Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x = cos x
513
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 69
In Exercises 67–74, rewrite each expression in terms of the given function or functions. ;
Guida verificata passo dopo passo1
Start by rewriting the given expression clearly: \(\frac{\cos x}{1 + \sin x} + \tan x \cdot \cos x\).
Recall that \(\tan x = \frac{\sin x}{\cos x}\), so substitute \(\tan x\) in the expression to get \(\frac{\cos x}{1 + \sin x} + \frac{\sin x}{\cos x} \cdot \cos x\).
Simplify the second term by canceling \(\cos x\) in numerator and denominator: \(\frac{\cos x}{1 + \sin x} + \sin x\).
To combine the terms, express \(\sin x\) with a common denominator \(1 + \sin x\): write \(\sin x = \frac{\sin x (1 + \sin x)}{1 + \sin x}\).
Now add the two fractions: \(\frac{\cos x}{1 + \sin x} + \frac{\sin x (1 + \sin x)}{1 + \sin x} = \frac{\cos x + \sin x (1 + \sin x)}{1 + \sin x}\). This is the expression rewritten in terms of sine and cosine functions.
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:
3mConcetti 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. They allow rewriting expressions in different forms, such as converting between sine, cosine, and tangent. Common identities include Pythagorean identities and quotient identities, which are essential for simplifying or rewriting expressions.
Video consigliato:
Percorso guidato
5:32
Fundamental Trigonometric Identities
Simplifying Complex Fractions
Simplifying complex fractions involves rewriting expressions with fractions in the numerator and denominator into simpler forms. This often requires finding common denominators, factoring, or multiplying numerator and denominator by conjugates. Mastery of this skill helps in expressing trigonometric expressions in terms of a single function or simpler combinations.
Video consigliato:
Percorso guidato
4:22Dividing Complex Numbers
Quotient and Reciprocal Relationships
The quotient identities relate tangent and cotangent to sine and cosine, such as tan x = sin x / cos x. Reciprocal identities express functions like sec x and csc x as reciprocals of cosine and sine, respectively. Understanding these relationships is crucial for rewriting expressions involving multiple trigonometric functions into a desired form.
Video consigliato:
Percorso guidato
03:40Quotients of Complex Numbers in Polar Form
Pratica correlata
Domanda del libro di testo
In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β + cos (α - β) sin β
954
views
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 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin 2x = cos x
596
views
Domanda del libro di testo
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 4 cos² x = 5 - 4 sin x
554
views
Domanda del libro di testo
In Exercises 69–74, rewrite each expression as a simplified expression containing one term. cos (α + β) cos β + sin (α + β) sin β
955
views