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 88
Capitolo 6, Problema 88

Verify that each equation is an identity.
sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)

Guida verificata passo dopo passo
1
Start by recalling the algebraic identity for the sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Here, let \(a = \sin \theta\) and \(b = \cos \theta\).
Rewrite the left side using the sum of cubes formula: \(\sin^3 \theta + \cos^3 \theta = (\sin \theta + \cos \theta)(\sin^2 \theta - \sin \theta \cos \theta + \cos^2 \theta)\).
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to simplify the expression inside the parentheses: replace \(\sin^2 \theta + \cos^2 \theta\) with 1.
After substitution, the expression becomes \((\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta)\), which matches the right side of the given equation.
Conclude that since both sides simplify to the same expression, the given equation is an 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 using known formulas like Pythagorean identities or algebraic manipulations.
Video consigliato:
Percorso guidato
5:32
Fundamental Trigonometric Identities

Algebraic Factorization

Algebraic factorization involves rewriting expressions as products of simpler factors. Recognizing patterns such as sum of cubes, a³ + b³ = (a + b)(a² - ab + b²), helps in breaking down complex trigonometric expressions to verify identities efficiently.
Video consigliato:
Percorso guidato
6:08
Factoring

Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, is fundamental in trigonometry. It allows substitution and simplification of expressions involving sine and cosine powers, which is essential when verifying or transforming trigonometric equations.
Video consigliato:
Percorso guidato
6:25
Pythagorean Identities
Pratica correlata
Domanda del libro di testo

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

x + y = 9, 2x + y = -1

817
views
Domanda del libro di testo

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

5x - 2y + 4 = 0, 3x + 5y = 6

638
views
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.

sin 162°

760
views