Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.84

Capitolo 6, Problema 5.84

Verify that each equation is an identity.
(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

Guida verificata passo dopo passo

Start by expressing all trigonometric functions in terms of sine and cosine. Note that \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Rewrite the right-hand side of the equation: \( 2 \csc^2 \theta - 2 \csc \theta \cot \theta - 1 \) becomes \( 2 \left(\frac{1}{\sin^2 \theta}\right) - 2 \left(\frac{1}{\sin \theta}\right)\left(\frac{\cos \theta}{\sin \theta}\right) - 1 \).

Simplify the expression: \( 2 \left(\frac{1}{\sin^2 \theta}\right) - 2 \left(\frac{\cos \theta}{\sin^2 \theta}\right) - 1 \) becomes \( \frac{2 - 2\cos \theta - \sin^2 \theta}{\sin^2 \theta} \).

Use the Pythagorean identity \( \sin^2 \theta = 1 - \cos^2 \theta \) to simplify further: \( \frac{2 - 2\cos \theta - (1 - \cos^2 \theta)}{\sin^2 \theta} \) simplifies to \( \frac{1 - \cos \theta}{1 + \cos \theta} \).

Compare the simplified right-hand side with the left-hand side \( \frac{1 - \cos \theta}{1 + \cos \theta} \) to verify that both sides are equal, confirming the identity.

Risposta video verificata per un problema simile:

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

Concetti chiave

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

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations.

Video consigliato:

Percorso guidato

5:32

Fundamental Trigonometric Identities

Cosecant and Cotangent Functions

Cosecant (csc) and cotangent (cot) are two of the six fundamental trigonometric functions. Cosecant is the reciprocal of sine (csc θ = 1/sin θ), while cotangent is the reciprocal of tangent (cot θ = cos θ/sin θ). Familiarity with these functions is essential for manipulating and transforming trigonometric equations.

Video consigliato:

Percorso guidato

6:22

Graphs of Secant and Cosecant Functions

Algebraic Manipulation in Trigonometry

Algebraic manipulation involves rearranging and simplifying expressions using algebraic techniques. In trigonometry, this includes factoring, combining like terms, and applying identities to transform one side of an equation to match the other. Mastery of these skills is necessary for verifying trigonometric identities effectively.

Video consigliato:

Percorso guidato

04:12

Algebraic Operations on Vectors

Pratica correlata

Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

1 + cot(-θ)/cot(-θ)

786

views

Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

csc θ - sin θ

825

views

Domanda del libro di testo

Identify the basic trigonometric function graphed, and determine whether it is even or odd.

<IMAGE>

views

Domanda del libro di testo

Find the remaining five trigonometric functions of θ.

cos θ = -1/4, sin θ > 0

views

Domanda del libro di testo