Skip to main content
Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 5 - Trigonometric IdentitiesProblema 70
Capitolo 6, Problema 70

Verify that each equation is an identity.
(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

Guida verificata passo dopo passo
1
Recall the tangent addition formula: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\).
Substitute \(\tan(\alpha + \beta)\) in the left-hand side (LHS) of the given equation with the formula: \(\frac{\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} - \tan \beta}{1 + \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \cdot \tan \beta}\).
Simplify the numerator by combining the terms over a common denominator: \(\frac{\tan \alpha + \tan \beta - \tan \beta (1 - \tan \alpha \tan \beta)}{1 - \tan \alpha \tan \beta}\).
Simplify the denominator similarly by combining terms: \(1 + \frac{(\tan \alpha + \tan \beta) \tan \beta}{1 - \tan \alpha \tan \beta}\), then write it as a single fraction over the common denominator \(1 - \tan \alpha \tan \beta\).
After simplifying numerator and denominator, divide the numerator by the denominator (which is equivalent to multiplying numerator by the reciprocal of denominator), and simplify the expression to show it equals \(\tan \alpha\), thus verifying 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 involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often by applying known formulas or algebraic manipulation.
Video consigliato:
Percorso guidato
5:32
Fundamental Trigonometric Identities

Tangent Addition Formula

The tangent addition formula states that tan(α + β) = (tan α + tan β) / (1 - tan α tan β). This formula is essential for expressing the tangent of a sum in terms of individual tangents, enabling simplification and verification of identities involving sums of angles.
Video consigliato:
Percorso guidato
3:17
Inverse Tangent

Algebraic Manipulation of Fractions

Simplifying complex fractions and expressions requires careful algebraic manipulation, such as combining terms, factoring, and canceling common factors. Mastery of these skills is crucial to transform one side of the identity into the other and verify the equation.
Video consigliato:
Percorso guidato
4:02
Solving Linear Equations with Fractions
Pratica correlata
Domanda del libro di testo

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

tan 72°

747
views
Domanda del libro di testo

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = (cot² x - 1)/(cot² x + 1)

731
views
Domanda del libro di testo

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = 1 - 2 sin² x

710
views
Domanda del libro di testo

Write each expression as a product of trigonometric functions. See Example 8.

sin 9x - sin 3x

37
views
Domanda del libro di testo

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = cos² x - sin² x

726
views
Domanda del libro di testo

Verify that each equation is an identity.

(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))

718
views