Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin x + cos x = 1
555
views
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
Capitolo 3, Problema 75
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin x cos x = √ 2 / 4
Guida verificata passo dopo passo1
Start with the given equation: \(\sin x \cos x = \frac{\sqrt{2}}{4}\).
Recall the double-angle identity for sine: \(\sin(2x) = 2 \sin x \cos x\). Use this to rewrite the left side of the equation.
Multiply both sides of the equation by 2 to express it in terms of \(\sin(2x)\): \(2 \sin x \cos x = 2 \times \frac{\sqrt{2}}{4}\), which simplifies to \(\sin(2x) = \frac{\sqrt{2}}{2}\).
Solve the equation \(\sin(2x) = \frac{\sqrt{2}}{2}\) for \$2x$ on the interval \([0, 4\pi)\), since $x$ is in \([0, 2\pi)\) and the argument is \$2x$.
Find all values of \(x\) by dividing the solutions for \$2x$ by 2, ensuring the solutions fall within the original interval \([0, 2\pi)\).
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:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, using the double-angle identity for sine, sin(2x) = 2 sin x cos x, simplifies the equation and helps solve for x efficiently.
Video consigliato:
Percorso guidato
5:32
Fundamental Trigonometric Identities
Solving Trigonometric Equations
Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. After applying identities, one must consider the periodic nature of sine and cosine to find all valid solutions in [0, 2π).
Video consigliato:
Percorso guidato
4:34How to Solve Linear Trigonometric Equations
Interval and General Solutions
When solving trig equations on a specific interval like [0, 2π), it is important to find all solutions within that range. Since trig functions are periodic, multiple angles can satisfy the equation, so understanding how to restrict solutions to the given interval is essential.
Video consigliato:
Percorso guidato
4:49Inverse Cosine
Pratica correlata
Domanda del libro di testo
In Exercises 67–74, rewrite each expression in terms of the given function or functions. ;
627
views
Domanda del libro di testo
In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x
659
views
Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x + 5 cos x + 3 = 0
542
views
Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin ( x + 𝝅/4) + sin ( x - 𝝅/4 ) = 1
443
views
Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin 2x cos x + cos 2x sin x = √ 2/2
478
views