Textbook Question
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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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.RE.43
In Exercises 43–44, express each product as a sum or difference. sin 6x sin 4x
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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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
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Textbook Question
In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x = 2 cos x tan x
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Textbook Question
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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Textbook Question
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.
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Textbook Question
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
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