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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 9
Capitolo 3, Problema 9

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. sin 6x + sin 2x

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1
Recognize that the expression is a sum of two sine functions: \(\sin 6x + \sin 2x\).
Recall the sum-to-product identity for sine functions: \(\sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)\).
Identify \(A = 6x\) and \(B = 2x\), then substitute these into the formula to rewrite the sum as a product.
Calculate the average and half-difference of the angles: \(\frac{6x + 2x}{2} = 4x\) and \(\frac{6x - 2x}{2} = 2x\).
Express the original sum as \(2 \sin 4x \cos 2x\), which is the product form of the given sum.

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Sum-to-Product Formulas

Sum-to-product formulas transform sums or differences of sine or cosine functions into products. For sine, the formula sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) is used to rewrite the expression as a product, simplifying calculations and enabling exact value determination.
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Verifying Identities with Sum and Difference Formulas

Angle Manipulation and Substitution

Understanding how to manipulate angles by addition, subtraction, and division is essential. In this problem, recognizing that 6x and 2x can be combined as (6x + 2x)/2 and (6x − 2x)/2 allows the use of sum-to-product formulas effectively.
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Solve Trig Equations Using Identity Substitutions

Exact Values of Trigonometric Functions

After expressing the sum as a product, finding the exact value requires knowledge of special angles and their sine and cosine values. Familiarity with exact values for angles like 30°, 45°, 60°, or their radian equivalents helps in evaluating the product precisely.
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Introduction to Trigonometric Functions
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