Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problema 59

Capitolo 2, Problema 59

In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. cot⁻¹ (cot 3π/4)

Guida verificata passo dopo passo

Recall the property of inverse functions: for any function \( f \) and its inverse \( f^{-1} \), \( f^{-1}(f(x)) = x \) holds true for all \( x \) in the domain of \( f \).

Identify the function and its inverse in the expression: here, \( \cot^{-1} \) is the inverse cotangent function, and \( \cot \) is the cotangent function.

Understand the domain and range of the inverse cotangent function: \( \cot^{-1}(x) \) typically returns values in the interval \( (0, \pi) \).

Since \( \cot^{-1}(\cot(\theta)) = \theta \) only if \( \theta \) is in the principal range of \( \cot^{-1} \), check if \( 3\pi/4 \) lies within \( (0, \pi) \).

Because \( 3\pi/4 \) is within \( (0, \pi) \), you can conclude that \( \cot^{-1}(\cot(3\pi/4)) = 3\pi/4 \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Inverse Trigonometric Functions

Inverse trigonometric functions reverse the effect of their corresponding trigonometric functions, returning an angle from a given ratio. For example, cot⁻¹(x) gives the angle whose cotangent is x. Understanding their domains and ranges is crucial to correctly evaluate expressions involving these inverses.

Video consigliato:

Percorso guidato

4:28

Introduction to Inverse Trig Functions

Properties of Inverse Functions

The key property f(f⁻¹(x)) = x holds for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f. This means applying a function and its inverse in succession returns the original input, but only when the input lies within the appropriate domain or range.

Video consigliato:

Percorso guidato

4:28

Introduction to Inverse Trig Functions

Principal Values and Domain Restrictions

Inverse trigonometric functions have restricted ranges (principal values) to ensure they are functions. For cot⁻¹(x), the principal value range is typically (0, π). When evaluating cot⁻¹(cot θ), the result is the principal value angle equivalent to θ within this range, not necessarily θ itself.

Video consigliato:

Percorso guidato

3:43

Finding the Domain of an Equation

Pratica correlata

Domanda del libro di testo

In Exercises 53–60, use a vertical shift to graph one period of the function. y = −3 sin 2πx + 2

832

views

Domanda del libro di testo

In Exercises 53–60, use a vertical shift to graph one period of the function. y = cos x + 3

766

views

Domanda del libro di testo

In Exercises 61–62, use the figures shown to find the bearing from O to A.

1054

views

Domanda del libro di testo

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = 3 cos x + sin x

615

views

Domanda del libro di testo

In Exercises 55–62, use the properties of inverse functions f(f⁻¹ (x)) = x for all x in the domain of f⁻¹ and f⁻¹(f(x)) for all x in the domain of f, as well as the definitions of the inverse cotangent, cosecant, and secant functions, to find the exact value of each expression, if possible. sec(sec⁻¹ 7π)

745

views

Domanda del libro di testo

In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. csc x = 1

864

views