Skip to main content
Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 42
Capitolo 1, Problema 42

Solve the following equations.
sin3x=22,0≤x<2π\(\sin\)3x=\(\frac{\sqrt{2}\)}{2},0\(\leq{x}\)\(\lt{2\pi}\)

Guida verificata passo dopo passo
1
First, recognize that the equation \( \sin(3x) = \frac{\sqrt{2}}{2} \) is a trigonometric equation where we need to find the values of \( x \) within the interval \( 0 \leq x < 2\pi \).
Recall that \( \sin(\theta) = \frac{\sqrt{2}}{2} \) corresponds to angles \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) in the unit circle.
Since we have \( \sin(3x) = \frac{\sqrt{2}}{2} \), set \( 3x = \frac{\pi}{4} + 2k\pi \) and \( 3x = \frac{3\pi}{4} + 2k\pi \) for integer values of \( k \).
Solve for \( x \) by dividing each equation by 3: \( x = \frac{\pi}{12} + \frac{2k\pi}{3} \) and \( x = \frac{\pi}{4} + \frac{2k\pi}{3} \).
Determine the values of \( k \) such that \( x \) remains within the interval \( 0 \leq x < 2\pi \). Calculate these values to find all possible solutions for \( x \).

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:
5m

Concetti chiave

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

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. The sine function, specifically, gives the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is crucial for solving equations involving angles, especially in the context of periodic functions like sine, which repeat their values in regular intervals.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsine, allow us to determine the angle that corresponds to a given sine value. For example, if \\sin(x) = y, then \\arcsin(y) = x. These functions are essential for solving equations where the angle is unknown, particularly when working with equations like \\sin(3x) = \\frac{\sqrt{2}}{2}, as they help find the specific angle solutions within a defined range.
Video consigliato:
06:35
Derivatives of Other Inverse Trigonometric Functions

Periodic Nature of Sine Function

The sine function is periodic with a period of 2π, meaning it repeats its values every 2π radians. This property is important when solving trigonometric equations, as it implies that there can be multiple solutions within a specified interval. For the equation \\sin(3x) = \\frac{\sqrt{2}}{2}, recognizing the periodicity allows us to find all possible values of x that satisfy the equation within the interval \\[0, 2π)\].
Video consigliato:
Percorso guidato
5:53
Graph of Sine and Cosine Function
Pratica correlata
Domanda del libro di testo

Solve the following equations.


{Use of Tech} sin2θ=15,0<θ<π2\(\sin\)2\(\theta\)=\(\frac\)15,0\(\lt{\theta}\)\(\lt{\frac{\pi}{2}\)}

329
views
Domanda del libro di testo

Solve the following equations.

tan22θ=1,0<θ<π\(\tan\)^22\(\theta\)=1,0\(\lt{\theta}\)\(\lt{\pi}\)

309
views
Domanda del libro di testo

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

a. ƒ(x+1)ƒ( x + 1)

272
views
Domanda del libro di testo

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>


f(2(x−1))f(2(x - 1))

236
views
Domanda del libro di testo

Solve the following equations.

cos3x=sin3x,0≤x<2π\(\cos\)3x=\(\sin\)3x,0\(\leq{x}\)\(\lt\)2\(\pi\)

501
views
Domanda del libro di testo

Composite functions and notation

Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

ƒ (√(x+4))

271
views