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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 64
Capitolo 6, Problema 64

Verify that each equation is an identity.
sin³ θ = sin θ - cos² θ sin θ

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1
Start by examining the right-hand side (RHS) of the equation: \(\sin \theta - \cos^{2} \theta \sin \theta\).
Factor out \(\sin \theta\) from the RHS to simplify the expression: \(\sin \theta (1 - \cos^{2} \theta)\).
Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\), which implies \(1 - \cos^{2} \theta = \sin^{2} \theta\).
Substitute \(1 - \cos^{2} \theta\) with \(\sin^{2} \theta\) in the factored expression: \(\sin \theta \cdot \sin^{2} \theta\).
Simplify the expression to get \(\sin^{3} \theta\), which matches the left-hand side (LHS), thus verifying the identity.

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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 identities like Pythagorean or angle sum formulas.
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Fundamental Trigonometric Identities

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This fundamental relationship allows substitution between sine and cosine terms, which is essential for simplifying and verifying trigonometric expressions.
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Pythagorean Identities

Algebraic Manipulation of Trigonometric Expressions

Simplifying trigonometric expressions often requires factoring, expanding, or rearranging terms. Recognizing patterns like factoring out common terms (e.g., sin θ) helps transform one side of the equation to match the other, confirming the identity.
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Simplifying Trig Expressions
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.

cos 18°

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Domanda del libro di testo

Verify that each equation is an identity.

2 cos² (x/2) tan x = tan x+ sin x

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Domanda del libro di testo

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

sin 102° - sin 95°

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Domanda del libro di testo

Verify that each equation is an identity. See Example 4.

tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)

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Domanda del libro di testo

Write each expression as a sum or difference of trigonometric functions. See Example 7.

8 sin 7x sin 9x

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Domanda del libro di testo

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

cos 5x + cos 8x

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