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 68

Capitolo 6, Problema 68

Write each expression as a product of trigonometric functions. See Example 8.
sin 9x - sin 3x

Guida verificata passo dopo passo

Recognize that the expression \( \sin 9x - \sin 3x \) is a difference of sines, which can be rewritten using the trigonometric identity for the difference of sines: \[ \sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \]

Identify \( A = 9x \) and \( B = 3x \) in the given expression.

Calculate the average of \( A \) and \( B \): \[ \frac{9x + 3x}{2} = \frac{12x}{2} = 6x \]

Calculate half the difference of \( A \) and \( B \): \[ \frac{9x - 3x}{2} = \frac{6x}{2} = 3x \]

Substitute these values back into the identity to express the original expression as a product: \[ \sin 9x - \sin 3x = 2 \cos 6x \sin 3x \]

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Sum-to-Product Identities

Sum-to-product identities transform sums or differences of sine and cosine functions into products. For example, the difference of sines can be expressed as 2 cos((A+B)/2) sin((A−B)/2). These identities simplify expressions and are useful in integration and solving equations.

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Percorso guidato

2:25

Verifying Identities with Sum and Difference Formulas

Trigonometric Function Properties

Understanding the basic properties of sine and cosine functions, such as periodicity and symmetry, helps in manipulating and simplifying expressions. Recognizing how angles combine and relate is essential for applying identities correctly.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Angle Manipulation and Substitution

Breaking down complex angles into sums or differences allows the use of identities effectively. For instance, expressing 9x and 3x in terms of their average and difference facilitates applying sum-to-product formulas to rewrite the expression as a product.

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Percorso guidato

04:42

Solve Trig Equations Using Identity Substitutions

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.

cot 18°

757

views

Domanda del libro di testo

Verify that each equation is an identity.

(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

733

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Domanda del libro di testo

Verify that each equation is an identity.

(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α

736