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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 70
Capitolo 3, Problema 70

In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β + cos (α - β) sin β

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Recognize that the given expression is of the form \(\sin(A) \cos(B) + \cos(A) \sin(B)\), where \(A = \alpha - \beta\) and \(B = \beta\).
Recall the sine addition formula: \(\sin(X + Y) = \sin X \cos Y + \cos X \sin Y\).
Apply the sine addition formula to the expression \(\sin(\alpha - \beta) \cos \beta + \cos(\alpha - \beta) \sin \beta\), which matches the pattern \(\sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B)\).
Substitute back the values of \(A\) and \(B\) to get \(\sin((\alpha - \beta) + \beta)\).
Simplify the argument inside the sine function: \((\alpha - \beta) + \beta = \alpha\), so the expression simplifies to \(\sin \alpha\).

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Angle Sum and Difference Identities

These identities express trigonometric functions of sums or differences of angles in terms of functions of individual angles. For example, sin(α - β) = sin α cos β - cos α sin β. Recognizing and applying these identities helps simplify complex expressions involving multiple angles.
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Verifying Identities with Sum and Difference Formulas

Trigonometric Function Properties

Understanding the basic properties and relationships of sine and cosine functions, such as their periodicity and symmetry, is essential. This knowledge aids in manipulating and combining terms to achieve simpler or more recognizable forms.
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Introduction to Trigonometric Functions

Expression Simplification Techniques

Simplifying trigonometric expressions often involves factoring, combining like terms, and substituting identities. Mastery of these algebraic techniques allows one to rewrite expressions as a single trigonometric term, making them easier to interpret or use in further calculations.
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