numbers.

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.

Range of Cotangent
The cotangent function, represented as \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\), can take any real number as its output. When you look at a graph of \(\text{cot}(x)\), you鈥檒l notice that it continuously decreases from \(+鈭瀄) to \(-鈭瀄) within each interval between its vertical asymptotes.
Cotangent Domain
Understanding the domain of the cotangent function is crucial. Since \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\), it is defined for all angles \( x \eq k\frac{\text{蟺}}{2}\), where \( k\) is any integer.
Vertical Asymptotes
Vertical asymptotes for cotangent occur at points where the function is undefined鈥攊n other words, where \( \text{sin}(x) = 0 \). This results in a division by zero, leading the function value towards \( \frac{1}{0} = 卤鈭�.\)

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

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\sec ^{2} 28^{\circ}-\tan ^{2} 28^{\circ}$$

A point on the terminal side of an angle \(\theta\) in standard position is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) $$

Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. \(19.99^{\circ}\)

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \cos 420^{\circ} $$

Problems \(63-66\) require the following discussion. Projectile Motion The path of a projectile fired at an inclination \(\theta\) to the horizontal with initial speed \(v_{0}\) is a parabola. See the figure. The range \(R\) of the projectile-that is, the horizontal distance that the projectile travels-is found by using the function $$ R(\theta)=\frac{2 v_{0}^{2} \sin \theta \cos \theta}{g} $$ where \(g \approx 32.2\) feet per second per second \(\approx 9.8\) meters per second per second is the acceleration due to gravity. The maximum height \(H\) of the projectile is given by the function $$ H(\theta)=\frac{v_{0}^{2} \sin ^{2} \theta}{2 g} $$ Find the range \(R\) and maximum height \(H\) of the projectile. Round answers to two decimal places. The projectile is fired at an angle of \(45^{\circ}\) to the horizontal with an initial speed of 100 feet per second.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.