Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos( β/2)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.RE.43
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.RE.43Capitolo 3, Problema 3.RE.43
In Exercises 43–44, express each product as a sum or difference. sin 6x sin 4x
Guida verificata passo dopo passo1
Recall the product-to-sum identity for the product of sines: \(\sin A \sin B = \frac{1}{2} [\cos (A - B) - \cos (A + B)]\).
Identify the angles in the problem: here, \(A = 6x\) and \(B = 4x\).
Substitute \(A\) and \(B\) into the identity: \(\sin 6x \sin 4x = \frac{1}{2} [\cos (6x - 4x) - \cos (6x + 4x)]\).
Simplify the expressions inside the cosine functions: \(\cos (6x - 4x) = \cos 2x\) and \(\cos (6x + 4x) = \cos 10x\).
Write the final expression as a sum or difference: \(\sin 6x \sin 4x = \frac{1}{2} [\cos 2x - \cos 10x]\).
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Product-to-Sum Identities
Product-to-sum identities transform products of sine and cosine functions into sums or differences of trigonometric functions. For example, the product sin A sin B can be expressed as a difference of cosines: (1/2)[cos(A - B) - cos(A + B)]. This simplifies integration and solving trigonometric equations.
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Percorso guidato
2:25
Verifying Identities with Sum and Difference Formulas
Trigonometric Angle Notation and Manipulation
Understanding how to handle angles in trigonometric expressions is essential. Here, angles like 6x and 4x represent multiples of a variable, and correctly applying operations like addition and subtraction on these angles is crucial when using identities.
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Percorso guidato
06:01i & j Notation
Simplification of Trigonometric Expressions
After applying identities, simplifying the resulting expressions by combining like terms or factoring is important. This step ensures the final answer is in its simplest sum or difference form, making it easier to interpret or use in further calculations.
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Percorso guidato
6:36Simplifying Trig Expressions
Pratica correlata
Domanda del libro di testo
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. sin 22.5°
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Domanda del libro di testo
In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x
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Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
b. cos(α﹣β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
992
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Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
843
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Domanda del libro di testo
In Exercises 35–38, find the exact value of the following under the given conditions:
c. tan(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2
723
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