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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblem 9
Chapter 3, Problem 9

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. sin 6x + sin 2x

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1
Recognize that the expression is a sum of two sine functions: \(\sin 6x + \sin 2x\).
Recall the sum-to-product identity for sine functions: \(\sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)\).
Identify \(A = 6x\) and \(B = 2x\), then substitute these into the formula to rewrite the sum as a product.
Calculate the average and half-difference of the angles: \(\frac{6x + 2x}{2} = 4x\) and \(\frac{6x - 2x}{2} = 2x\).
Express the original sum as \(2 \sin 4x \cos 2x\), which is the product form of the given sum.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sum-to-Product Formulas

Sum-to-product formulas transform sums or differences of sine or cosine functions into products. For sine, the formula sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2) is used to rewrite the expression as a product, simplifying calculations and enabling exact value determination.
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Verifying Identities with Sum and Difference Formulas

Angle Manipulation and Substitution

Understanding how to manipulate angles by addition, subtraction, and division is essential. In this problem, recognizing that 6x and 2x can be combined as (6x + 2x)/2 and (6x − 2x)/2 allows the use of sum-to-product formulas effectively.
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Exact Values of Trigonometric Functions

After expressing the sum as a product, finding the exact value requires knowledge of special angles and their sine and cosine values. Familiarity with exact values for angles like 30°, 45°, 60°, or their radian equivalents helps in evaluating the product precisely.
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Related Practice
Textbook Question

Use the given information to find the exact value of each of the following: tan2θ\(\tan\)2\(\theta\)

cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)

2
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Textbook Question

Use the given information to find the exact value of each of the following: sin2θ\(\sin\)2\(\theta\)

sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

1
views
Textbook Question

Use the given information to find the exact value of each of the following: tan2θ\(\tan\)2\(\theta\)

sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

6
views
Textbook Question

Use the given information to find the exact value of each of the following: cos2θ\(\cos\)2\(\theta\)

sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)

Textbook Question

Use the given information to find the exact value of each of the following: cos2θ\(\cos\)2\(\theta\)

cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)

1
views
Textbook Question

Use the given information to find the exact value of each of the following: sin2θ\(\sin\)2\(\theta\)

cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)

8
views