Domanda del libro di testo
Find all solutions of each equation. cos x = ﹣1/2
539
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 15
In Exercises 12–18, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x + cos x = 1
Guida verificata passo dopo passo1
Start by rewriting the given equation: \(2 \sin^{2} x + \cos x = 1\).
Use the Pythagorean identity \(\sin^{2} x = 1 - \cos^{2} x\) to express the equation entirely in terms of \(\cos x\). Substitute to get: \(2(1 - \cos^{2} x) + \cos x = 1\).
Simplify the equation: \(2 - 2 \cos^{2} x + \cos x = 1\). Then rearrange to form a quadratic equation in \(\cos x\): \(-2 \cos^{2} x + \cos x + 1 = 0\).
Multiply the entire equation by \(-1\) to make the quadratic standard: \(2 \cos^{2} x - \cos x - 1 = 0\). Now solve this quadratic equation for \(\cos x\) using the quadratic formula \(\cos x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) where \(a=2\), \(b=-1\), and \(c=-1\).
After finding the values of \(\cos x\), determine the corresponding values of \(x\) in the interval \([0, 2\pi)\) by using the inverse cosine function and considering the cosine sign in different quadrants.
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:
12mConcetti 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, the Pythagorean identity sin²x + cos²x = 1 is essential to rewrite sin²x in terms of cos x, simplifying the equation for easier solving.
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. This requires understanding how to manipulate the equation algebraically and use inverse trigonometric functions to find exact or approximate angle values.
Video consigliato:
Percorso guidato
4:34How to Solve Linear Trigonometric Equations
Interval Notation and Solution Sets
The problem restricts solutions to the interval [0, 2π), meaning all solutions must be found within one full rotation of the unit circle. Understanding how to interpret this interval and identify all valid solutions within it is crucial for providing a complete answer.
Video consigliato:
Percorso guidato
06:01i & j Notation
Pratica correlata
Domanda del libro di testo
In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°
970
views
Domanda del libro di testo
Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin 75°
963
views
Domanda del libro di testo
Find all solutions of each equation. tan x = 0
660
views
Domanda del libro di testo
Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. sin x + sin 2x
651
views
Domanda del libro di testo
Verify each identity. cos² θ (1 + tan² θ) = 1
865
views