Domanda del libro di testo
Verify that each equation is an identity.
sin(x + y) + sin(x - y) = 2 sin x cos y
667
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Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- 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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 6, Problema 60
Write each expression as a sum or difference of trigonometric functions. See Example 7.
5 cos 3x cos 2x
Guida verificata passo dopo passo1
Recall the product-to-sum identity for cosine functions: \(\cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)]\).
Identify the angles in the expression: here, \(A = 3x\) and \(B = 2x\).
Apply the product-to-sum formula to \(\cos 3x \cos 2x\):
\(\cos 3x \cos 2x = \frac{1}{2} [\cos(3x + 2x) + \cos(3x - 2x)]\).
Simplify the angles inside the cosine functions:
\(\cos(3x + 2x) = \cos 5x\) and \(\cos(3x - 2x) = \cos x\).
Multiply the entire expression by 5 (from the original expression \(5 \cos 3x \cos 2x\)) to get the sum or difference form:
\(5 \cos 3x \cos 2x = 5 \times \frac{1}{2} [\cos 5x + \cos x]\).
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Concetti chiave
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Product-to-Sum Formulas
Product-to-sum formulas convert products of sine and cosine functions into sums or differences, simplifying expressions. For example, the formula for cos A cos B is (1/2)[cos(A+B) + cos(A−B)]. These formulas are essential for rewriting products like 5 cos 3x cos 2x as sums or differences.
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Percorso guidato
2:25
Verifying Identities with Sum and Difference Formulas
Trigonometric Function Properties
Understanding the properties of cosine and sine functions, such as periodicity and symmetry, helps in manipulating and simplifying expressions. Recognizing how angles combine in sums or differences is crucial when applying product-to-sum identities.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Algebraic Manipulation of Trigonometric Expressions
Skill in algebraic manipulation, including factoring constants and rearranging terms, is necessary to correctly apply identities and rewrite expressions. For instance, factoring out the constant 5 before applying product-to-sum formulas ensures clarity and accuracy.
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Percorso guidato
6:36Simplifying Trig Expressions
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