Domanda del libro di testo
Verify that each equation is an identity.
sin³ θ = sin θ - cos² θ sin θ
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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 64
Write each expression as a product of trigonometric functions. See Example 8.
cos 5x + cos 8x
Guida verificata passo dopo passo1
Recognize that the expression is a sum of two cosine functions: \(\cos 5x + \cos 8x\).
Recall the sum-to-product identity for cosine: \(\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).
Identify \(A = 5x\) and \(B = 8x\) in the given expression.
Apply the identity by substituting \(A\) and \(B\): \(\cos 5x + \cos 8x = 2 \cos \left( \frac{5x + 8x}{2} \right) \cos \left( \frac{5x - 8x}{2} \right)\).
Simplify the arguments inside the cosine functions to express the sum as a product of cosines.
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Sum-to-Product Identities
Sum-to-product identities transform sums or differences of trigonometric functions into products. For example, the sum of cosines can be expressed as a product using the formula: cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2). This simplifies expressions and aids in solving equations.
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Percorso guidato
2:25
Verifying Identities with Sum and Difference Formulas
Trigonometric Function Properties
Understanding the basic properties and periodicity of trigonometric functions like cosine is essential. Recognizing how angles combine and how cosine behaves under addition helps in applying identities correctly and simplifying expressions.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Angle Manipulation and Substitution
Manipulating angles by adding, subtracting, or factoring is crucial when applying identities. Substituting expressions like (A+B)/2 and (A−B)/2 allows rewriting sums as products, making complex expressions more manageable.
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Percorso guidato
04:42Solve 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.
cos 18°
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Domanda del libro di testo
Verify that each equation is an identity.
2 cos² (x/2) tan x = tan x+ sin x
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Domanda del libro di testo
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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Domanda del libro di testo
Write each expression as a sum or difference of trigonometric functions. See Example 7.
8 sin 7x sin 9x
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Domanda del libro di testo
Verify that each equation is an identity.
sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)
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