Textbook Question
Verify that each equation is an identity.
sin(x + y) + sin(x - y) = 2 sin x cos y
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 62
Write each expression as a sum or difference of trigonometric functions. See Example 7.
8 sin 7x sin 9x
Verified step by step guidance1
Recall the product-to-sum identity for sine functions: \(\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]\).
Identify \(A\) and \(B\) in the given expression: here, \(A = 7x\) and \(B = 9x\).
Apply the identity to rewrite \(\sin 7x \sin 9x\) as \(\frac{1}{2} [\cos(7x - 9x) - \cos(7x + 9x)]\).
Simplify the arguments inside the cosine functions: \(7x - 9x = -2x\) and \(7x + 9x = 16x\).
Multiply the entire expression by 8 (from the original expression \(8 \sin 7x \sin 9x\)) to get \(8 \times \frac{1}{2} [\cos(-2x) - \cos(16x)]\), which simplifies to \(4 [\cos(-2x) - \cos(16x)]\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product-to-Sum Formulas
Product-to-sum formulas convert products of sine and cosine functions into sums or differences, simplifying expressions and integrals. For example, the formula for the product of two sine functions is sin A sin B = 1/2 [cos(A - B) - cos(A + B)]. This is essential for rewriting 8 sin 7x sin 9x as a sum or difference.
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Verifying Identities with Sum and Difference Formulas
Trigonometric Function Properties
Understanding the properties of sine and cosine functions, such as their periodicity and symmetry, helps in manipulating and simplifying expressions. Recognizing how angles combine in sums and differences is crucial when applying product-to-sum identities effectively.
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6:04Introduction to Trigonometric Functions
Algebraic Manipulation of Trigonometric Expressions
Algebraic skills are necessary to factor constants and apply identities correctly. For instance, factoring out the constant 8 and applying the product-to-sum formula requires careful distribution and simplification to express the product as a sum or difference.
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
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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Textbook Question
Write each expression as a sum or difference of trigonometric functions. See Example 7.
5 cos 3x cos 2x
Textbook Question
Verify that each equation is an identity.
2 cos² θ - 1 = (1 - tan² θ)/(1 + tan² θ)
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Textbook Question
Write each expression as a product of trigonometric functions. See Example 8.
cos 5x + cos 8x
Textbook Question
Verify that each equation is an identity.
sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)
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