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. sin 2x = √ 3 sin x
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 3, Problem 3.RE.44
In Exercises 43–44, express each product as a sum or difference. sin 7x cos 3x
Verified step by step guidance1
Recall the product-to-sum identity for sine and cosine: \(\sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)]\).
Identify the angles in the problem: here, \(A = 7x\) and \(B = 3x\).
Substitute \(A\) and \(B\) into the identity: \(\sin 7x \cos 3x = \frac{1}{2} [\sin(7x + 3x) + \sin(7x - 3x)]\).
Simplify the expressions inside the sine functions: \(\sin(7x + 3x) = \sin 10x\) and \(\sin(7x - 3x) = \sin 4x\).
Write the final expression as a sum: \(\sin 7x \cos 3x = \frac{1}{2} (\sin 10x + \sin 4x)\).
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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 of trigonometric functions. For example, sin A cos B can be expressed as ½[sin(A + B) + sin(A - B)], simplifying integration or further manipulation.
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Verifying Identities with Sum and Difference Formulas
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. Understanding these identities, such as angle addition and subtraction formulas, is essential for transforming and simplifying expressions.
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Angle Notation and Manipulation
Recognizing and manipulating angles in expressions like sin 7x and cos 3x requires understanding how to handle multiples of variables within trigonometric functions. This skill is crucial for correctly applying formulas and simplifying results.
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Related Practice
Textbook Question
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.
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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. cos 2x = -1
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Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions: b. cos(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:
a. sin(α + β)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.
1
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Textbook Question
In Exercises 35–38, find the exact value of the following under the given conditions:
d. sin 2α
sin α = 3/5, 0 < α < 𝝅/2, and sin β = 12/13, 𝝅/2 < β < 𝝅.
1
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