/*! 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 37 Find all solutions of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 37

Find all solutions of the equation. $$\cos (\ln x)=0$$

Short Answer

Expert verified
The solutions are \( x = e^{\frac{\pi}{2} + k\pi} \) for integer \( k \).

Step by step solution

01

Understand the Equation

The given equation is \( \cos(\ln x) = 0 \). Our job is to find the values of \( x \) that satisfy this equation. We know that \( \cos(\theta) = 0 \) when \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Thus, we need to find \( x \) such that \( \ln x = \frac{\pi}{2} + k\pi \).
02

Solve for \( x \)

Starting from \( \ln x = \frac{\pi}{2} + k\pi \), we solve for \( x \) by exponentiating both sides: \( e^{\ln x} = e^{\frac{\pi}{2} + k\pi} \), which simplifies to \( x = e^{\frac{\pi}{2} + k\pi} \).
03

Conclude the Solutions

The solutions to the equation \( \cos (\ln x) = 0 \) are given by \( x = e^{\frac{\pi}{2} + k\pi} \) for integer values of \( k \). These represent the infinite set of solutions where each value corresponds to a different \( k \).

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.

Trigonometric Functions
Trigonometric functions are mathematical functions that relate angles of triangles to the lengths of sides in a right triangle. The most common trigonometric functions are sine (\(\sin\theta\)), cosine (\(\cos\theta\)), and tangent (\(\tan\theta\)). These functions are fundamental in studying periodic phenomena, like waves and oscillations.

In the equation \(\cos(\ln x) = 0\), we specifically deal with the cosine function, which equals zero at certain points. The cosine function is periodic with a period of \(2\pi\).
  • \(\cos\theta = 0\) at angles \(\theta = \frac{\pi}{2} + k\pi\), where \(k\) is any integer.
This periodic property helps us find when the natural logarithm of \(x\) equals one of those angles to satisfy the equation.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. It is widely used in mathematical calculations, especially in calculus and when dealing with continuous growth or decay processes.

The natural logarithm transforms multiplicative processes into additive ones, which is crucial in solving equations like \(\cos(\ln x) = 0\). In this equation, we need to find when the \(\ln x\) value results in cosine equalling zero.
  • Logarithms change multiplication into addition: \(\ln(a \times b) = \ln a + \ln b\).
  • Exponentiating cancels out the logarithm, as seen with: \(e^{\ln x} = x\).
Utilizing these properties, we convert from the logarithmic form to potential solutions when aligned with cosine's zero points.
Exponential Functions
Exponential functions involve expressions where the variable appears in the exponent. A basic form is \(y = a^{x}\), but the most important one in advanced mathematics is \(e^{x}\). The function \(e^{x}\) is unique because its derivative is the same as the function itself; \(\frac{d}{dx}(e^{x}) = e^{x}\).

In solving our given equation \(\cos(\ln x) = 0\), we found that exponentiating helped convert the natural logarithm back to its number form.
  • Starting from \(\ln x = \frac{\pi}{2} + k\pi\), we express it in terms of \(x\) by exponentiating: \(e^{\ln x} = e^{\frac{\pi}{2} + k\pi}\).
  • This simplifies to \(x = e^{\frac{\pi}{2} + k\pi}\), offering a set of potential solutions as \(k\) varies among all integers.
Exponential functions provide a way to transform logarithmic equations back into their original numeric settings, offering clarity in complex solutions like ours.

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 the graph of \(f\) to find the simplest expression \(g(x)\) such that the equation \(f(x)=g(x)\) is an Identity. Verify this identity. $$f(x)=\frac{\sin ^{3} x+\sin x \cos ^{2} x}{\csc x}+\frac{\cos ^{3} x+\cos x \sin ^{2} x}{\sec x}$$

n designing a collector for solar power, an important consideration is the amount of sunlight that is transmitted through the glass into the water being heated. If the angle of incidence \(\theta\) of the sun's rays is measured from a line perpendicular to the surface of the glass, then the fraction \(f(\theta)\) of sunlight reflected off the glass can be approximated by $$f(\theta)=\frac{1}{2}\left(\frac{\sin ^{2} \alpha}{\sin ^{2} \beta}+\frac{\tan ^{2} \alpha}{\tan ^{2} \beta}\right) $$where$$\alpha=\theta-\gamma, \quad \beta=\theta+\gamma, \quad \text { and } \quad \gamma=\sin ^{-1}\left(\frac{\sin \theta}{1.52}\right)$$ Graph \(f\) for \(0<\theta<\pi / 2,\) and estimate \(\theta\) when \(f(\theta)=0.2\)

Complete the statements. (a) As \(x \rightarrow 1^{-}, \sin ^{-1} x \rightarrow\text{____}\) (b) As \(x \rightarrow-1^{+}, \cos ^{-1} x \rightarrow\text{____}\) (c) As \(x \rightarrow-\infty, \tan ^{-1} x \rightarrow\text{____}\)

Approximate the solution to each inequality on the interval \([0,2 \pi]\). $$\sin x<-0.6$$

Verify the Identity. $$2 \cos ^{-1} x=\cos ^{-1}\left(2 x^{2}-1\right), 0 \leq x \leq 1$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.