/*! 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 198 $$ \tan ^{-1} x+2 \cot ^{-1} x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 198

$$ \tan ^{-1} x+2 \cot ^{-1} x=\frac{2}{3} \pi $$

Short Answer

Expert verified
The solution to the equation is approximately \(x 鈮� 3.7321\).

Step by step solution

01

Identify and Use Inverse Trig Identity

Recognize that \(\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}\) for \(x 鈮� 0\). So substitute this identity into the equation: \(\frac{\pi}{2} + 2\cot^{-1}(x) = \frac{2\pi}{3}\).
02

Isolate \(\cot^{-1}(x)\) term

Subtract \(\frac{\pi}{2}\) from both sides of the equation: \(2\cot^{-1}(x) = \frac{2\pi}{3} - \frac{\pi}{2} = \frac{\pi}{6}\). Then divide both sides by 2 to solve for \(\cot^{-1}(x)\) : \(\cot^{-1}(x) = \frac{\pi}{12}\). This means that \(\cot(x) = \frac{1}{\tan(x)} = \tan\left(\frac{\pi}{12}\right)\).
03

Find the Value of \(x\)

Since \(\cot(x) = \tan\left(\frac{\pi}{12}\right)\), we therefore have \(x = \frac{1}{\tan\left(\frac{\pi}{12}\right)}\). We know \(\tan\left(\frac{\pi}{12}\right)\) is approximately 0.26794. So \(x 鈮� \frac{1}{0.26794} 鈮� 3.7321.\)

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.

Inverse Trig Identity
In trigonometry, inverse functions play a crucial role when reversing the process of a trig function to find an angle. An important identity involving the inverse trigonometric functions is that the sum of the inverse tangent and inverse cotangent of the same variable equals \( \frac{\pi}{2} \). More formally:
  • \( \tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2} \) for \( x eq 0 \)
This identity can be quite handy when solving equations that involve both \( \tan^{-1}(x) \) and \( \cot^{-1}(x) \).
In particular, understanding and using these identities correctly allows for simplification of complex trigonometric equations, as demonstrated in the provided exercise solution.
Cotangent Function
The cotangent function, or \( \cot(x) \), is the reciprocal of the tangent function.
Functionally, it is defined as:
  • \( \cot(x) = \frac{1}{\tan(x)} \)
  • \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)
The inverse cotangent, \( \cot^{-1}(x) \), is used when finding angles whose cotangent is \( x \).
This function often appears in geometric and trigonometric solutions where answers are derived from angles, especially when the tangent function is involved.
Tangent Function
The tangent function, represented by \( \tan(x) \), is a fundamental trigonometric function.
It is defined as the ratio of the sine and cosine of the angle:
  • \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
The inverse tangent, \( \tan^{-1}(x) \), is the angle whose tangent gives the number \( x \).
This is especially useful when determining unknown angles from known values.
In this context, \( \tan^{-1} \) helps to find angles for a given tangent value, fitting perfectly into solving equations and applying inverse identities in trigonometric contexts.

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

$$ 4 \cos ^{3} \frac{x}{2}+3 \sqrt{2} \sin x=8 \cos \frac{x}{2} $$

$$ \frac{1+\sin x}{1+\cos x}=\frac{1}{2} $$

If \(a \leq \cos 2 x+5 \sin x+6 \leq b\), find \(a\) and \(b\).

\(\frac{\cos x}{1+\cos 2 x}=0\)

Show that the value of \(\sec ^{2} \theta+\cos ^{2} \theta\) is never less than 2 .
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

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.